diffraction modelling for solar coronagraphy · solar coronagraph aspiics •associtation de...
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Diffraction model for the external occulterof the solar coronagraph ASPIICS
Raphaël Rougeot
14/05/2018 R.Rougeot 1
OCA, Nice – 14/05/2018
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Outline
1) Proba-3 mission and ASPIICS
2) Diffraction from external occultersa) How to compute diffractionb) Diffraction patterns for several occulters
3) Penumbra profile
4) Conclusion
14/05/2018 R.Rougeot 2
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Proba-3 mission and ASPIICS
14/05/2018 R.Rougeot 3
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ESA Proba-3 mission• In-orbit demonstration of precise Formation Flying
• Two spacecraft flying 150m apart, controlled with a millimeter accuracy
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Occulter Spacecraft
Coronagraph Spacecraft
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ESA Proba-3 mission• In-orbit demonstration of precise Formation Flying
• Two spacecraft flying 150m apart, controlled with a millimeter accuracy
• The formation will be co-aligned with the Sun during the 6-hours apogee phase
14/05/2018 R.Rougeot 5
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Solar coronagraph ASPIICS
• Associtation de Satellites Pour l’Imagerie et l’Interférométrie de la Couronne Solaire
• A 1,42m diameter occulting disk carried by the Occulter SpacecraftA 5cm Lyot-style coronagraph on the Coronagraph Spacecraft
• Observation of the K-corona- Findings on the heating process- Alven’s waves, dynamics of the plasma- Coronal Mass Ejections
14/05/2018 R.Rougeot 6
ASPIICS in a nutshell
White light [540nm ; 570nm]2,8 arcsec/pixelHigh cadenceLamy, 2010
Renotte, 2015
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Coronagraphy and diffraction• The corona of the sun is much fainter than the solar disk itself
Observation in white light requires perfect eclipse conditions
14/05/2018 R.Rougeot 7
𝐵𝑐𝑜𝑟𝑜𝑛𝑎~10−6 𝑡𝑜 10−9𝐵𝑠𝑢𝑛
Allen, 1997Cox, 2000
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Coronagraphy and diffraction
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A CO’B DO
L1 L2
Lyot stop Focal planePupil
L3
External occulter
Focal plane
Internal Occulter
• The hybrid externally occulted Lyot solar coronagraph ASPIICS
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Coronagraphy and diffraction
14/05/2018 R.Rougeot 9
A CO’B DO
L1 L2
Lyot stop Focal planePupil
L3
External occulter
Focal plane
Internal Occulter
• The hybrid externally occulted Lyot solar coronagraph ASPIICS
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Coronagraphy and diffraction
14/05/2018 R.Rougeot 10
A CO’B DO
L1 L2
Lyot stop Focal planePupil
L3
External occulter
Focal plane
Internal Occulter
• The hybrid externally occulted Lyot solar coronagraph ASPIICS
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Coronagraphy and diffraction
14/05/2018 R.Rougeot 11
A CO’B DO
L1 L2
Lyot stop Focal planePupil
L3
External occulter
Focal plane
Internal Occulter
• The hybrid externally occulted Lyot solar coronagraph ASPIICS
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Diffraction by an external occulter
14/05/2018 R.Rougeot 12
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Diffraction by an external occulter
Point source at ∞ External occulter Plane of the entrance aperture
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…
Ψ𝑧(𝑥, 𝑦)Ψ0(𝑥, 𝑦)1
Propagation
λ=550nm R=710mm
z=144,348m
𝑁𝑓 =𝑅2
𝜆𝑧≈ 6400
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• Major issue in solar coronagraphy: the Sun is an extended light source
• The diffraction pattern must be known over a large spatial extent≠ stellar coronagraphy
Diffraction by an external occulter
Rsun=16,2’
14/05/2018 R.Rougeot 14
z=144,348mUmbra of 38mm
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How to compute diffraction?
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Apodisation No Axis-symmetry
Brute force 2D FFT √ √
Analytical Hankel transform √ X
Vanderbei et al. Approach √ O (periodicity)
Lommel series X X
Maggi-Rubinowicz representationBoundary diffraction integral
X √
Aime, 2013
Vanderbei, 2003
Aime, 2013
Cady, 2012Born & Wolf
Rougeot & Aime, 2018
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How to compute diffraction
• Brute force 2D FFT
The occulter is padded in a 2D arrays
Condition from the Fresnel filter: 𝜎 >𝐾𝑧
𝜆
Consequence: K of very large size
In Rougeot & Aime 2018, we tried 156000 x 156000, not sufficient for petalized shape
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Fresnel filterOcculter
Sampling σ
Size K
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How to compute diffraction
• Maggi-Rubinowicz representation
Requires a binary mask (1 or 0)
Ψ𝑧 𝑥, 𝑦 = −Ψzd
x, y in the geometrical shadow
Ψ𝑧 𝑥, 𝑦 = Ψ0(𝑥, 𝑦) − Ψzd
x, y otherwise
Boundary diffraction integral Ψzd=
1
4𝜋𝑊��ׯ 𝑑𝑙
Sampling of the occulter edge must be carefully chosen
14/05/2018 R.Rougeot 17
Ψ𝑧𝑑(𝑥, 𝑦)𝑊(𝑥, 𝑦; 𝑙)
Cady, 2012Born & WolfRougeot & Aime, 2018
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Diffraction by an external occulter
• The sharp-edged occulting disk
Occulting ratio of 1,05 solar radius at z=144m
R=710mm
14/05/2018 R.Rougeot 18
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Diffraction by an external occulter
• The sharp-edged occulting disk
The bright spot of Arago (or Poisson… demonstrated by Fresnel)
Geometrical umbra
14/05/2018 R.Rougeot 19
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Diffraction by an external occulter
• The sharp-edged occulting disk
Bright spot of Arago Transition shadow/light
14/05/2018 R.Rougeot 20
710mm
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Diffraction by an external occulter
• The apodized occulting disk
Variable radial transmission
R=710mm
Δ
14/05/2018 R.Rougeot 21
Δ
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Diffraction by an external occulter
• The apodized occulting disk
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Δ = 20𝑚𝑚
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Diffraction by an external occulter• The serrated (or petalized) occulter
In stellar coronagraphy, the reasonning starts from the ideal apodized occulter
The petalized occulter is the discrete substitute
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𝜏𝑎𝑝𝑜𝑑 𝑟 = ∫ 𝜏𝑝𝑒𝑡𝑎𝑙 𝑟, 𝜃 𝑑𝜃
Cady, 2006Vanderbei et al., 2007
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Diffraction by an external occulter• The serrated (or saw-toothed) occulter
In solar coronagraphy, the reasonning is well different!
The diffraction occurs perpendicularly to the edgeA toothed disc rejects the light outside the central region
14/05/2018 R.Rougeot 24
Boivin’s radius
Boivin (1978) predicted the radius of the dark inner region of the diffraction pattern based on geometrical considerations
Koutchmy, 1988
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
R=710mm
Δ
Nt teeth Nt = 1024 ; Δ=20mm
14/05/2018 R.Rougeot 25
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
Occulter: 512-teeth, 10mm Modulus Ψ𝑧 𝑥,𝑦 Phase ∠Ψ𝑧 𝑥,𝑦
14/05/2018 R.Rougeot 26
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
A
A Dark inner region
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
A
A Dark inner region
Intermediate regionBB
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
A
A Dark inner region
Intermediate region
Fully illuminated region
B
C
B
C
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Diffraction by an external occulter
• We numerically verified the geometrial predictions of Boivin (1978)
Size of teeth ↗
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Rougeot & Aime, 2018
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• The serrated (or saw-toothed) occulter
Diffraction by an external occulter
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Penumbra profile
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• Convolution of the diffraction pattern Ψ𝑧 𝑥, 𝑦 2 with the solar diskCentre-to-limb darkening function
Penumbra profiles
Rsun=16,2’
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z=144,348m Umbra of 38mm
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Penumbra profiles
• The sharp-edged occulting disk
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Diffraction
Purely geometrical
𝐼𝑧(𝑟 = 0) ≈ 10−4Isun
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Penumbra profiles
• The serrated (or saw-toothed ) occulter
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Δ=20mm
Δ increasesNt increases
Nt=464
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Penumbra profiles
• The serrated (or saw-toothed ) occulter
14/05/2018 R.Rougeot 37
Integrated illumination over the pupil, normalized to the sharp-edged disk cas
Boivin radius 𝑁𝑡, Δ > 𝑟𝑠𝑢𝑛 =671mm
𝑝𝑢𝑝𝑖𝑙𝐼𝑠𝑒𝑟𝑟𝑎𝑡𝑒𝑑𝑑𝑠
𝑝𝑢𝑝𝑖𝑙𝐼𝑠ℎ𝑎𝑟𝑝𝑑𝑠
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Conclusion
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Conclusion
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• Model of diffraction for external occulters in solar coronagraphy- Limits of Fresnel diffraction integrals using 2D FFT- Maggi-Rubinowicz representation
• Assessement of the theoretical performance of serrated occulters
• References- Aime C. 2013, A&A, 558, A138- Rougeot R., Flamary R., Galano D., Aime C. 2017, A&A, 599, A2- Rougeot R., Aime C. 2018, A&A, 612, A80
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Other works
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• Propagation of the diffracted wavefronts inside the Lyot-style coronagraph- end-to-end performance in straylight rejection- impacts of the size of the internal occulter and the Lyot stop- PSF in the vignetting zone
• On-going/future works:- optical aberrations of the optics- effects of surface roughness scattering
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Questions?
14/05/2018 R.Rougeot 41
Thank you for your attention!
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Propagation inside the coronagraph
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Propagation inside the coronagraph
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A CO’B DO
L1 L2
Lyot stop Focal planePupil
L3
External occulter
Focal plane
Internal Occulter
• The hybrid externally occulted Lyot solar coronagraph ASPIICS
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Propagation inside the coronagraph
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• Propagation of the diffracted wavefront from one plane to the next one:- Fourier optics formalism- ideal optics- perfect axis-symetric geometry
• Numerical implementation: successive FFT with arrays of large size
• Objective: - estimate the level and spatial distribution of the residual diffracted sunlight- address the rejection performance of the coronagraph
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Propagation inside the coronagraph
• Intensity in plane O’, where the internal occulter is set
14/05/2018 R.Rougeot 45
With external occulterWithout external occulter
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Propagation inside the coronagraph
14/05/2018 R.Rougeot 46
• Intensity in plane O’, where the internal occulter is set
With external occulter
Without external occulterSolar disk image (out-of-focused)
Rougeot et al., 2017
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Propagation inside the coronagraph
14/05/2018 R.Rougeot 47
• Intensity in plane O’, where the internal occulter is set
With external occulter
Without external occulterSolar disk image (out-of-focused)
Internalocculter
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Propagation inside the coronagraph
• Intensity in plane C, where the Lyot stop is set
14/05/2018 R.Rougeot 48
With external occulterWithout external occulter
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Propagation inside the coronagraph
14/05/2018 R.Rougeot 49
• Intensity in plane C, where the Lyot stop is set
With external occulter
Without external occulter(Lyot coronagraph)
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Propagation inside the coronagraph
14/05/2018 R.Rougeot 50
• Intensity in plane C, where the Lyot stop is set
With external occulter
Without external occulter(Lyot coronagraph)
Lyot stop
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Propagation inside the coronagraph
• Intensity in plane D, final focal plane with the detector
14/05/2018 R.Rougeot 51
With external occulter
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Propagation inside the coronagraph
14/05/2018 R.Rougeot 52
• Intensity in plane D, final focal plane with the detector
With external/internal occultersand Lyot stop
Without external occulter(Lyot coronagraph)
No occulter and stopSolar disk image
Just the external occulterNo internal occulterNo Lyot stop
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Propagation inside the coronagraph• Impact of sizing the internal occulter and the Lyot stop
Intensity on plane D, the final focal plane
Fixed Lyot stop Fixed internal occulter
14/05/2018 R.Rougeot 53
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Propagation inside the coronagraph• Impact of sizing the internal occulter and the Lyot stop
14/05/2018 R.Rougeot 54
Residual diffracted sunlight @ 1.3𝑅⊙
Better rejection
Closer to solar edge
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Propagation inside the coronagraph
• PSF in the vignetted zone
14/05/2018 R.Rougeot 55
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Annex on diffraction formulations
14/05/2018 R.Rougeot 56
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2D FFT technique
The Fresnel filter exp 𝑖𝜋𝜆𝑧𝑢2 has its phase varying as u2
At the edge of the array, 𝑢𝑐 = 1/2𝜎
We impose that the (maximum) phase variation at the edge of the array is <π
𝜎 >𝜆𝑧
𝐾
Consequence: 𝜎 ↘⟹ 𝐾 ↗
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2D FFT technique
Very sensitive to numerical sampling: impact of the size of the array
14/05/2018 R.Rougeot 58
2D FFT computation
K↗
Reference curve (Hankel)
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2D FFT technique
Very sensitive to numerical sampling: impact of sampling
14/05/2018 R.Rougeot 59
Sampling meeting the condition
Sampling too small regarding Fresnel filter’s condition
Sampling too large to correctly sample the occulter
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The Hankel transformation
Fourier wave optics formalism
Fresnel free-space propagation
Axis-symmetric (apodized) occulter
Ψ𝑧 𝑥, 𝑦 = 1 − 𝑓 𝑟 ⊛1
𝑖𝜆𝑧exp
𝑖𝜋
𝜆𝑧𝑥2 + 𝑦2
Ψ𝑧 𝑟 =𝜑𝑧 𝑟
𝑖𝜆𝑧න0
𝑅
2𝜋𝜌 × 𝑓 𝜌 × exp𝑖𝜋𝜌2
𝜆𝑧× 𝐽0
2𝜋𝜌𝑟
𝜆𝑧𝑑𝜌
Lommel series – decomposition into series (Aime, 2013)
Radial apodizationDiffraction at zRadial function
12/02/2018 R.Rougeot 60
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The Hankel transformation
Fresnel free-space propagation
Axis-symmetric (apodized) occulter
Ψ𝑧 𝑥, 𝑦 = 1 − 𝑓 𝑟 ⊛1
𝑖𝜆𝑧exp
𝑖𝜋
𝜆𝑧𝑥2 + 𝑦2
Ψ𝑧 𝑟 =𝜑𝑧 𝑟
𝑖𝜆𝑧න0
𝑅
2𝜋𝜌 × 𝑓 𝜌 × exp𝑖𝜋𝜌2
𝜆𝑧× 𝐽0
2𝜋𝜌𝑟
𝜆𝑧𝑑𝜌
Lommel series: decomposition into series (Aime, 2013)
Radial apodization
12/02/2018 R.Rougeot 61
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Vanderbei et al. approah
Based on Fresnel diffraction theoryFor serrated or petal-shaped occulter, i.e. a periodic pattern by rotation
Ψ𝑧 𝑟, 𝜃 = Ψ𝑧𝑎𝑝𝑜𝑑
𝑟 +
𝑗=1
∞
𝑓1 𝑗, 𝑁𝑡 ×න0
𝑅+Δ
𝑓2 𝑗, 𝜌 × 𝐽𝑗𝑁𝑡2𝜋𝑟𝜌
𝜆𝑧𝜌𝑑𝜌
In stellar coronagraphy:𝑁𝑡 ≈ 20, and very small working angles: j=1 dominates
In solar coronagraphy:𝑁𝑡 ≈ 100 − 1000, and large region (671mm): the computation is very heavy
14/05/2018 R.Rougeot 62
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Penumbra and Boivin radii
14/05/2018 R.Rougeot 63
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Penumbra for serrated occulters
Convolution of the diffraction intensity |Ψ𝑧 𝑥, 𝑦 |2 with the solar stenope imageIncludes limb darkening function
Penumbra:∫ Diffraction x Solar image
14/05/2018 R.Rougeot 64
Rsun=16,2’rsun=671mm
Solar image
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Penumbra for serrated occulters
14/05/2018 R.Rougeot 65
Diffraction patternSolar image
𝐼 𝑥 = 0
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Penumbra for serrated occulters
14/05/2018 R.Rougeot 66
Diffraction patternSolar image
𝐼 𝑥1 ≃ 𝐼(0)
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Penumbra for serrated occulters
14/05/2018 R.Rougeot 67
Diffraction patternSolar image
𝐼 𝑥2 > 𝐼(0)
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Penumbra for serrated occulters
14/05/2018 R.Rougeot 68
Diffraction patternSolar image
𝐼 𝑥 = 0 ↘
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Penumbra for serrated occultersWe can predict the penumbra depth for serrated occulters:
14/05/2018 R.Rougeot 69
The deepest umbra is achieved when:
Boivin radius 𝑁𝑡 , Δ > 𝑟𝑠𝑢𝑛
The second parameter is the intensity level of the diffraction pattern
→ Large number of teeth preferred!
𝑟𝑠𝑢𝑛