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Hale COLLAGE (CU ASTR-7500) “Topics in Solar Observation Techniques” Spring 2016, Part 1 of 3: Off-limb coronagraphy & spectroscopy Lecturer: Prof. Steven R. Cranmer APS Dept., CU Boulder [email protected] http://lasp.colorado.edu/~cranmer/ Lecture 4: Diffraction around an occulter Brief overview Goals of Lecture 4: 1. Understand why we need to block out the bright solar disk. 2. Derive basic physics of diffraction around an “occulter.” Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Intensity above the limb • Visible-light emission from the solar corona: • K-corona due to Thomsonscattering of free electrons • F-corona due to scattering of inner heliospheric dust Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Intensity above the limb • Visible-light emission from the solar corona: • K-corona due to Thomsonscattering of free electrons • F-corona due to scattering of inner heliospheric dust • Compare with sky brightness: • typical hazy sky • clear day sky (mountain-top?) • eclipse totality sky Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 F corona = Zodiacal light Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Separating the K & F coronae • Near the Sun, K corona is linearly polarized; F corona is unpolarized (more about this soon…) • K corona follows magnetic structure; F corona is ~spherically symmetric • Original meanings: F = Fraunhofer, K = kontinuierlich (“continuous”) • F-corona: dust is cold; scatters full solar spectrum, absorption lines & all. • K-corona: free electrons are hot; Doppler broadening smears out lines in the spectrum. Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 How do we go about observing the corona? Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Off-limb geometry • The Sun’s disk subtends solid angle Ω, angular radius = 959” = 0.267o • Our goal: observe light coming from larger elongation angles Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Off-limb geometry • For observations not too far from the Sun, “observing rays” are nearly parallel: x = Line of sight (LOS) impact parameter in the “plane of sky” (POS) heliocentric radius of any point along the LOS Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 What can an “artificial eclipse” achieve? i.e., why do we need to block out the Sun? • Even if every ray is separable from every other ray, the huge dynamic range of intensities is usually too much for a detector to handle. • Of course, the rays get mixed up by diffraction. geometrical optics physical optics ! Trade-offs: • The more we occult the Sun, the better we can beat down diffraction. • More over-occulting → we can’t see regions nearest to the Sun. Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Basic optical paths (linear cut) • We want to block rays from the solar disk, while letting in coronal light: Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Basic optical paths (linear cut) • In reality, some fraction of the disk light is diffracted around the occulter edge and be reflected by the mirror, thus contaminating the coronal light. Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Diffraction • Born & Wolf: Principles of Optics, Hecht: Optics, ASTR-5550, PHYS-4510 • Interference of electromagnetic waves when they interact with “obstacles” Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 The Huygens-Fresnel principle • Point-sources of radiation emit spherical wave-fronts. Huygens: • Every point on a spherical wave-front (surface of constant phase) acts as a point-source for secondary spherical waves. Fresnel: • E-field at any point can be constructed as the superposition of all “incoming” wave-fronts. Kirchhoff: • Above is a direct consequence of the wave equation derivable from Maxwell’s equations. Lecture 4: Diffraction around an occulter In vacuum, the wave-fronts coalesce in predictable ways… Hale COLLAGE, Spring 2016 Diffraction: aperture geometry • Goal: Given E(x,y,z,t) at source P0 , compute E(x,y,z,t) at observation point P. • Note: distances r & s vary a lot for different points Q inside the aperture; distances r′ & s′ are known & fixed. Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • A spherical wave starts at P0 • The field at point Q in the aperture (at distance r from P0) is: • Collect time-dependence into amplitude: Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • Fresnel found that the differential amount of electric field that reaches point P after passing through a differential area (dA) of the aperture is: i.e., • But what is the normalization constant K ? Units: 1 / Length • It tells us the relative efficiency of the secondary-wave generation. • Two ways to derive it: 1. Forget Huygens-Fresnel, use Maxwell’s eqns! 2. Assume dA covers all space… EP = unobstructed E Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • In any case, χ = “scattering angle” (prevents secondary waves going back to the source) In telescopes, these angles are usually small; assume χ ≈ 0 . • If the angles are all small, then that means the aperture diameter (call it D) is << the distances r, r′, s, s′ • Thus, can we can replace r & s by r′ & s′ ? Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • In any case, χ = “scattering angle” (prevents secondary waves going back to the source) In telescopes, these angles are usually small; assume χ ≈ 0 . • If the angles are all small, then that means the aperture diameter (call it D) is << the distances r, r′, s, s′ • Thus, can we can replace r & s by r′ & s′ ? Lecture 4: Diffraction around an occulter In denominator: Yes In exponent: No! Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • We need to specify distances in terms of a known coordinate system… Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • Remember: (x,y) tell us the location of point P (the detector) (ξ,η) tell us the location of point Q (“inside the aperture”) • We’ve already assumed that D (the maximum extent of ξ and η) is << r′ & s′ • Thus, all ratios above are << 1 Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Follow the waves: P0 → Q → P • We’re getting closer to useful expressions… • The exp argument depends on the difference between actual & reference paths: Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Simplify some things • Assume P0 is so astronomically distant that waves at Q are plane waves. • The reference distance s′ (from aperture to detector) is often called R x ≈ Rθ D R • Thus, if original plane wave was unobstructed, it would reach detector with: and thus… Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fraunhofer vs. Fresnel diffraction • Fraunhofer: for “small” apertures, quadratic terms in ξ and η can be neglected. → eikf integral becomes a spatial Fourier transform of the aperture → appropriate for slits, pinholes, and finite-sized optics in telescopes • Fresnel: need to keep all terms. Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fresnel diffraction: rectangular aperture • Sneaky trick: take point P along the z-axis, but move around the aperture: y Thus, we care only about η2 x η1 ξ1 for which ξ2 Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fresnel integrals Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fresnel integrals “Cornu spiral” Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fresnel diffraction: straight-edge • Expand 3 of the 4 sides of the rectangular aperture to infinity… y η2 x η1 ξ1 ξ2 v1 < 0 : observer (at x=0) in direct sunlight v1 > 0 : observer in the shadow of occulter Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fresnel diffraction: straight-edge Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Fresnel diffraction: straight-edge If the calculation is re-done with a circular occulter or a circular aperture… Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Far inside the shadow… Let’s work it out for a simple idea for a solar coronagraph… • Visible light (λ = 500 nm) • Typical distance between occulter & primary mirror (R = 2 m) • Try a range of θ corresponding to coronal “elongation angles” Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Far inside the shadow… Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Far inside the shadow… What are our options? λ = 500 nm R=2m • We have to move the occulter much further away (like the distance of the Moon?), • or we need to experiment with other shapes (other than a straight-edge), • or we need to just deal with the diffracted light after it enters the telescope. Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016 Next time • How do real coronagraphs solve (or at least minimize) the problem of diffraction? Lecture 4: Diffraction around an occulter Hale COLLAGE, Spring 2016