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Space Op(cs (2) AstrOpt2016 Derive Etendu . AΩ J. B. Breckinridge Adjunct professor Caltech & College of Op(cal Sciences Transmiance, throughput, & vigneng • How bright is my image? • Can I record it? • Parameters that describe the ability of the op(cal system to transmit power • What is the diameter of my op(cal elements? • Can they be fabricated or just designed! 4/15/14 Etendu & radiometry 2 Kirchoff’s Laws If a body of mass is at thermal equilibrium with its surrounding environment, conservaon of energy requires that Φincident = Φabsorbed + Φreflected + Φtransmitted By dividing both sides by , we write Φincident α + r + t = 1, where α is absorbtance, r is reflectance, and t is transmi]ance. For an opaque body where there is no transmi]ance (t = 0), the radiaon is either absorbed or reflected. Therefore, watts absorbed = α ⋅ E ⋅area = ε ⋅ M ⋅area = watts radiated. 4/12/12 Etendu & radiometry 3 Minimize reflectance loss to maximize transmi]ance • An(reflec(on coat has limitaons I λ R Reflected ( ) i (λ) = n n n I λ 1 2 3 Incident ( ) If n2 (λ) = n1(λ)n3(λ) Then reflectance is zero & transmiance is maximized Ocen a physical material with just the right n2 (λ) does not exist 4/15/14 Etendu & radiometry 4 Power at the focal plane • Determined by • Transmi]ance • Etendu or “through-put” • Polarizaon (discussed later) 4/15/14 Etendu & radiometry 5 Étendú or throughput [pp103] • Calculated using tools of 1st order opcs • Expresses the geometric ability of an op(cal system to pass radiaon from object space to image space • Ray trace can indicate an excellent image but if no light gets through the system, there is no image – SNR=0.0! • Consider 2 general rays, pass them through the op(cal system, then look at pupil and image planes 4/15/14 Etendu & radiometry 6 Étendú or throughput Consider 2 general rays Plane 1 Plane 2 φ Plane 3 2 n n1 2 y1 y2 Red u2 u1 u1 y1 y u 2 From Ch 2: 2 n2u2 = n1u1 − y1φ2 n2u2 = n1u1 − y1φ2 4/15/14 Etendu & radiometry 7 Étendú or throughput Consider 2 general rays n2u2 = n1u1 − y1φ2 n2u2 = n1u1 − y1φ2 (n1 − n2 ) Recall that: φ2 = (n1 − n2 )C2 = R The opcal power is the same for both rays2 (n2u2 − n1u1 ) (n2u2 − n1u1 ) = φ2 = y1 y1 Re-group the terms Then we discover that there is an invariant between any two planes in the op(cal system Invariant on n1u1y1 − n1u1y1 = n2u2 y1 − n2u2 y1 = H 4/15/14 refrac(on Etendu & radiometry 8 Étendú, Helmholtz, LaGrange Invariant n u y − u y = n u y − u y = H 1 ( 1 1 1 1 ) 2 ( 2 2 2 2 ) Rewrite this equaon with the object plane on the LHS y 0 1 = And and the pupil plane on the right hand side y 0 2 = Then nuy nuy H 1 = 2 = H has units of angle × distance, e.g., radians × cen(meters. 4/15/14 Etendu & radiometry 9 Area solid angle product Pupil plane +y +y +z Object plane chief AP ΩO ΩP A AOΩP = APΩO O 4/15/14 Etendu & radiometry 10 Confusion? • Transmi]ance, transmission, transmissivity - dimensionless • Throughput, etendu, Op(cal Invariant • Units of solid angle × Area, when calculang op(cal system capacity to transmit radiave power • Units of radians × length for op(cal ray trace design Etendu & radiometry 11 Useful relaonships π Ω = 4( f #)2 Etendu & radiometry 12 Vigneng: Étendú is not conserved at field points 1 2 For no vigneng, the radius Rk ≥ yk + yk . of the kth surface must be Etendu & radiometry 13 Space Op(cs (2) AstrOpt2016 Geometric Aberraon Theory J. B. Breckinridge Adjunct professor Caltech & College of Op(cal Sciences Class outline • The challenges of space op(cs • Derive Etendu, throughput, transmi]ance – Power to the focal plane • Geometric aberraons: thermal, structural, metrology, tolerancing & A/O • Scalar wave image formaon • Vector-wave image formaon: polarizaon aberraons – par(al coherence & correlated wave fields • Hubble trouble Image locaon, size, orientaon 16 Space op(cs differ from ground op(cs • No atmospheric turbulence • High angular resolu(on astronomy • Diffrac(on limited important • Aberraon really make a difference – Metering structure • Thermal (radiave, conduc(on, convecon), • Dynamics – Op(cal surfaces – • Stac & dynamic wavefront control 1st order opcs • Two planes, normal to the axis are important in an op(cal system – Pupil plane • Entrance pupil • Exit pupil – Image plane – A complicated op(cal system can have several image planes, pupil planes, but ONLY one each: • Entrance pupil • Exit pupil Aberraon Theory (1) 18 Two rays define locaon of image & pupil planes marginal Image plane chief Pupil plane Y-Z Plane is called Object plane the Meridional plane • The volume required for an op(cal system is defined by both the chief ray and the marginal ray radius of the clear aperture = r(z) = y + y Aberraon Theory (1) 19 A fan of rays provides an es(mate of image quality marginal chief Image plane Pupil plane Object plane Spot diagram Aberraon Theory (1) 20 Tolerancing an op(cal system Sources of errors Mechanical & structural 21 Coordinate system for geometric aberraon analysis 4/12/12 Aberraon Theory (2) 22 Sign convenons Aberraon Theory (1) 23 Sign convenon 4/12/12 Aberraon Theory (2) 24 Rays and Geometric Waves reference ray path – ray path OPD ρx′, ρ′y W , ( ) , (ρx′ ρ′y ) = = λ λ Independent of λ Aberraon Theory (1) 25 Defini(on of Wavefront aberraon W The equaon for a spherical wave converging to a point at a distance f from the pupil (lens) is ⎡ k ⎤ 2π u′ x, y = u x, y ⋅ exp – j x2 + y2 , k = ( ) ( ) ⎢ 2 f ( )⎥ ⎣ ⎦ λ And x and y are Cartesian coordinates across the pupil plane. An expression that includes the wave aberraon term W. k 2 2 E = – j (x + y )⎣⎡1 + W (x, y)⎦⎤ , 2 f Now what is W (x, y)? Aberraon Theory (1) 26 Geometrical wavefront error The geometrical wavefront error is independent of the wavelength. If one wanted to calculate how much material [as a funcon of posion] is needed to be removed From a mirror surface, then one need to mulply by wavelength. Aberraon Theory (1) 27 Expand the wavefront error • Zernike polynomials – Power series expansions on the unit circle • Seidel aberraons – Defocus, (lt, spherical, coma, as(gmasm, field curvature, Petzval Zernike polynomials Zernike polynomials are a set of orthogonal polynomials defined on a unit circle. They are expressed in either Cartesian (x, y) or polar ( ) ρ,ψ coordinates. There are both even and odd Zernike polynomials: m m Zn (ρ,ψ ) = Rn (ρ) cos(mψ ) −m m Zn (ρ,ψ ) = Rn ρ sin mψ ( ) ( ) 4/12/12 Aberraon Theory (2) 29 Zernike polynomials where the radial polynomials are defined as n – m k 2 –1 n – k ! Rm ( ) ( ) n – 2k n (ρ) = ∑ ρ k = 0 ⎡(n + m) ⎤ ⎡(n – m) ⎤ k!⎢ – k ⎥ ! ⋅ ⎢ – k ⎥! 2 2 ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ We will not use the Zernike polynomials here, but rather will analyze using the Seidel aberraons, which lend themselves to an intui(ve understanding of the physical origins of the aberraons. 4/12/12 Aberraon Theory (2) 30 Coordinate system for geometric aberraon analysis 4/12/12 Aberraon Theory (2) 31 The wavefront series expansion for the aberraons yields the expression m W a hk n cos , = ∑ knm ⋅ ρ ( ψ ) k, n, m By conven(on, first-order aberraon terms are those for which k + n – 1 = 1, Third-order aberraon terms are those for which k + n + 1 = 3, and Fich-order aberraon terms are those for which k + n + 3 = 5 The third-order aberraon coefficients are: a040, a220, a400, a131, a311, and a222. The fich-order aberraon coefficients are: a060, a240, a420, a600, a151, a331, a511, a242, and a422. Aberraon Theory (1) 32 Aberraons independent of the op(cs Defocus a020 • The aberraon a020 is introduced by an axial change in the focus and is called defocus. For defocus, the aberraon is corrected by translang the image point along the axis un(l the sphere corresponding to the aberrated image point is superposed onto the reference sphere. This aberraon, like a111, is corrected by reposi(oning the focal plane; it is not necessary to correct by refiguring any opcal element. Aberraon Theory (1) 33 Aberraons independent of the op(cs Tilt a111 • The term a111 represents a lateral shic. The aberrated wave is (lted about the vertex of the reference wave. This aberraon can be corrected by reposi(oning the focal plane. The aberraon term a111 represents a (lt in the wavefront in the y direcon. This aberraon can be corrected by reposi(oning the focal plane; it is not necessary to correct by refiguring any opcal element. Aberraon Theory (1) 34 Defocus and Tilt The actual surface is the dashed black sphere. Standing at the blue point looking lec you have created the blue reference surface. To super-impose your reference surface to the actual surface you need to move to the right (+), or refocus the system to make: a020 = 0. To super-impose your reference surface to the actual surface you need to (lt your reference surface up, or lt your focal plane the system to make: a111 = 0. Aberraon Theory (1) 35 The 3rd order (Seidel) terms We will examine in detail the first and third order aberraon m W a hk n cos = ∑ knm ⋅ ρ ( ψ ) k, n, m Aberraon Theory (1) 36 A point in object space is imaged … A point in object space is imaged into several points in image space by a geometric aberraons Aberraon Theory (1) 37 Tool for analysis: the spot diagram Aberraon Theory (1) 38 CAD ray-trace objec(ves • For each point [in object space] • Use trigonometry to trace rays from a point in object space – through a grid (hexapolar) on the physical surface of the lens, – refract and – then translate to the next surface un(l it – penetrates the focal plane at a point.
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