Astronomical Observing Techniques Lecture 6: Op:cs

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1 Astronomical Observing Techniques Lecture 6: Op:cs Christoph U. Keller

2 Outline 1. Geometrical Op<cs 1. Defini<ons 2. Aberra<ons 2. Diffrac<on Op<cs 1. Fraunhofer Diffrac<on 2. PSF, MTF 3. SR & EE 4. high contrast imaging Astronomical Observing Techniques: Telescopes 2

3 Spherical and Plane Waves light source: collec<on of sources of spherical waves astronomical sources: almost exclusively incoherent lasers, masers: coherent sources spherical wave origina<ng at very large distance can be approximated by plane wave

4 Ideal Op:cs ideal op<cs: spherical waves from any point in object space are imaged into points in image space corresponding points are called conjugate points focal point: center of converging or diverging spherical wavefront object space and image space are reversible

5 Ideal Op:cal System ideal op<cal system transforms plane wavefront into spherical, converging wavefront

6 Azimuthal Symmetry Optical Axis most op<cal systems are azimuthally symmetric axis of symmetry is op<cal axis

7 Locally Flat Wavefronts rays normal to local wave (loca<ons of constant phase) local wave around rays is assumed to be plane wave

$refracted according to Fresnel equ.$

8 Rays geomtrical op<cs works with rays only rays reflected and refracted according to Fresnel equ. phase is neglected (incoherent sum)

9 Finite Object Distance object may also be at finite distance also in astronomy: reimaging within instruments and telescopes

10 Geometrical Op:cs Example: SPEX

11 Aperture and Field Stops telescope camera Aperture stop: determines diameter of light cone from axial point on object. Field stop: determines the field of view of the system.

12 Entrance and Exit Pupils telescope camera Entrance pupil: image of aperture stop in object space Exit pupil: image of aperture stop in image space

13 Marginal and Chief Rays telescope camera Marginal ray: ray from object point on op<cal axis that passes at edge of entrance pupil Chief ray: ray from an object point at the edge of the field, passing through the center of the aperture stop.

14 Images and Pupils image every object point comes to a focus in an image plane light in one image point comes from pupil posi<ons object informa<on is encoded in posi<on, not in angle pupil all object rays are smeared out over complete aperture light in one pupil point comes from different object posi<ons object informa<on is encoded in angle, not in posi<on

15 Speed/F- Number/Numerical Aperture f D Θ n Speed of op<cal system described by numerical aperture (NA) or F number: f NA = n sinθ and F = D 1 2(NA) Generally, fast op0cs (large NA) has a high light power, is compact, but has 0ght tolerances and is difficult to manufacture. Slow op0cs (small NA) is just the opposite.

16 Étendue and AxΩ Geometrical étendue (fr. `extent ) is product of area A of source <mes solid angle Ω of op<cal system's entrance pupil as seen from source. Etendue is maximum beam size that instrument can accept. Hence, the étendue is also called acceptance, throughput, or A Ω product. The étendue never increases in any op<cal system. A perfect op<cal system produces an image with the same étendue as the source. Shrinking the field size A makes the beam faster (Ω bigger). Area A=h 2 π; Solid angle Ω given by marginal ray.

17 Aberra:ons Aberra<ons are departures of the performance of an op<cal system from the predic<ons of paraxial op<cs. Two categories of aberra<ons: 1. On- axis aberra<ons (defocus, spherical aberra<on) 2. Off- axis aberra<ons: a) Aberra<ons that degrade the image: coma, as<gma<sm b) Aberra<ons that alter the image posi<on: distor<on, field curvature

18 Reference sphere S with radius R for off- axis point P and aberrated wavefront W. Aberrated ray from object intersects image plane at P. Ray aberra<on is P P. Wave aberra<on is n QQ Wave and Ray Aberra:ons For small FOVs and radially symmetric aberrated wavefront W(r) we can approximate intersec<on with image plane: r i = R n i W r ( r)

19 Defocus (Out of Focus) defocused image focused image Depth of focus: δ = 2 2 λ 1 2λF = 2 NA Usually refers to op<cal path difference of λ/4.

20 Spherical Aberra:on Rays further from the op<cal axis have a different focal point than rays closer to the op<cal axis: ϕ is the wave aberration Θ is the angle in the pupil plane ρ is the radius in the pupil plane ξ = ρ sinθ; η = ρ cosθ

21 Hubble Trouble

22 HST Primary Mirror Spherical Aberra:on Null corrector for measuring mirror shape was incorrectly assembled (one lens misplaced by 1.3 mm). A management problem: Mirror manufacturer had analyzed surface with other null correctors, which indicated the problem, but test results were ignored because they were believed to be less accurate. Null corrector cancels non- spherical por<on of aspheric mirror figure. When correct mirror is viewed from point A, combina<on looks precisely spherical.

23 Coma Varia<on of magnifica<on across entrance pupil. Point sources will show a cometary tail. Coma is an inherent property of telescopes using parabolic mirrors. ϕ is the wave aberration Θ is the angle in the pupil plane ρ is the radius in the pupil plane ξ = ρ sinθ; η = ρ cosθ y 0 = position of the object in the field

24 As:gma:sm From off- axis point A lens does not appear symmetrical but shortened in plane of incidence (tangen<al plane). Emergent wave will have a smaller radius of curvature for tangen<al plane than for plane normal to it (sagikal plane) and form an image closer to the lens. ϕ is the wave aberration Θ is the angle in the pupil plane ρ is the radius in the pupil plane ξ = ρ sinθ; η = ρ cosθ y 0 = position of the object in the field

25 Field Curvature Only objects close to op<cal axis will be in focus on flat image plane. Close- to- axis and far off- axis objects will have different focal points due to OPL difference. ϕ is the wave aberration Θ is the angle in the pupil plane ρ is the radius in the pupil plane ξ = ρ sinθ; η = ρ cosθ y 0 = position of the object in the field

26 Distor:on (1) Straight lines on sky become curved lines in focal plane. Transversal magnifica<on depends on distance from op<cal axis. Chief ray undistorted Chief ray ϕ is the wave aberration Θ is the angle in the pupil plane ρ is the radius in the pupil plane ξ = ρ sinθ; η = ρ cosθ y 0 = position of the object in the field pincushiondistorted

27 Distor:on (2) Two cases: 1. Outer parts have smaller magnifica<on à barrel distor<on 2. Outer parts have larger magnifica<on à pincushion distor<on

28 Summary: Primary Wave Aberra:ons Dependence on pupil size Dependence on image size Spherical aberration ~ρ 4 const. Coma ~ρ 3 ~y Astigmatism ~ρ 2 ~y 2 Field curvature ~ρ 2 ~y 2 Distortion Defocus ~ρ ~ρ 2 ~y 3 const.

$Chroma:c Aberra:on Refrac<ve index n(λ), focal length of lens f(λ); different wavelengths have different foci.$

29 Chroma:c Aberra:on Refrac<ve index n(λ), focal length of lens f(λ); different wavelengths have different foci. (Mirrors are usually achroma<c). Mi<ga<on: use two lenses of different material with different dispersion à achroma<c doublet

30 Fresnel and Fraunhofer Diffraction Fresnel diffraction = near-field diffraction When a wave passes through an aperture and diffracts in the near field it causes the observed diffraction pattern to differ in size and shape for different distances. For Fraunhofer diffraction at infinity (far-field) the wave becomes planar. Fresnel: Fraunhofer: F F = = 2 r 1 d λ 2 r << 1 d λ An example of an optical setup that displays Fresnel diffraction occurring in the near-field. On this diagram, a wave is diffracted and observed at point σ. As this point is moved further back, beyond the Fresnel threshold or in the farfield, Fraunhofer diffraction occurs. (where F = Fresnel number, r = aperture size and d = distance to screen).

31 Fraunhofer Diffraction at Pupil Circular pupil function G(r) of unity within A and zero outside. Theorem: When a screen is illuminated by a source at infinity, the amplitude of the field diffracted in any direction is the Fourier transform of the pupil function characterizing the screen A. Mathematically, the amplitude of the diffracted field can be expressed as (see Lena book pp. 120ff for details): i2π ( θ θ ) V 1 ( θ, t) 1 = λ E A screena r G e λ 1 0 r λ dr 2 λ

32 Imaging and Filtering V(Θ 0 ), V(Θ 1 ): complex field amplitudes of points in object and image plane K(Θ 0 ;Θ 1 ): transmission of the system Then the image of an extended object can be described by: V i2πθ λ ( θ1 ) = V0 ( θ0 ) K( θ1 θ0 ) dθ0 where K( θ ) = G( r) e 2 object convolution In Fourier space: FT{ V( θ )} FT{ V ( θ )} FT{ K( θ )} = FT{ V ( )} G( r) 1 = θ0 The Fourier transform of the image equals the product of Fourier transform of the object and the pupil function G, which acts as a linear spatial filter. r dr λ

Point Spread Function (1) When the circular pupil is illuminated by a point source then the resulting PSF is described by a 1 st order Bessel function: I 1 ( θ ) ( 2π r θ / λ) 2 2J1 0 = 2π r0θ / λ

33 Point Spread Function (1) When the circular pupil is illuminated by a point source then the resulting PSF is described by a 1 st order Bessel function: I 1 ( θ ) ( 2π r θ / λ) 2 2J1 0 = 2π r0θ / λ This is also called the Airy function. The radius of the first dark ring (minimum) is at: r r.22λf or α1 = f 1 1 = 1 = The PSF is often simply characterized by the half power beam width (HPBW) or full width half maximum (FWHM) in angular units. λ D I 0 ( θ ) = δ ( θ ) According to the Nyquist-Shannon sampling theorem I(Θ) (or its FWHM) shall be sampled with a rate of at least: Δθ = 1 2 ω c

34 Point Spread Function (2) Most real telescopes have a central obscuration, which modifies our simplistic pupil function The resulting PSF can be described by a modified Airy function: I 1 ( θ ) = 1 ( 2 1 ε ) 2 ( 2π r θ / λ) 2J ( 2π r εθ / λ) 2 2J1 0 2π r0θ / λ 2 ε where ε is the fraction of central obscuration to total pupil area. G ( r) = Π( r / 2r ) π r εθ / λ Astronomical instruments sometimes use a phase mask to reduce the secondary lobes of the PSF (from diffraction at hard edges ). Phase masks introduce a position dependent phase change. This is called apodisation.

35 Op:cal/Modula:on Transfer Func:on Remember: so far we have used the Rayleigh criterion to describe resolution: two sources can be resolved if the peak of the second source is no closer than the 1st dark Airy ring of the first source. sin Θ =1.22 A better measure of the resolution that the system is capable of is the optical transfer function (OTF): λ D (a) Input (b) output MTF ( f ) = C C ( f ) 0, where C = I I max max + I I min min

36 Op:cal/Modula:on Transfer Func:on (2) The Optical Transfer Function (OTF) describes the spatial signal variation as a function of spatial frequency. With the spatial frequencies (ξ,η), the OTF can be written as where the Modulation Transfer Function (MTF) describes its magnitude, and the Phase Transfer Function (PTF) the phase. Example: OTF MTF PTF ( ξ, η) = MTF( ξ, η) PTF( ξ, η) ( ξ, η) = OTF( ξ, η) ( ) i2πλ( ξ, η ) ξ, η = e

37 Strehl ratio is convenient measure to assess quality of optical system. Strehl ratio (SR) is the ratio of the observed peak intensity of the PSF compared to the theoretical maximum peak intensity of a point source seen with a perfect imaging system working at the diffraction limit. Using the wave number k=2π/λ and the RMS wavefront error ω one can calculate that: Examples: Strehl Ratio SR = e k 1 k A SR > 80% is considered diffraction-limited à average WFE ~ λ/14 A typical adaptive optics system delivers SR ~ 10-50% (depends on λ) A seeing-limited PSF on an 8m telescope has a SR ~ %. 2 ω 2 2 ω 2

38 Encircled Energy Q: What is the maximum concentration of light within a small area? The fraction of the total PSF intensity within a certain radius is given by the encircled energy (EE): EE r λf π 2 1 πr λf 2 ( r) = 1 J J 0 Note that the EE depends strongly on the central obscuration ε of the telescope: F is the f/# number

Lecture 2: Geometrical Optics. Geometrical Approximation. Lenses. Mirrors. Optical Systems. Images and Pupils. Aberrations.

Lecture 2: Geometrical Optics. Geometrical Approximation. Lenses. Mirrors. Optical Systems. Images and Pupils. Aberrations. Lecture 2: Geometrical Optics Outline 1 Geometrical Approximation 2 Lenses 3 Mirrors 4 Optical Systems 5 Images and Pupils 6 Aberrations Christoph U. Keller, Leiden Observatory, keller@strw.leidenuniv.nl