1 Introduction

• Telescopes: collect light
  • focusing telescopes: almost all wavelength
    • Sensitivity = primary mirror aperture (regardless the complexity of the telescope)
  • non-focusing telescopes: γ-ray and some radio and X-ray
• Telescopes are multi-elements systems: different optical designs and focal arrangements
• Theoretical background:
  • Geometrical optics provides the basis for telescope design
  • Diffraction theory and Fourier optics are needed to analyse the performance (aberrations, etc)
  • Paraxial approximation
    • paraxial optical rays: sinθ=θ+{\cal O}(θ³)
    • paraxial optical system:
      • all elements are infinitely thin
      • rotation symetry
      • centered on a common, optical axis
      • analytical formula can be derived: first-order optics
      • higher-order optics: off-axis image formation, aberrations, distortion effects on curved focal surfaces

2 Telescopes: basic quantities

• Focal length: f (depends on the optical design); physical size in the focal plane: d=α_rad f
• Focal scale or plate scale:
  • S(arcsec/mm) = 206265/f(mm)
  • the size (in mm) in the focal plane of an object (at infinity) of size α (arcsec) is d=α_arcsec/S
  • other usual (but instrument-dependent) units: arcsec/pixel
• Focal ratio (aperture ratio, numerical aperture, F-number): F=|f|/D
  • D the unobstructed aperture diameter of the lens;
  • e.g. a telescope with |f|/D=10 is said to have f/10 focal ratio
  • low F-number are called fast telescopes: why ?
    • image of size d: energy deposited per pixel is ∝ 1/d² ∝ 1/f²; collected photons ∝ D²; energy deposited per unit time per pixel is ∝ (D/f)²=1/F²; assumes that the object is spatially resolved!
• Field of view (FoV)
  • either the focal plane area with a good quality;
  • or the sky area over which illumination of the primary is (sufficiently) uniform (the area over which a source can be displaced without affecting the measured intensity)
  • limited by coma aberration

• Examples:
  • ESO/NTT (La Silla), D=3.58m, Nasmyth focus: f/11; show that S=5.3"/mm; what would be the plate scale of a CCD with 15mic pixels ? Compare with the ESO-3.6m telescope;
  • plate scale of the VLT is 1.894"/mm (Cassegrain focus): what is the aperture ratio F? assuming 24mic pixels, what is the plate scale in "/px ? What is the FoV of a 2048² CCD with such pixels?

2.1 Basic relations from geometrical optics

• The thin lens equation (thickness ≪ focal length):

  1/f = 1/s + 1/s'

  • where: f=focal length, s=object distance, s'=image distance
  • s and s' are algebraic quantities: refractive system, s (s') is positive if the object is located before (after) the lens (an oriented optical axis is needed)
  • object and image points are conjugate points: all rays leaving one point will (at least in the paraxial approx.) reach the other, conjugate, point
  • s→∞ for astronomical sources: conjugate image lies in the focal plane
  • generalization to account for different refractive index between the object and image space
  • for a spherical mirror of radius R in the paraxial approx.: same expressions with f=R/2

• Magnification:
  • transverse (or lateral): m = -s'/s = h'/h
  • longitudinal: m²;
  • angular: M=tan u' / tan u = s/s' = h/h' with u (u') the angle of the incident (refracted) ray
• Conservation of energy: h.n.tan(u) = h'.n'.tan(u') (called Lagrange invariant)
  • n, n': refractive index in object and image spaces, respectively
  • valid for systems with arbitrary number of lossless refracting or reflecting surfaces
  • alternative formulation: conservation of the étendue AΩ

2.2 Refracting telescopes

• objective forms an image that is magnified by the eyepiece
• entrance pupil: objective; exit pupils=focal plane of the eyepiece
• object at infinity and eyepiece such that s=f_o+f_e ⇒ output rays parallel, virtual image at infinity: afocal telescope
• chief ray: passes through the center of the system (entrance and exit pupils)
• magnification: m=θ'/θ

• not used effectively but recalled here for historical (and pedagogical) reasons
• inherent problems of refractors:
  • chromatic aberration: n≡ n(λ) ⇒ f≡ f(λ)
  • energy absorption
  • size and f-ratio (largest refractor: f/46 unusable in practice !); largest scientific refractor was at Yerkes Observatory (Chicago University, Wisconsin): D=1.02m, f/D=19

2.3 Reflecting telescopes

• All telescopes today
• Mirrors vs lens:
  • lighter: simpler and more reliable mechanical setup
  • can be polished with extreme precision (better than λ/10)
  • better sensitivity
  • no chromatic aberration
• Reflecting telescopes can also reach large focal ratios depending on the optical configuration (see below):
  • prime focus: F=f/D=1.2 to 2.5
  • Cassegrain and Nasmyth: F=7-30
  • Coudé: F>50

3 Aberrations and real telescopes

• Spherical mirrors are not stigmatic: spherical aberrations
  • reflected rays incident at different radii intercept the optical axis at different location
  • angular diameter of the blur circle (circle of least confusion): β = 1/(128 F³) [rad], or 1.6" for a f/10 telescope, but 60" for a f/3 telescope;
  • the more open the telescope, the larger the circle of least confusion;
  • famous example: HST

Geometrical aberrations

• Fermat's principle:
  • for any pair of conjugate object and image points, there exists a conic surface that gives a perfect image, ind. of the paraxial approx. being applicable or not;
  • for any other pair of conjugate, the surface would lead to aberrations
  • parabola: it is rigorously stigmatic for objects located at infinity;
• Conic sections are the ideal shapes:
  • paraboloid: a point at infinity is imaged to a point at finite distance
  • ellipsoid: a point at finite distance is imaged to a point at finite distance
  • hyperboloid: a point at finite distance is imaged into a virtual focus behind the mirror
• Image blurring:
  • Geometrical aberrations
  • Diffraction

3.1 Seidel aberrations

Five, intrinsic primary aberrations inducing wavefront errors at low spatial frequency:

1. spherical aberration: depends on r; different focus for off-axis and paraxial rays; wavefront bend too much; more pronounced for low f/D systems; ex.: HST
2. coma (depends on α)
3. astigmatism (r and α): different response in x and y
4. field curvature (r and α): image plane is curved; off-axis image is defocus
5. field distortion (α): image scale changes with α; (distorded image of a grid)

More general characterization: Zernike polynomials. See general Optics textbooks for more details.

Spherical aberration, coma, astigmatism

• as the inclination angle θ increases, θ³ terms grow in the sinθ development

Spherical aberration

Coma

Astigmatism

Spherical aberration, coma, astigmatism

3.2 Two-mirrors stigmatic optical configurations

3.2.1 Two-mirror telescopes: Cassegrain and Gregorian configurations

• Main configurations
  • Gregorian (1663): paraboloid primary + hyperboloid secondary (image reversed)
  • Cassegrain (1672): paraboloid primary + ellipsoid secondary
  • Newtonian (1668): paraboloid + prime focus
• Advantages of two-mirrors systems
  • Secondary mirror: magnification; can increase the telescope focal length f by changing only secondary
  • reduce (or suppress) aberrations of the primary

3.2.2 Coma vs astigmatism limited FoV

• FoV of classical Cassegrain (or Gregorian) telescopes (left): limited by off-axis deformation (coma)
• RC telescopes (right): FoV limited by astigmatism

Cassegrain configuration

• Advantages:
  • long effective focal length and compactness: reduced mechanical loading
  • focal region is "easy" to access
  • Large focal length in a compact structure; advantage of compactness: less flexure
  • Zero spherical aberration by proper combination of M1 and M2;

• Performance limited by coma (images become comet-like as moving away from the paralactic approximation).
  • Field curvature;
  • Most appropriate for small FoV.
• Relation between basic quantities:
  • transverse magnification of secondary: m=-s₂'/s₂, m>0 for Cassegrain, m<0 for Gregorian
  • primary-mirror focal ratio: F₁=|f₁|/D
  • telescope focal ratio: F=|f|/D
  • m=f/f₁ and F=|m|F₁

Gregorian configuration

• Advantages:
  • Similar to Cassegrain
  • Image inverted in Cassegrain, erected in Gregorian
  • Secondary (ellipsoid) is easier
• Limitations:
  • Longer than Cassegrain: more expensive to support and house
  • Also limited by coma an field curvature
  • Secondary larger for Gregorian than Cassegrain, more blocking;

3.2.3 Ritchey-Chrétien optical layout

• Aplanat: a configuration with zero spherical and coma aberrations
• Examples: Aplanatic Cassegrain or Ritchey-Chrétien (RC) telescopes
• RC: two hyperbolic mirrors
  • most popular configuration for professional telescopes
  • large field of view
  • astigmatism and field curvature are reduced;
  • increased field curvature;
  • but larger secondary hence smaller aperture
• Field of view:
  • limited by coma for non aplanatic telescopes (Cassegrain or Gregorian)
  • limited by astigmatism for aplanatic telescopes (RC)
• HST, VLT, NTT, Herschel, Keck, Subaru

3.2.4 Comparison of optical designs

• Symbols:
  • \(k\): ratio of secondary/primary diameters ⇒ \(k^2\) gives the (minimum) fractional obstruction of M1 by M2;
  • \((1-k)f_1\): distance between M1 and M2 ⇒ compactness
  • \(mkf_1\): distance between M2 and focal surface
  • angular aberrations: transverse coma (TC), angular astigmatism (AS), distortion (DI)
  • κ_m R₁ and κ_p R₁: related to the image field curvature
• Results:
  • CC vs CG: CG less compact and more obscuration: CC overall better than CG
  • Aplanatic config. (RC, AG): zero coma (by def.)
  • RC vs CC:
    • larger image curvature
    • larger astigmatism

3.3 Focal plane and instrumentations

Telescope foci

• prime focus
• Cassegrain focus
• coudé or Nasmyth (off-axis) focus; allow for heavy instruments; coudé=fixed point; Nasmyth: moves with the azimuth axis
• coudé: no motion (heaviest instruments, high-resolution spectrometers)

3.4 Telescope gallery

The Palomar (Hale) 5m telescope

• The 200-inch (5.1-meter) Hale Telescope: built in 1948
• a single piece 5m paraboloid of Pyrex (14.5 tons)
• the largest until 1993; still used (exoplanet imaging)

3.4.1 Telescope gallery: CFHT

• Cassegrain optical configuration (see here)
• prime focus (f/3.77, f=13533mm), Cassegrain (f/8), and coudé focus
• Primary: 3.58 m

3.4.2 Telescope gallery: HST

• launched April 24, 1990
• 340 miles from Earth, 15 orbits/day (8 km/s)
• Primary: 2.4m; Secondary: 30.5 cm
• Primary curvature is wrong by 1mic ! Corrective optical system COSTAR, 1993

3.4.3 Telescope gallery: VLT

• Paranal, Chile; 2635m altitude
• One (of the four) 8.2m unit
• Primary: 8.2 m; secondary (with chopping capability 0.1-5 Hz): 1.1 m

3.4.4 Telescope gallery: The Keck Telescope

• 10m primary in a Ritchey-Chrétien configuration
• Performance: 80% in 0.4" diameter; FWHM≈ 0.2"
• f/1.6 primary; f/15 secondary (also f/25 and f/40 chopping secondary mirrors for infrared obs.)
• Nasmyth and Cassegrain foci

4 Physical optics and image formation

• Geometric optics: basic image formation and optical design
• Diffraction: needed to understand image formation (Huygens-Fresnel theory)
  • Huygens principle: starting from the wavefront seen as a surface where all points have the same phase (rather than the same distance to a source, as in the Fermat's point of view), it is postulated that, at any time during the propagation of the wavefront, any point on the wavefront acts as a source of secondary spherical wavelets; the envelope of these wavelets, at any later time, defines the new primary wavefront. In addition, the secondary wavelets propagate at the same speed and frequency as the primary wavefront. Successful in explaining refraction, reflection, and straight line propagation. But failed in explaining the diffraction.
  • Fresnel's added the superposition principle to allow for the secondary wavelets to interfere, namely the wavelets add in amplitude and phase.
  • Note: formally, diffraction = interference
• Astronomy: source in the far-field, Fraunhofer diffraction theory

4.1 Imaging: diffraction

Overview

• Perfect system: incident plane wave transforms into image wavefront forming the diffraction pattern in the focal plane
• Image wavefront: 2D map of the phase (wrt reference surface)
• Real systems: wavefront deformations due to phase errors
  • associated with off-axis (distance r from axis) and non-paraxial rays (incident angle α)
  • optical aberrations
  • reduce energy concentration, spatial resolution, and contrast
  • maximum r: aperture stop
  • maximum α: FoV
• Piston (constant phase shift), tip-tilt (linear gradient in x and/or y)
• Defocus: lowest order aberration; reduces image contrast and energy concentration; fast telescopes (f/D~3) are very sensitive to defocus

4.1.1 The Airy pattern

• Image of a point source by a single, perfectly round and uniformly illuminated aperture is the Airy pattern;

• This is the squared modulus of the Fourier transform of a circular aperture of radius \(r\):

  I(θ) = I₀ [2J₁(x)/x]², \(x=kr\sin\theta\)

  • J₁: first-kind Bessel function of order one
  • \(k=2\pi/\lambda\): wavenumber
  • θ: angle from the peak position
  • I₀ = P₀ π r² / (λ² R²), R=aperture-screen distance
• First null: x=3.83, θ=1.22λ/D; FWHM=1.028λ/D

Angular resolution

• First null of the Airy pattern is for

  sinθ_R = 1.22 λ/D

• Rayleigh criterion: two point sources will be separated, or just resolved, if the peak of one Airy pattern coincides with the first minium of the other:

  δθ = θ_R

• Two point-sources are fully resolved if the distance between the photocenters of the Airy patterns is

  δθ = 2θ_R

• First nulls: w=x/π=1.220, 2.233, 3.238 (for the unobstructed Airy pattern)

4.1.2 Fourier optics: contrast and PSF

• Imaging system characterized by modulation transfer function
  • input: object=intensity distribution; output=image
  • point source: for diffraction limited telescopes, its image is the Airy pattern
  • real telescopes: image of a point source = PSF, for point spread function; this is simply the response to an impulse (as in electronics, or signal treatment theory)

  • treating all points from the image as a collection of incoherent sources, their intensity add linearly; superposition theorem gives

    I(x,y) = ∫∫ PSF(x-u,y-v) I₀(u,v) du dv = I_object ⊗ PSF

  • in the Fourier plane, this becomes a multiplication of the FT of the brightness distribution by the FT of the PSF

    FT(I_image) = FT(PSF) × FT(I_object)

Fourier optics: PSF, OTF, MTF, PTF

• image intensity = object intensity ⊗ PSF and FT(I_image) = FT(PSF) × FT(I_object)
• The FT of the PSF is called the Optical Transfer Function (OTF)
  • if PSF is even, OTF is real (it is then called the modulation transfert function, MTF);
  • in general, OTF is a complex function: OTF(ν) = MTF(ν) exp[iΦ(ν)]
• Modulation Transfer Function: MTF = |OTF|
  • MTF(ν) = C_image/C_object = contrast in image / contrast in object
  • T(ν)→1 if ν→0 (uniform object); T(ν)→0 if ν below cutoff frequency ν_c=1/λF (F=f/D)
• Phase transfer function: PTF = angle(OTF)
  • Aberration such as coma would shift lateraly the image by an amount δ, corresponding to a phase shift Φ=2πδ/p, in the FT of the object brightness:
• Use of OTFs is extremely useful: deformations acting on the wavefront (atmosphere, aberrations, etc) can be treated separately and multiplied together to form the overall, or system, transfer function;

PSF and OTF: Spatial domain

• Consider an object with a 1D-sinusoidal intensity distribution:
  • period p₀, spatial frequency ν₀=1/p₀; note that for an object at infinity, p₀ and ν₀ have angular units
  • contrast: C₀ = (I_max - I_min) / (I_max + I_min), assumed ind. of ν₀; C₀ ∈ [0:1]
  • the object is a collection if incoherent sources: their intensity add together;
• Spatial-domain in incoherent imaging: I_image = PSF ⊗ I_object
  • assuming uniform magnification m, the image of the object is still a sinusoid, with p=mp₀ or ν=ν₀/m; spatial frequencies are unchanged
  • the image of the high-frequency object has a reduced contrast

PSF and OTF: Frequency-domain

• Frequency-domain view: TF(I_image) = MTF × TF(I_object)

• Cutoff frequency: limited pupil leads to a cutoff frequency that is the highest frequency transmitted by the system:

  ν_c = 1/(λ F) = D/(λ f) in m^-1, or D/λ in rad^-1

• Note: sampling w/o aliasing requires θ_sampl = 2θ_c

4.1.3 Influence of the aberrations on the MTF

4.2 Image quality

• Different quantities to characterize the image quality
  • Full Width at Half Maximum (FWHM): usually a Gaussian fit to the observed PSF
  • Encircled energy (EE): radius of the circle encompassing 50% (EE50) or 80% (EE80) of the total energy; diffraction-limited circular aperture has EE80=1.38 λ/D;
  • Ellipticity
  • Optical Path Difference (OPD): difference of path length between ideal and effective image wavefront; OPD<λ/4 are diffraction-limited; if OPD increases, EE increases (more energy is distributed in secondary maxima)

  • Strehl ratio (S): ratio of effective PSF peak intensity to the diffraction-limited PSF; for a rms wavefront deformation σ,

    S=exp[-(2πσ/λ)²]

• Diffraction limited
  • aberration-free telescope; an incoming spherical wave is transformed into an emerging spherical wave by the system; in reality, aberrations will generally deform an incoming spherical wave.
  • Note: S=0.80 is conventionnaly admitted as the limit for diffraction-limited telescopes. This corresponds to rms wavefront deformations of λ/13.4.

5 Improving reflecting mirrors

• motivation: time to reach given signal-to-noise is ∝ D^-2
• Note: for background limited, ∝ D^-4 (less background !)
• How to improve image quality and sensitivity ?
  • Increase aperture stop (diameter of primary): wavefront deformations must remain smaller than λ
  • Tracking and focus
• Need to control the deviations of the reflecting surfaces from their nominal shape: temperature fluctations, structure and mirror deformations induced by pointing at different elevation
• How to increase D ?
  • cost, risk, manufacturing and shipping, large structure deflections and mirror deformations
• Passive telescopes: tracking and focussing are controlled
• Active optics: computer-assisted control of the alignement and shape of the mirrors; seeing-limited telescopes
• Adapative optics: correct wavefront deformations due to the atmosphere; diffraction-limited telescopes (see D. Mouillet's lecture)

5.1 Segmented mirrors

• Different configurations and/or geometries:
  • independant telescope arrays
  • independant telescopes on a common mount
  • random subapertures within a common primary
  • annular, hexagonal segmentation

• Advantage of segmented mirrors: active control system

5.1.1 The ESO/E-ELT project

• the Extremely Large Telescope: 39m primary; f/17.48 Nasmyth focus, f/60 coudé focus;
• Segmented, elliptical, f/0.93, primary mirror: 931 segments (365 kg each)
• A new, 5-mirror, optical design was prefered to a Gregorian design
• Cerro Armazones, Chile's Atacama Desert, 3060m altitude
• median seeing is 0.67 arcsec at 500 nm with a median coherence time of 3.5 ms

5.2 Timescales of perturbations

• Different timescales associated to different perturbations
• Active optics: minute
• Adaptive optics: milli-second

• atmospheric timescale depends on wavelength
• wavefront propagating through the atmosphere: piston and tilt, separate images or speckles in the focal plane causing scintillation; averaged over t>τ₀, speckle ∼ Gaussian (seeing disk): FWHM≈ 25" λ(mic)/r₀(cm)
• Fried's parameter (~size of atmospheric turbulent eddy) is r₀ ∝ λ^6/5
• seeing PSF: α_seeing∼λ/r₀ ∝ λ^-1.5 is ≈ 1" at 0.5mic
• r₀ < D: optical telescopes are turbulence limited (contrary to radio telescopes)
• coherence time: τ₀ ∼ r₀/v ∝ λ^6/5; λ=0.5mic: r₀ ≈ 20cm and τ₀ ∼ 10ms (Mauna Kea, Chile)

Active optics in the visible

The active optics system of the VLT primary mirror consists of 150 tripod supports

• four 8.2m mirrors of 23t each
• thickness: 175mm: highly flexible

Active optics: the Keck

• top panels: lowest freq. modes; defocus (left) and astigmatism (right); small size discontinuities (most difficult)
• bottom panels: highest freq. modes; large discontinuities (simpler)

Active optics in the radio domain: the GBT

• NRAO's 300-Foot telescope (100m) completed in 1962
• foreseen for 10 years operation
• actually 26 years

• Structural failure in Nov. 15th 1988: collapse !
• 1989: new GBT project: 100-m, unblocked, off-axis, closed-loop active optics
• Offset paraboloid primary: 2004 rectangular panels, 2mx2m, 2209 actuators at corners
• Homologous structure: remains paraboloid to about 1mm RMS (10ppm) for 5-90^o elevation