Observational Astronomy Chap 3

Observational Astronomy Chap 3 Telescope optics........................................................................................................................................ 2

Conic surfaces ....................................................................................................................................... 2 Stops and pupils .................................................................................................................................... 4 Primary Aberrations.............................................................................................................................. 5

Wave front errors (WFE) .................................................................................................................. 8 Analytic representation ..................................................................................................................... 9

Diffraction........................................................................................................................................... 11 Image formation.................................................................................................................................. 12

Diffraction limited systems............................................................................................................. 15 Mirror Systems.................................................................................................................................... 16

Single mirror systems...................................................................................................................... 16 Two-mirror systems ........................................................................................................................ 17 Three or more mirror systems......................................................................................................... 20 Spherical mirrors............................................................................................................................. 21 Auxiliary optics............................................................................................................................... 22

Optical error budget ............................................................................................................................ 23 Criteria for image quality................................................................................................................ 24

2

Telescope optics

Conic surfaces Large reflecting telescopes are composed of conic surfaces: paraboloids - ellipsoids – hyperboloids

Formed by rotating conic section about their axes

• Fundamental properties: normal at a given point bisects the angle formed by two radii joining that point to the two foci

All optic rays issuing from a focus will converge at the other focus forming a perfect image

Ex: parabola (degenerate ellipse) with one focus at infinity

Two-mirrors: second conic surface (secondary) is placed at the focus of the primary

• Gregorian: secondary = ellipsoids

• Cassegrain: secondary = hyperboloid

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Three-mirrors: primary + secondary + tertiary (multiple mirrors – fold flat mirrors are never counted)

Equivalent (effective) focal length: any system composed of several mirrors is equivalent to a single mirror Focal ratio (f/# or N): ratio between the effective focal length ( f ) and the diameter of the primary mirror ( D ):

f/# fND

= =

Plate scale (P): scale of image, in arcsecond, at focal plane, where the effective focal length (f) is in meter ( 0.206 is conversion factor)

0.206 arsecond 0.206 arsecond m m

Pf NDµ µ

= =

Practically, one can determine the plate scale by observing a binary star for which the separation is known For CCD, multiply by size of pixel in mµ to get P in arcsec/pixel Example:

For 1mD = , 7.5fND

= = and 15pm

pixelµµ =

arcsec0.4pixel

P =

The Field of View (FOV) is then easy to calculate:

FOV #pixelP= ×

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Stops and pupils Two stops limit the ray bundle passing through a telescope:

1. aperture (limit the ray that enter the telescope) 2. field (limit extent of image – FOV) stops

Vignetting: fading of the image near the edges of the field

• Happens when the intermediate optic components are not large enough to accept all oblique rays entering the aperture

• Object space: in front of first optical element of the telescope • Image space: after the last optical element of the telescope • Entrance pupil: image of the aperture in the object space • Exit pupil: image of aperture in the image space • Pupil: any intermediate image

The pupils contain all the rays that will reach the image, whatever the field angle ray bundles ⇒ different field angles do not shift over the pupil as a function of the field angle ⇒ ray fans (comprised of all field angles) passes through a minimum diameter at the pupil Optimal location for deformable mirrors, fine steering mirrors (minimizes size of these elements) or filters (guarantees that their characteristics remain unchanged as a function of the field angle)

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Primary Aberrations Oblique or off-axis aberrations: for any mirror, the image quality deteriorates with increasing angular distance of the source formed on the axis Aberrations no exact point to point correspondence between source and image

There are five primary aberrations: spherical, coma, astigmatism, field curvature and distortion

1. Spherical (aberration of spherical mirrors): rays issuing from a source at infinity on axis do not converge at the same point

• The effect is proportional to 3

3 DNf

− =

• Eliminated by using paraboloids • Two-mirror systems: spherical aberration happens when the primary and secondary do not have

the same conic constant (HST + any large ground based telescope)⇒ tests for two mirrors at the same time is costly and very difficult

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2. Coma: rays issuing from off-axis source do not converge at the same point in the focal plane

Creates a blur resembling a comet

a. Dominant aberration of a Cassegrain system used off-axis

• Effect is field dependent and increases with off-axis angle θ • Small N (fast mirrors) are more affected

b. Secondary mirror axis is not coaxial with primary mirror axis (collimation) • The effect is field dependent, but the amplitude is the same throughout the field

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3. Astigmatism: focus of rays in the plane containing axis and off-axis source (tangential plane) is different from focus of rays in perpendicular plane (sagittal plane)

• The effect scale as 2θ and 1N

affect fast mirror

4. Field curvature: image focus on a curved surface

• Without astigmatism, the image focus at the Petzval surface with the sag that scale with the astigmatism – same dependence as astigmatism

5. Distortion: plate scale is not constant, varying with the field angleθ

• Effect can be calibrated out easily (measured and removed from actual 2-D data) • Scale as 3θ

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Wave front errors (WFE) Deviation of wave front (WF) from a plane or sphere describes aberrations

• Light beam can be thought as a WF propagating through the optical system • Incoming (at infinite) - flat and perpendicular to the direction of propagation • Exiting - spherical with center located at the image optical path of all the rays are of the same

length • WFE is measured as the rms of the deviation over the entire surface • Expressed in nanometers or as a fraction of wavelength (wave) A system is considered

nearly perfect if the rms of WFE < 14λ

• During the actual use the WFE vary with time higher terms are more stable than those with low spatial frequency

• Lowest terms (tip-tilt – guiding – focus) vary easily because of temperature changes and

gravity effects • Misalignment mirror – thermal deformation Coma • Deformation of mirror due to improper support or thermal effects Astigmatism • Axial displacement of secondary Third order spherical aberration • Higher terms -- build into optics during figuring no action can be taken to correct them

(except adaptive optics)

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Analytic representation Zernike orthogonal polynomials Polar coordinates: - r = normalized radius (ratio of running radius on wave front to outer radius) - ϕ = polar angle

( ) ( ), ,n nn

W r a Z rϕ ϕ=∑

Where ( ),nZ r ϕ is Zernike polynomial, na is coefficient of the WFE

For an array of values ( ),r ϕ , in matrix form: W Za= where ( ) 1T Ta Z Z Z W−

=

• Values of ( ),nZ r ϕ depend on shape of the aperture • The higher the Zernike term the higher the frequency and lower the amplitude of the WFE • Advantage of ( ),nZ r ϕ : easily related to classical aberration + high spatial frequency

defects in mirror surface + also describes atmospheric WF distortion

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• 20 terms are sufficient to describe the WFE due to misalignment, mechanical and thermal deformation and errors in the optics

• But cannot describe very high spatial frequency = air turbulence or micro roughness in

mirrors – because too many terms would be necessary

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Diffraction Effect due to the wave nature of light: spreading of light as it passes at edges of aperture or any obstacle in path ray (secondary + supporting vanes)

• Circular aperture: pattern is a bright disk surrounded by alternating dark and bright rings (Airy

function) intensity falls like 3

1r

• Linear obstacle: spike in the image independent of position and perpendicular to the vane • Polygonal aperture: structures of cores and spikes (Ex. imaging with Keck)

Other diffracting degradation

1. Dust on the mirror: scattered incoming light produces faint halo around the image • Size of dust is several tens of microns

Wide angles (order of degrees) diffracting or scattering

2. Mirror surface defects at high frequency - roughness or periodic pattern (internal structures)

Produce faint halo with angular size determined by spatial frequency of defects

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Image formation Point Spread Function (PSF): distribution of light intensity in image depends on shape of aperture + obstruction + geometrical aberration + diffraction effects Complex Pupil Function (CPF): ( ) ( ),, ikW rCPF P r e ϕϕ= ( ),P r ϕ = transmittance of the aperture (unobscured 1P = , obscured 0P = )

2k πλ

= = wave number

( ),W r ϕ = Wave front Error (WFE) Theorem: PSF of the image of point source at infinity is proportional to 2D Fourier transform of CPF

Analysis with Fast Fourier Transform (FFT) Perfect optics Airy function = PSF of unobstructed circular aperture and monochromatic light:

( ) 21J x

I Cx

=

Where I = intensity, C = constant, ( )1J x = first Bessel function

Dx π θλ

= , where D = diameter of the aperture, and θ = angular coordinate of image spread

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Obstruction by secondary: ( ) ( ) 21 12J x J x

I Cx x

εε

ε

= −

Where ε = obscuration ratio = diameter central obstructiondiameter of aperture

• Obstruction has significant effect for 10 15%ε > −

For actual telescope, PSF is more complex: aberration of optics + various shape aperture + complex pattern of diffraction use Fourier transform method (MACOS – JPL TIM or Tiny TIM – STSI) To include effects like atmospheric seeing + mirror quilting + mirror roughness + dust, statistical models are used (reference Hasan and Burrows)

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HST pupil function and PSF. Top = pupil amplitude and phase maps taken with WFPC2 -- include shadows of secondary mirror, support vanes and three primary mirror support pads, as well as WFPC2 secondary obscuration and support vanes (offset from those of HST) bottom left = defocused PSF in UV (λ = 170 nm) middle and right corresponding models with and without error maps

The graphic shows the variations of the ratio of the HST PSF to the diffraction-limited PSF, as a function of wavelength and distance from the center of the PSF

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Diffraction limited systems Rayleigh quarter wavelength rule: as long as the WF in the image space remains between 2 concentric spheres separated by ¼ of λ the image quality is not affected The problem is that the effect of the WFE on the PSF varies significantly according to the type of aberration Rule of Maréchal: a system is essentially perfect when the normalized peak intensity of the image is equal to 0.8 of a perfect image Strehl ratio: ratio between normalized peak intensity of actual PSF to that of perfect image

Strehl ratio = ( )2

221 πλ

− ∆Φ

where ∆Φ is the rms WFE in wavelength (expression is valid only for

value > 0.5) requiring a Strehl ratio = 0.8 is equivalent to a rms WFE < λ/14 Quasi perfect optical systems are called diffraction limited Angular resolution: ability of optical systems to distinguish details in the image quantified by smallest angle between 2 points sources, for which separate recognizable images are produced Rayleigh criterion (diffraction limited system): angle for which the central peak of one image fall upon the first minimum of the other

1.22 Dλθ∆ =

Sparrow criterion: resolution limit = angular separation when the continued pattern of two sources

has no minimum between the two centers: Dλθ∆ =

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Depth of focus: tolerance on the axial position of the detector relative to the best optical focus (diffraction limited focus)

• Light beam near focus has a tunnel shape due to diffraction effects

• WFE at distance z∆ from geometrical focus (first order): 28z N∆

Depth of focus = 22 Nλ±

Mirror Systems

Single mirror systems

• Very small FOV • Determinant aberration = coma angular length of chromatic image on the sky (radians):

21

3Coma16 N

θ=

Where θ is the semi field angle (angle in radians from optical axis), and 1 1N f D= Since acceptable coma on the ground due to seeing is 0.5 arcsecond 2

10.044Nθ = in arcmin Need very slow mirrors (long f), Ex. for θ = 1 arcminute f/5, θ = 10 arcsecond f/2

• With small refractive optics (correctors) near the focal plane it is possible to reach 1 degree but this precludes use in the infrared

• Advantages: 1) purity 2) maximum throughput 3) no degradation due to misalignment • Disadvantages 1) limit mass and size of instrumentation 2) access to focus is difficult

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Two-mirror systems

• Most widely used o Focus access highly improved o Minimum reflection and central obstruction losses o Compact

• Classic types = parabolic + conical secondary -- perfectly astigmatic (on-axis) and less coma

because equivalent f/# is much slower than primary due to secondary magnification

Cassegrain Gregorian Shorter tubes Easier to baffle (real exit pupil)

• Typical f/8 or slower, with FOV (diameter – seeing limited) of 5-20 arcminutes 1910 - Ritchey-Chrétien (RC): departs slightly from paraboloid/hyperboloid configuration two hyperboloids aplanic optics, free of spherical aberration and coma

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Generalized Schwarzshild theorem: for any geometry with reasonable separations between the optical elements it is possible to correct n primary aberrations with n powered elements

Two mirrors can correct for spherical and coma aberrations two other mirrors would be needed to correct for astigmatism and field curvature

• Dominant remaining aberration of R-C telescope is astigmatism: for normalized back focal

distance 0β = , astigmatism scales as 2

12Nθ the field of good quality is proportional to square

root of the primary mirror focal ratio and becomes too small for fast primaries • Correction for astigmatism can be achieved using: 1) a refractive element like a single lens near

focus 2) for instruments used at fixed off-axis position (HST) correction can be build into reflective relay optics of each instruments

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• Image quality of R-C is sensitive to misalignment error: tilt + de-centered secondary produce coma (corrected by collimation) - longitudinal (piston) errors produce spherical aberration and plate scale variations

• Tolerance and de-centered secondary is proportional to cube of primary focal ratio play

against fast primaries • Both tilt and de-centering introduce same kind of coma - tilt can be corrected by

appropriate re-centering (Meinel design system insensitive to wind disturbances)

Off-axis telescope ⇒ when cleanliness is important (infrared, coronagraph) no obstruction of primary ⇒ inherently longer designs: for given f/#, diameter of parent, rotationally symmetric mirror twice that of off-axis mirror

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Three or more mirror systems

• Needed for beam steering or WF correction (better to use small mirrors ⇒ can be oriented or deformed rapidly with minimal negative dynamics effects

• Deformable or fine steering mirror located at pupils

⇒ R-C telescope = problem, since pupil is virtual and in front of the secondary

⇒ need to re-image pupil with single powered tertiary mirror (slightly off-axis) to create real and accessible one or use two judiciously placed extra powered mirrors (tertiary and quaternary) to remain on-axis

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Spherical mirrors

• Advantage: low cost fabrication • Disadvantage: correction of massive spherical aberration over a reasonable FOV is not trivial

Ex. six-mirror system proposed for a 100-meter telescope (two of the mirrors are simple flat)

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Auxiliary optics

1. Steering (tip-tilt) output beam to correct for image motion 2. Correcting WFE due to atmospheric seeing or internally generated aberrations (AO) 3. Improving imaging performance at the focus (field flatteners and correctors) 4. Matching given telescope design to specific detector (focal reducer) 5. Relying beam (Offner relay) 6. Capturing part of the output to provide signals for guiding + WF correction (dichroic)

Field derotator: for Coudé configuration in equatorial mount or on alt-az mount the field rotates when the field of the instrument is not too large (~ 1 arcminute) ⇒ Can be de-rotated by rotating a set of flat mirrors placed in the beam (real mirrors or total reflection prism) careful adjustment is required to keep the central image from wandering off-axis and keep the beam properly collimated with respect to the instrument ⇒ For wider field, field de-rotation is unpractical one then needs to rotate the focal plane of the instrument

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Optical error budget PSF is function of two factors:

1. Aperture (intrinsic to telescope design): outer shape of the primary + gaps mirror segments + central hole + support for secondary + obstruction in the beam

2. Wave front errors (mirror fabrication, misalignment, mechanical-thermal effects): imperfect

optics + atmosphere

a. Low spatial frequencies: spatial wavelength D to D/10 (classical aberrations) b. Mid spatial frequencies: D/10 to D/1000 c. High spatial frequencies: D/1000 down to fraction of wavelength of light

Traditional approach to budgeting WFE: specify the upper limit to mid-high frequency errors so that the impact is negligible

Full error budget is allocated on the low spatial frequencies -- justified because low spatial frequencies are most difficult to control

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Criteria for image quality Image quality is characterized by the PSF = 2D function that can be very complex ⇒ not practical Practical metrics:

1. MTF: Modulation Transfer Function

• Based on Fourrier analysis: an optical object can be represented as the sum of an infinite series of sinusoidal components - as each component is transmitted through the optics the spatial frequency is unchanged, but the amplitude decrease

o MTF measures the degradation of the amplitude with the frequency (like a filter

function applied on the object)

o Modulation or contrast: max min

max min

I IMI I

−=

+, where maxI , minI are the maximum and

minimum intensities respectively

o The MTF = ratio of modulation of image iM to that of object oM : i

o

MMTFM

=

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• MTF is usually used as a 1D function averaged azimuthally

o For a perfect system with circular aperture: ( ) ( )2 cos sinMTF ν φ φ φπ

= − , where

arcosDλνφ = , λ = wavelength of light, ν = spatial frequency and D the diameter of

the aperture ⇒ MTF = 0 at the cutoff spatial frequency cDνλ

= (the ultimate resolution

of the system Dλ )

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• Normalized frequency: nc

ννν

=

Figure: MTF of HST optics (solid) compared to ideal optics

• Optical Transfer Function (OTF): Inverse Fourrier transform of the PSF

o The amplitude = MTF ⇒ MTF contains same information as PSF ⇒ MTF of a system is the product of the MTF of its components (optics + detector + atmosphere), at least when their WFE are uncorrelated

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2. 80% encircled energy (EE)

• Defined as the angular diameter containing 80% of the energy • For perfect system (no aberration and no atmosphere) 80% of the light is contain in a

diameter 1.8Dλ∼

• This criterion represent the practical angular size of the image point source - it is an excellent measure of performance of large telescope because it relates to the two main astronomically meaningful parameters = sensitivity + resolution

• Disadvantage: λ dependent – must be set for prime λ for which the observatory is intended

3. FWHM (full width half maximum)

• Width averaged diameter of the PSF at half intensity • A good measure of image size but without including the wings (as for the EE)

4. Strehl ratio • Ratio of the peak intensity in the actual image to the peak of the theoretical diffraction

intensity • The Strehl ratio is proportional to the area under the MTF curve • Maréchal rule: a diffraction limited system has a Strehl ratio of 0.8 • Good image quality measure for near diffraction limited telescope

o But do not capture the features of the PSF beyond the core – Ex. strong mid-optical

frequencies in the WFE can seriously degrade the sensitivity, because they create a halo around the PSF core, while the height of the core and thus the Strehl ratio remain unaffected

5. WFE rms (∆Φ )

• Since the Strehl ratio is defined as ( )2

221 πλ

− ∆Φ

, one can take ∆Φ , the rms of the WFE,

as a metrics for the quality of the image o Disadvantage same as taking the Strehl ratio, but this can be alleviated by specifying

the rms WFE of the low mid and high spatial frequencies • ∆Φ is convenient for optical error budgeting the various components of the WFE can be

broken down or recombined using the rule of sum of squares

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6. Central intensity ratio (CIR) • (Introduced by Dierickx) quantifies the image quality of ground based telescopes - where the

degradation of the atmosphere turbulence is a dominant factor

• 0

SCIRS

= where

o 0S is the Strehl ratio of a telescope assumed perfect, taking into account only the atmospheric turbulence

o S is same quantity but taking into account the WFE • 0 1CIR≤ ≤ , reaching 1 when the telescope is limited only by the atmospheric seeing • The CIR is wavelength and seeing dependent, it depends on the rms of the WF slope error

• To the first order: 2

0

1 2.9CIR σθ

= −

where σ is the rms WF slope error, 0θ is the seeing

angle, 00

0.98rλθ = , where 0r is the Fried parameter - Ex. for the VLT CIR > 0.82 at λ =

5000 Å and for a seeing angle 0.2 arcsecond

7. Sharpness (Ψ )

• Image quality figure of merit for the detection of point source in background-limited mode (introduced by Burrows):

2ijPΨ =∑

Where ijP is the intensity in each pixel of the normalized PSF ( 1ijP =∑ ) • Second moment of the pixelized PSF – extract the maximum information from the image -

one would weight the importance of each pixel in the image according to the square of its intensity

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• Best image quality criterion for near-diffraction-limited telescope - used primarily for background limited observations because it is directly related to the astronomical performance of the telescope:

S IN B

Ψ=

Where I is the total number of photons from the source, B is background/pixel (includes sky + telescope emission + detector RN + dark current)

• Although this is the ultimate metric for background mode observation, it is not used for very

faint extended object, because it assumes fitting a model to actual image which is a highly uncertain process

Conclusions:

1. The MTF is the only criterion offering quasi-complete description of the image - for convenience sake, the two most global or single number measures used are the CIR (ground-based telescope) and the 80% EE (space-diffracted system) - for error budgeting purposes, rms WFE is used

2. For users – important to request the best image possible, but exquisite image quality has a cost

compromised has to be found 3. Performances specifications should be the result of thorough study on how best to meet the

scientific goals within cost and scheduled constraints 4. Because of variety of goals of observations, the best approach is an empirical one:

a. Establish clear scientific goals b. Models the proposed telescope with various choices of image quality c. Evaluates how each of the choices performs in the extraction of the scientific

parameters of interest

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Ex: study image quality for James Webb telescope:

1. Field of early time galaxies was modeled on purely scientific forms 2. 32 PSF were created exploring range of WFE with low and high spatial frequencies (simulated

images) 3. Evaluate images using observers processing software to extract parameters relevant to the study

(photometric redshifts, size of galaxies)

Allowed to pinpoint the most relevant image quality figure of merit (80% EE) and the wavelength where it should be defined