( ) = First Bessel function, x = π Dθ
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1 Observational Astronomy Image formation Complex Pupil Function (CPF): (3.3.1) CPF = P( r,ϕ )e ( ) ikw r,ϕ P( r,ϕ ) = Transmittance of the aperture (unobscured P = 1, obscured P = 0 ) k = π λ = Wave number W ( r,ϕ ) = Wave front Error (WFE) Theorem: the PSF of the image of a point source at infinity is proportional to the D Fourier transform of the CPF Analysis can be done using Fast Fourier Transform (FFT) For a perfect optics PSF = Airy function I For an unobstructed circular aperture and monochromatic light: ( ) (3.3.) I = C J x 1 x J 1 x ( ) = First Bessel function, x = π Dθ angular coordinate of image spread λ, where D = diameter of the aperture, and θ = 1
2 On the right, energy encircled by Airy function: more than 80% within! 1.λ D Including the Obstruction by secondary: ( ) (3.3.3) I = C J x 1 x ( ) ε J εx 1 εx diameter central obstruction Where ε = obscuration ratio = diameter of aperture Left: effect of obscuration on encircled energy of PSF o Obstruction has significant effect for ε > 10 15% For actual telescope, the PSF is much more complex Aberration of optics + various shape aperture + complex pattern of diffraction Atmospheric seeing + mirror quilting + mirror roughness + dust, statistical models are used (reference Hasan and Burrows) Must use Fourier transform method (MACOS JPL TIM or Tiny TIM STSI) o Ex. James Webb Space Telescope PSF Simulation Tool: WebbPSF
3 Ex. HST pupil function and PSF Top = CPF amplitude and phase maps taken with WFPC o Include shadows of secondary mirror, support vanes and three primary mirror support pads, as well as WFPC secondary obscuration and support vanes (offset from those of HST) Bottom left = defocused PSF in UV (λ = 170 nm) o 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 3
4 Quality of the image Diffraction limited systems Quasi- perfect optical systems are called diffraction limited Their performance are limited by intrinsic nature of light Two different criteria: 1. Rayleigh quarter wavelength rule: As long as the WF in the image space remains between concentric spheres separated by ¼ of λ (WFE ~ λ/14) 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, as given by the Strehl ratio Strehl ratio: This is the ratio between the normalized peak intensity of actual PSF to that of perfect image (3.3.4) 1 π λ ( ΔΦ) Where ΔΦ 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 4
5 Optical error budget When planning a telescope, PSF study must be done to determine quality needed to reach scientific goals PSF is function of two main factors: 1) Aperture (intrinsic to telescope design): outer shape of the primary + gaps mirror segments + central hole + support for secondary + obstruction in the beam ) Wave front errors (mirror fabrication, misalignment, mechanical- thermal effects): imperfect optics + atmosphere Graded components by spatial frequencies: Low spatial frequencies: spatial wavelength D to D/10 (classical aberrations) Mid spatial frequencies: D/10 to D/1000 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 5
6 Criteria for image quality Image quality is characterized by the PSF = D function that can be very complex Not practical In practice one can use different metrics: 1) Modulation Transfer Function (MTF) 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 The MTF measures the degradation of the amplitude with the frequency (like a filter function applied on the object) Modulation or contrast: (3.3.5) M = I max I min I max + I min Where I max, I min are the maximum and minimum intensities respectively The MTF = ratio of modulation of image M i to that of object M o : (3.3.6) MTF = M i M o 6
7 MTF is usually used as a 1D function averaged azimuthally For a perfect system with circular aperture: (3.3.7) MTF ν ( ) = ( φ cosφ sinφ) π Where φ = arcos λν D the aperture, λ = wavelength of light, ν= spatial frequency and D is the diameter of MTF = 0 at the cutoff spatial frequency: (3.3.8) ν c = D λ The ultimate resolution of the system is λ D 7
8 (3.3.9) ν n = ν ν c Example: MTF of HST optics (solid) compared to ideal optics (dash) The Normalized frequency: Optical Transfer Function (OTF): The Inverse Fourrier transform of the PSF, for which the amplitude is the MTF The MTF contains therefore the same information as the PSF Furthermore, the MTF of a system is the product of the MTF of its components (optics + detector + atmosphere), at least when their WFE are uncorrelated 8
9 ) 80% encircled energy (EE) Defined as the angular diameter containing 80% of the energy For a perfect system (no aberration and no atmosphere) 80% of the light is contain in a diameter 1.8 λ D (includes the first ring, almost between 1. and.3 λ/d ) 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 wavelength for which the observatory is intended 3) Full width half maximum (FWHM) Defined as width averaged diameter of the PSF at half intensity A good measure of image size but without including the wings of PSF (as for the EE) Practical when observing 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 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 9
10 5) WFE rms (ΔΦ) The Strehl ratio being defined as: 1 π λ metrics for the quality of the image ( ΔΦ) one can take ΔΦ, the rms of the WFE, as a Using ΔΦ 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 Disadvantage: the same as taking the Strehl ratio Can be alleviated by specifying the rms WFE of the low mid and high spatial frequencies 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 (3.3.10) CIR = S S 0 Where S 0 is the Strehl ratio of a telescope assumed perfect, taking into account only the atmospheric turbulence S is same quantity but taking into account the WFE Usually 0 CIR 1, 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: (3.3.11) CIR = 1.9 σ θ 0 Where σ is the rms WF slope error, θ0 is the seeing angle, and θ 0 = 0.98 λ r 0, where r 0 is the Fried parameter o Ex. for the VLT CIR > 0.8 at λ = 5000 Å and for a seeing angle 0. arcsecond 10
11 7) Sharpness (Ψ) (Introduced by Burrows) Image quality figure of merit for the detection of point source in background- limited mode: (3.3.1) Ψ = P ij Where P ij is the intensity in each pixel of the normalized PSF ( P ij = 1) This corresponds to the second moment of the pixelized PSF Extract the maximum information from the image One could weight the importance of each pixel in the image according to the square of its intensity 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 (3.3.13) N = I Ψ B 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 11
12 Conclusions: The MTF is the only criterion offering quasi- complete description of the image For convenience sake, the two most global or single number metrics used are the CIR (ground- based telescope) and the 80% EE (space- diffracted system) For error budgeting purposes, rms WFE is used Remarks for users Important to request the best image possible, but since exquisite image quality has a cost compromises has to be found Performances specifications should be the result of thorough study on how best to meet the scientific goals within cost and scheduled constraints Because of variety of goals of observations, the best approach is an empirical one: Establish clear scientific goals Models the proposed telescope with various choices of image quality Evaluates how each of the choices performs in the extraction of the scientific parameters of interest Ex: study image quality for James Webb telescope Field of early time galaxies was modeled on purely scientific forms (scientific goal) 3 PSF were created exploring range of WFE with low and high spatial frequencies (simulated images quality) Evaluate images using observers processing software to extract parameters relevant to the study (photometric redshifts, size of galaxies) This allowed to pinpoint the most relevant image quality figure of merit (80% EE) and the wavelength where it should be defined 1
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