Point Spread Function (PSF)

• The 2D light distribution in an image of a point source is called "Point Spread Function" (PSF).

Perfect Optical System

• Airy Pattern: The tightest, most ideal PSF is achieved

Real Optical System

• Small manufacturing surface errors degrade PSFs.
• Turbulence in Earth's atmosphere causes large wavefront corrugations for astronomical telescopes.
• Rapid wavefront changes due to turbulence cause constantly varying, shifting instantaneous PSF.
• Gaussian shaped seeing PSF can be created by integrating instataneous PSFs over long exposure.
• Ground-based telescopes usually create PSFs that are blurrier than the expected Airy pattern, with resolution usually not better than 1".
• For a telescope with D = 1.5m, 1" is 10x worse than the theoretical 0.1" Rayleigh resolution; Astronomers might say, "the seeing is 1 arcsecond tonight”.

Strehl Ratio

• The Strehl Ratio indicates the light intensity at the PSF's center peak divided by the theoretical Airy disk's peak intensity for the same diameter D & focal ratio n.
• Formula: Strehl = I_PSF(actual) / I_Airy.
• 1.0 implies a perfect value (actual PSF = Airy), .8 is diffraction limited.
• Most large telescopes have low Strehl ratios (<< 0.1) due to atmospheric turbulence.
• "Adaptive Optics" can yield large telescope Strehl ratios up to 0.3.

Enclosed/Encircled Energy Ratio (EER)

• The EER indicates the ratio of actual to theoretical integrated brightness (energy) within a given PSF radius, often expressed as a percentage (%).
• EER < 100% in actual systems.
• Compares actual and theoretical performance.
• Strehl examines brightness at its peak
• EER analyzes the 2-d light spread.

The Imaging Equation (O(x,y))

• O(x,y), the observed image can be considered the underlying sharp image I(x,y) convolved with the PSF P(x,y) + random noise N(x,y).
• Resulting in the "imaging equation": O(x,y) = I(x,y) * P(x,y) + N(x,y).
• For a star (point source) I(x,y) is represented as a delta-function without thickness, no dimensionality.
• O(x,y) = P(x,y).
• Closely spaced stars/points result in overlapping PSFs.
• MTF is the basic principle

Modulation Transfer Function (MTF) & Spatial Frequency

• For any imaging instrument, the MTF assesses how the PSF affects resolution/contrast at all spatial scales: how well object details (modulation) transfer to the output image.
• MTF is how contrast varies with spatial frequency.

MTF Procedure Measurement

• Use an optical system, image line pairs of increasing closeness like a USAF-1951 resolution test chart.

• Spatial frequency is in periods/mm or cycles/mm

• One black + one white = one cycle; line pairs per mm (lp/mm) = lines per mm (lpm)

• Original object line pattern is a square-wave; the modulation of the object/test chart is within this wave pattern.

• The system's PSF convolves with the input pattern.

• Output image lines become rounded with reduced contrast: pure white > light grey, pure black > dark grey.

• The incoming sharp modulation degrades – output not fully transferred.

• Measuring degradation across different line spacings gives us the function, Modulation Transfer Function.

• Intensity cross-sections of the output images are examined to measure contrast/modulation:

• Modulation = (Imax - Imin) / (Imax + Imin)

• modulation (contrast) typically decreases as spatial frequencies increase

• Imperfections displace the MTF trace downwards, resulting in lower contrast given a spatial frequency.

• Cutoff spatial frequency defines when the optical system cannot resolve sources because they are at zero contrast (MTF = 0.0)

• The cutoff frequency is Dawes Limit: defined with respect to stars which have zero dip (contrast) points between peaks.

• Cutoff frequency formula: f_cutoff = 1 / (λn) lp/mm

• Wavelength (λ) is in mm, and n is the focal ratio as in Eqn [1.2] (not present in the text).

• Example: For a system with n=8, and light ≈ 5.5 x 10^-4 mm green light, f_cutoff = 227 lp/mm.

• Theoretical/ideal MTF corresponds to Rayleigh limit at MTF ~ 0.09.

• Dawes & Rayleigh limits are typically expressed in terms of angular resolution, they can be located as a linear/spatial resolution on the focal plane by using the MTF diagram.

Spot Size & Off-Axis Performance

• Spot size is the linear diameter of a faint star image on the focal plane. Theoretically:
• Spot size (in µm) ≈ n, i.e proportional to n f-ratio (assumes visible light)
• Smaller than Airy disk size: 1.34n
• Empirically: Inverse of the line pairs/mm for which the MTF = (50% contrast). Example:
• If 50 lp/mm, the spot size is 1/(50 lp/mm) = 0.02 mm or 20µm
• Assumes light source/double star is positioned optimally aligned with telescope's optical axis
• Desired: Spot size should be maintained across the whole FOV

Practical Application

• Control off-axis optical aberrations
• <20µm is considered high performance
• Faster systems often fail at meeting theoretical spot size because of optical aberration
• A telescope with a centered obstruction (secondary mirror) has a low contrast drop at the low & middle MTF frequencies, when compared to an unobstructed system that shows subtle planetary details.
• Diffraction around the obstruction moves light outside of the Airy disk's center into the inner diffraction rings
• Central obstruction > 15% causes image modification
• Most obstructed telescopes fall in between the 25-40% range.
• Calculating the obstructed system's aperture via subtracting from the telescope's true aperture indicates the equivalent contrast for lower frequencies.
• Formula: D_equiv = D - D_obstruction
• Obstruction has almost no effect on on high frequencies. Loss of light collection is another issue.
• Effective aperture area light gathering: A_eff = π(D/2)^2 - π(D_obs/2)^2.
• Effective aperture diameter: Deff = 2√(A_eff/π) Deff = √D^2-Dobs^2