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Astronomy · Seeing faint and small details

Diffraction limits and detector sampling

Distinguish the finite optical response to a point source from detector angular spacing, using an ideal circular-pupil model with clearly labelled radius, diameter and sampling scales.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

  • Interpret the ideal circular-aperture first-minimum scale and Rayleigh convention.
  • Calculate a small angular scale from wavelength and pupil diameter.
  • Explain what finer sampling or additional shifted observations can contribute beyond a resampled display.
01

A finite opening produces a wave-optical response

Geometric rays can meet at an ideal focus, but electromagnetic waves passing through a finite aperture produce diffraction. For an ideal monochromatic, uniformly illuminated, unobscured circular pupil with no aberration, a point source produces a bright central region and surrounding rings.

The pupil is the opening that shapes the wave entering the imaging system. Let its diameter be D and wavelength λ, expressed in the same length unit. At small angles the first intensity minimum occurs at angular radius θ ≈1.22λ/D radians from the central peak.

The coefficient 1.22 is rounded. This is the first dark-ring radius, not the full central-disk diameter or the FWHM of the pattern. The central disk’s diameter is twice that radius, and additional diffraction structure extends beyond it.

The point-spread function, or PSF, is the system’s image response to a point source. Its width does not measure the physical radius of an unresolved star. Real pupils, wavelength ranges, aberrations and detector response can change the recorded PSF.

02

Calculate the radius before choosing a resolution convention

For a synthetic ideal D =1 m pupil and λ =500 nm =5 ×10⁻⁷ m, the rounded first-minimum formula gives θ ≈6.1 ×10⁻⁷ rad. A radian is an angular unit;1 rad is about 206265 arcseconds, so this radius is approximately 0.126 arcseconds.

The corresponding central-disk diameter is approximately 0.252 arcseconds. The figure evaluates the ideal analytic pattern, whose more accurate first-zero radius is about 0.125787 arcseconds; both calculations round to 0.126 arcseconds. That small distinction reflects the rounded 1.22 coefficient.

The plotted curve is a normalised cut through the two-dimensional point-response pattern. Its peak value is 1. It is not an integrated one-dimensional flux distribution, a photon count record or a measurement of a real star. The finite displayed angular window does not contain every outer ring.

For two incoherent point sources, the Rayleigh criterion conventionally places one central maximum at the other’s first minimum. This defines a useful separation scale for that model. It is not a universal impossibility boundary independent of source brightness ratio, signal, sampling and inference method.

A diffraction scale and a pixel interval

Scroll sideways to see the full diagram. Its explanation is below.

A diffraction scale and a pixel intervalOriginal ideal monochromatic model for a uniformly illuminated unobstructed circular pupil of diameter D = 1 m at wavelength λ = 500 nm. Panel A plots a signed radial cut through its two-dimensional point-response intensity, normalised to the central peak: I(θ)/I(0) = [2J1(u)/u]², where θ is converted to radians in u = πDθ/λ, and the limit at θ = 0 is 1. The horizontal coordinate is angular offset from −0.30 to +0.30 arcseconds; the vertical coordinate is normalised intensity from 0 to 1. This is neither intensity integrated over the perpendicular coordinate nor a flux-density or total-flux measurement. The first dark ring lies at angular radius 0.1257874869 arcseconds, with the symmetric cut zeros at ±0.1257874869 arcseconds. Its exact coefficient within this model is 1.2196698913 times λ/D; 1.22 is a rounded version. One dimension bar runs from the central peak direction to the positive first zero and labels the radius approximately 0.126 arcseconds. Another spans both first zeros and labels the full central-disk diameter approximately 0.2516 arcseconds. The radius, full diameter and full width at half maximum are distinct; the conventional Rayleigh separation uses the first-minimum radius for the stated ideal point-source model, not a universal distinguishability limit. Panel B is separate: three empty geometric pixel-grid strips use the same angular horizontal scale as Panel A and the same −0.30 to +0.30 arcsecond extent. Their cell widths are 0.100, 0.050 and 0.025 arcseconds, giving 6, 12 and 24 cells. The strips contain no photon counts, brightness shading or point samples of the curve. A separate geometric bar shows the ideal critical uniform angular sampling interval λ/(2D) = 0.05156620156 arcseconds. Thus 0.100 arcseconds is coarser than that interval, while 0.050 and 0.025 are finer. The critical sampling interval differs from the first-zero radius. These statements retain an ideal monochromatic incoherent imaging and uniform-sampling model; they do not guarantee reconstruction or actual object detection in finite, noisy, aberrated or atmospheric data. No real source, instrument data, stellar size, total-flux or sensitivity claim is shown.A diffraction scale and a pixel intervalIdeal clear circular pupil · D = 1 m · λ = 500 nmA · Radial intensity cut through a 2D point responseNormalised to the peak; this curve is not an integrated flux.I(θ)/I(0)00.250.50.751−0.3−0.2−0.100.10.20.3Angular offset θ (arcsec)Radius ≈ 0.126 arcsecFull central-disk diameter ≈ 0.2516 arcsec1.22 λ/D is rounded; it is neither the full diameter nor FWHM.B · Angular pixel grids: geometry onlySame horizontal scale; no counts or samples of the curve.p = 0.100 arcsec · coarser than the critical intervalp = 0.050 arcsec · finer than the critical intervalp = 0.025 arcsec · finer than the critical intervalCritical interval λ/(2D) ≈ 0.05156620 arcsecIdeal monochromatic model and uniform angular sampling.First-zero radius, pixel spacing and FWHM are different.The Rayleigh separation is a convention for the stated ideal case.Finite noisy data, real pupils and atmosphere need further limits.No measured source, pixel signal or sensitivity is shown.
Original ideal monochromatic model for a uniformly illuminated unobstructed circular pupil of diameter D = 1 m at wavelength λ = 500 nm. Panel A plots a signed radial cut through its two-dimensional point-response intensity, normalised to the central peak: I(θ)/I(0) = [2J1(u)/u]², where θ is converted to radians in u = πDθ/λ, and the limit at θ = 0 is 1. The horizontal coordinate is angular offset from −0.30 to +0.30 arcseconds; the vertical coordinate is normalised intensity from 0 to 1. This is neither intensity integrated over the perpendicular coordinate nor a flux-density or total-flux measurement. The first dark ring lies at angular radius 0.1257874869 arcseconds, with the symmetric cut zeros at ±0.1257874869 arcseconds. Its exact coefficient within this model is 1.2196698913 times λ/D; 1.22 is a rounded version. One dimension bar runs from the central peak direction to the positive first zero and labels the radius approximately 0.126 arcseconds. Another spans both first zeros and labels the full central-disk diameter approximately 0.2516 arcseconds. The radius, full diameter and full width at half maximum are distinct; the conventional Rayleigh separation uses the first-minimum radius for the stated ideal point-source model, not a universal distinguishability limit. Panel B is separate: three empty geometric pixel-grid strips use the same angular horizontal scale as Panel A and the same −0.30 to +0.30 arcsecond extent. Their cell widths are 0.100, 0.050 and 0.025 arcseconds, giving 6, 12 and 24 cells. The strips contain no photon counts, brightness shading or point samples of the curve. A separate geometric bar shows the ideal critical uniform angular sampling interval λ/(2D) = 0.05156620156 arcseconds. Thus 0.100 arcseconds is coarser than that interval, while 0.050 and 0.025 are finer. The critical sampling interval differs from the first-zero radius. These statements retain an ideal monochromatic incoherent imaging and uniform-sampling model; they do not guarantee reconstruction or actual object detection in finite, noisy, aberrated or atmospheric data. No real source, instrument data, stellar size, total-flux or sensitivity claim is shown.
03

Pixels sample the delivered image

An angular pixel interval describes how far neighbouring detector samples are separated on the sky. Real pixels integrate radiation over finite areas with their own responses. The pixel interval is therefore a different quantity from the optical first-zero radius or disk diameter.

For the ideal monochromatic incoherent-image model, the optical spatial-frequency cutoff gives a critical uniform angular sampling interval λ/(2D). This is an ideal bandwidth-based scale, not a general claim that any two pixels across a loosely defined “spot” guarantee a faithful image.

For 500 nm and 1 m, λ/(2D) =2.5 ×10⁻⁷ rad, approximately 0.0516 arcseconds. A 0.10 arcsecond pixel interval is coarser than that ideal scale;0.05 and 0.025 arcsecond intervals are at least as fine. Finer pixels do not change the pupil’s diffraction pattern.

At a 0.05 arcsecond interval, the first-minimum radius spans about 2.52 pixels and the central-disk diameter about 5.03 pixels in this model. Radius, full diameter, FWHM and pixel spacing need distinct labels before any “pixels across” comparison.

A broad observing band combines wavelength-dependent responses. The shortest wavelengths impose the finest ideal sampling requirement in this simple model. Obstructions, aberrations, pixel integration and noise require additional care for practical measurements.

Original ideal 500 nm,1 m example. The optical first-minimum radius is about 0.126 arcseconds; ideal critical sampling interval is about 0.0516 arcseconds.
Angular pixel intervalFirst-minimum radius in pixelsRelation to ideal critical interval
0.10 arcseconds1.26Coarser
0.05 arcseconds2.52At least as fine
0.025 arcseconds5.03At least as fine
04

Additional measurements can differ from interpolation

If sampling is too coarse, different spatial patterns can become difficult to distinguish in the recorded data. Simply interpolating one image onto a finer grid rearranges the available information; it does not acquire the missing measurements.

Dithering means taking additional observations with deliberate offsets. Suitable offsets can provide new sampling phases, help handle detector defects and support a better-sampled combined image. The result depends on actual offsets, response calibration, noise and reconstruction.

Combining images also introduces resampling choices and possible artefacts. A more detailed-looking output grid is not by itself proof that all fine structure is reliable. Check the input sampling, PSF and processing assumptions before making a source-structure claim.

The retained sampling and aperture Foundations records provide the definitions behind these limits. The next lesson adds atmospheric seeing and adaptive correction, which affect the delivered image before detector sampling is considered.

Worked examples

Convert the model angle into arcseconds

Using the rounded 1.22 coefficient, calculate the first-minimum angular radius for λ =500 nm and D =1 m.

  1. Convert wavelength:500 nm =5 ×10⁻⁷ m.
  2. θ≈1.22 ×5 ×10⁻⁷ =6.1 ×10⁻⁷ rad.
  3. Multiply by 206265 arcseconds/rad to obtain about 0.126 arcseconds.

A first-minimum radius of about 0.126 arcseconds; the full central-disk diameter is about 0.252 arcseconds.

Name the scale before dividing

For this ideal pattern and 0.05 arcsecond pixels, how many pixels span the first-minimum radius and full central-disk diameter?

  1. Radius divided by interval gives about 0.1258/0.05 =2.52.
  2. The full diameter is twice the radius, giving about 5.03 pixels.
  3. Neither number defines the pattern’s FWHM.

About 2.52 pixels for the radius and 5.03 for the full central disk. The aperture response remains unchanged by choosing that sampling.

Try the reasoning

  1. Is 1.22λ/D the ideal central disk’s full angular diameter?

    Show the answer and reasoning

    No.

    It gives the first-minimum radius at small angle in the stated circular-pupil model.

  2. At fixed wavelength in this ideal model, what happens to the first-minimum scale when D doubles?

    Show the answer and reasoning

    It halves.

    The scale is proportional toλ/D, under the declared pupil assumptions.

  3. Does making ten interpolated pixels from one recorded pixel guarantee ten times finer source detail?

    Show the answer and reasoning

    No.

    A new display grid supplies no new independent observations.

  4. Why can multiple shifted observations provide information that interpolation of one exposure cannot?

    Show the answer and reasoning

    They can measure the image at additional sampling phases.

    Those are new measurements, although their benefit still depends on calibration, noise and the reconstruction model.

What to carry forward

  • A finite pupil gives a point-response pattern with separately defined radius, diameter and width measures.
  • Optical response and angular detector sampling are different scales.
  • New observations can improve sampling; a finer output grid alone does not guarantee recovered structure.

References

Teaching resources and research supporting this lesson’s explanations and examples.

  1. University Physics Volume 3, 4.5 Circular Apertures and ResolutionSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source
  2. 2.71 Optics: Lecture14, Wave Optics,31 March2014Nick Fang / Massachusetts Institute of Technology OpenCourseWareCourse notes · PDFocw.mit.eduRead source
  3. The Effect of Undersampling on 2MASS PSFsK. A. Marsh / IPAC, 29 May 2001Reference guideirsaproxy1.ipac.caltech.eduRead source
  4. Space Telescope Science Institute / JWST User Documentation2 sources
    1. JWST Dithering OverviewSpace Telescope Science Institute / JWST User DocumentationReference guidejwst-docs.stsci.eduRead source
    2. Resampling ArtifactsSpace Telescope Science Institute / JWST User DocumentationReference guidejwst-docs.stsci.eduRead source