Encircled EnergyEdit
Encircled energy is a fundamental concept in imaging, optics, and astronomy that describes how light from a source is distributed as it is collected by an instrument. In practice, it is the fraction of total energy that falls inside a circular region centered on the image of the source. Engineers and scientists use encircled energy to quantify optical performance, calibrate measurements, and compare different systems.
Intuitively, encircled energy tells you how concentrated a source’s light is after it has passed through an optical system. A system with light tightly concentrated near the center will have a steep encircled energy curve, reaching high fractions of total energy at small radii. Conversely, a system with noticeable blur or aberrations will spread light farther from the center, producing a more gradual curve. The concept is widely used in fields from telescope design to fluorescence microscopy, and it underpins practical tasks such as aperture photometry and PSF modeling.
Definition and mathematical framework
Let I(x, y) be the image intensity distribution of a source in the imaging plane (after any detector effects are accounted for). Center the coordinate system on the image of the source. The encircled energy E(r) is defined as the fraction of the total energy contained within a circle of radius r:
- If the PSF is radially symmetric: E(r) = ∫0^r 2πρ I(ρ) dρ / ∫0^∞ 2πρ I(ρ) dρ
- In general, for a circle of radius r centered on the source: E(r) = ∫∫_{x^2 + y^2 ≤ r^2} I(x, y) dx dy / ∫∫ I(x, y) dx dy
For digital data, the integrals become sums over pixels. The function E(r) is often called the encircled energy curve or the encircled energy fraction (EEF). Typical convenience radii are r50 (the radius enclosing 50% of the energy), r80, and so on. Encircled energy curves are commonly plotted to summarize how rapidly energy is gathered as the aperture grows.
Encircled energy is closely related to the point spread function point spread function and to related metrics such as the full width at half maximum Full width at half maximum and the Strehl ratio Strehl ratio, which together describe how an optical system distributes light from a point source. In practice, measurements of encircled energy are often derived from observed stars or calibrated point sources, with careful treatment of background, noise, and background subtraction.
Practical computation and data handling
- Build a radial profile: From an image of a point source, compute the azimuthally averaged radial profile I(ρ) by averaging intensity in concentric rings around the centroid.
- Integrate to obtain E(r): Use the definition above to accumulate energy within circles of increasing radius, normalizing by the total source energy.
- Address real-world issues: Detector readout noise, background sky or bias, and nearby sources must be accounted for before computing the encircled energy curve. Wavelength dependence is important, as the PSF and thus E(r) can vary with color.
- Use in photometry: Encircled energy informs aperture corrections in aperture photometry, where a measured flux within a finite radius is adjusted to estimate the total flux aperture photometry; it also supports choosing an optimal aperture size for different science goals.
- Compare instruments and conditions: Encircled energy curves enable apples-to-apples comparisons of optical quality across telescopes, cameras, or observing conditions, and can help gauge the impact of aberrations or misalignment.
links: photometry, aperture photometry, point spread function
Applications across disciplines
- Astronomy and astrophysics: Encircled energy is central to characterizing telescope optics, camera detectors, and observing performance. It helps in quantifying image quality, defining photometric apertures, and performing calibration for surveys using fixed or adaptive apertures. It is also used to describe the PSF across a field of view and to model how energy from stars distributes within different instrument configurations.
- Optical engineering: In designing lenses, mirrors, and optical benches, encircled energy curves are part of the performance budget. They guide choices about coatings, aberration control, and alignment tolerances by illustrating how much light remains concentrated in the region of interest.
- Microscopy and fluorescence imaging: Encircled energy concepts apply to light collection efficiency and to evaluating objectives and detectors, especially when comparing different imaging modalities or numerical apertures. terms: optics, telescope, detector (instrumentation)
Variations, standards, and controversies
- Radial vs. nonradial PSFs: The standard encircled energy metric assumes a centered, approximately radially symmetric distribution. In the presence of significant asymmetries (due to coma, astigmatism, or field-dependent aberrations), the interpretation of E(r) can be more nuanced and may require localized or anisotropic energy metrics.
- Choice of aperture: Different observational goals favor different encircled energy radii. For precision photometry, smaller radii reduce background noise but require larger aperture corrections; for total flux estimates, larger radii capture more energy but increase contamination risk.
- Field dependence and wavelength: PSFs vary across the field of view and with wavelength, so encircled energy curves must be understood in the context of specific instruments, filters, and positions on the detector.
- Standards and cross-calibration: As with other instrument performance metrics, there is ongoing work to standardize how encircled energy is measured and reported, to enable robust comparisons across instruments such as telescope platforms and imaging detectors.