1 Definition and basic concepts

1.1 Energy fraction within a radius

Encircled energy is the portion of an optical intensity distribution that falls within a circle of radius \(r\) centered at a chosen reference point. For a given image of a point source, it quantifies “how concentrated” the signal is around that center by reporting the cumulative energy inside the circle as a fraction (or percentage) of the total energy.

1.2 Relationship to the point spread function (PSF)

In imaging optics, the intensity pattern produced by a point source is described by the point spread function (PSF). Encircled energy is commonly computed directly from the PSF intensity, treating it as the spatial distribution whose cumulative sum over radius is evaluated.

1.3 Encircled energy radius as a performance metric

Because encircled energy curves typically increase monotonically with radius, optical performance can be summarized by the radius required to reach a chosen energy fraction. Smaller radii achieving the same fraction indicate tighter concentration, which is often associated with improved resolution or reduced blur.

1.4 Total energy normalization conventions

Practical definitions vary in how “total energy” is obtained. Total energy may be taken as the integral over an infinite plane (ideal normalization) or approximated using a finite measurement window (cropped normalization). When the measurement does not capture all energy—especially in long-tailed PSFs—reported encircled energy fractions depend on the normalization convention.

2 Mathematical formulation

2.1 Radial integration of intensity

2.1.1 Circular aperture model

Let \(I(\mathbf{x})\) be the PSF intensity at image-plane coordinates \(\mathbf{x}\). Encircled energy within radius \(r\) is \[

E(r)=\frac{\int_{\mathbf{x}-\mathbf{x}_0\le r} I(\mathbf{x})\, dA}{\int_{\text{all}} I(\mathbf{x})\, dA},

\] where \(\mathbf{x}_0\) is the center reference (commonly the image centroid). The denominator sets the normalization.

2.1.2 Coordinate system and centroid reference

The choice of \(\mathbf{x}_0\) matters. A centroid reference aligns the integration circle with the intensity-weighted center, while alternative references (such as the nominal optical axis or detected peak position) can shift the circle and change the cumulative result, particularly for asymmetric or aberrated PSFs.

2.2 Encircled energy as a cumulative distribution

Encircled energy can be interpreted as a cumulative distribution function for the radial intensity. Differentiating with respect to radius yields the radial energy density—often helpful for understanding how energy is distributed across spatial scales.

2.3 Discrete vs continuous measurements

In experiments and digital imaging, intensities are sampled on pixels. The continuous integrals are approximated by summing sampled intensity values within the pixels whose centers lie inside the radius \(r\), using an appropriate pixel-area factor if required. Interpolation schemes may be used to reduce aliasing when radii do not align neatly with pixel boundaries.

2.4 Units, scaling, and plotting practices

Encircled energy is dimensionless when expressed as a fraction, or expressed as percent. Radii are typically reported in image-plane units (e.g., micrometers or millimeters), in angular units on the sky (for telescopes), or converted to detector pixels. Plotting conventions include:

  • \(E(r)\) versus \(r\) (often on linear axes for radii and logarithmic axes for energy when tails are important),
  • and reporting key points such as the half-energy radius on the same curve or in tables.

3 Practical measurement and estimation

3.1 Imaging from point sources

A practical PSF measurement begins by imaging a point source under controlled conditions. Methods include using a pinhole, fiber-coupled light source, or a star image for astronomical systems. The encircled energy curve is then computed from the acquired intensity distribution.

3.2 Instrumentation and sampling considerations

Adequate sampling is crucial. If the PSF core is undersampled, the peak may be biased and the cumulative energy at small radii can be inaccurate. Conversely, overly large pixel sizes can blur the measured intensity distribution due to pixel integration. System magnification, detector point-spread effects, and optical attenuation levels influence both the shape and the extent of the recorded intensity map.

3.3 Noise, background subtraction, and dynamic range

Real measurements include noise, stray light, and background offsets. Background subtraction is typically performed using regions sufficiently far from the PSF core, but care is needed: an incorrect background level can distort the long-radius behavior where energy may be small. Limited dynamic range can also cause truncation of faint wings, affecting the total captured energy and therefore the normalization.

Rather than computing radii at discrete sampled points only, practitioners may fit a smooth model to \(E(r)\) or to an analytic PSF approximation. From such fits, characteristic radii (e.g., the radius at which \(E(r)\) reaches 50%) can be extracted more robustly than by interpolation on noisy data. Connections to other summary measures (such as full width at half maximum and second-moment radius) can guide initial guesses, but the fitted encircled-energy function remains distinct in meaning and sensitivity.

4 Interpretation in optical systems

4.1 System blur and optical quality

Encircled energy reflects the degree of blur by showing how rapidly energy accumulates near the center. A system with sharper imaging yields a steeper rise at small radii, reaching any given energy fraction at a smaller radius. The curve’s overall form—core sharpness versus extended wings—helps distinguish moderate blur from significant stray light or aberration-induced spreading.

4.2 Comparison across designs and configurations

Encircled energy curves enable comparisons between optical designs, coatings, alignment states, or operating configurations, provided consistent definitions are used. Comparability requires:

  • consistent centering method,
  • consistent normalization approach (same measurement window or same total-energy estimate),
  • and similar acquisition conditions such as wavelength, bandwidth, and detector setup.

4.3 Throughput vs spatial concentration trade-offs

In some systems, improvements in spatial concentration can reduce throughput due to aperture stops, baffling, or design choices that limit stray paths. Encircled energy alone does not quantify efficiency; it measures distribution within a normalization. For complete evaluation, systems often pair encircled-energy metrics with throughput, modulation transfer, or total collected power.

4.4 Effect of aberrations on encircled energy curves

Aberrations shape the PSF and therefore the encircled energy curve. Common trends include:

  • wavefront errors that broaden the core increase the radius needed for a fixed energy fraction,
  • asymmetric aberrations can produce slower accumulation due to centroid shifts or skewed wings,
  • and certain aberrations create characteristic wing patterns that can be visible as nonstandard curve curvature.

5.1 Half-energy radius (HE radius)

The half-energy radius (often abbreviated HE radius) is the radius \(r_{50}\) where \(E(r)=0.5\). It provides a single-number summary of central concentration and is widely used because it is less sensitive to the very faintest wings than high-percentage radii.

5.2 EE at fixed percentages (e.g., 50%, 80%, 90%)

Other reported quantities include \(r_{80}\) or \(r_{90}\), corresponding to radii where the encircled energy reaches 0.8 or 0.9. These emphasize how much energy is retained near the center versus distributed into wings. Higher thresholds are useful for applications that require low stray-light contamination close to the target.

Although encircled energy, FWHM (full width at half maximum), and second-moment radius are related through the PSF shape, they generally do not coincide because each metric emphasizes different parts of the distribution:

  • FWHM focuses on the core around the peak,
  • second-moment radius reflects a balance between core and wings weighted by distance squared,
  • encircled energy directly tracks cumulative fraction with radius.

In practice, correlations can be system-dependent, especially when the PSF has non-Gaussian tails.

5.4 Encircled energy vs ensquared energy

Ensquared energy is a related metric computed over a square region rather than a circular one. It is often used when detectors have square pixels or when performance requirements are naturally framed by bounding boxes in image analysis. Because geometry differs, ensquared and circular encircled measures can yield different characteristic sizes even for the same underlying PSF.

6 Sensitivity analyses and error sources

6.1 Alignment and pointing errors

Misalignment between the true PSF center and the chosen integration center can shift energy out of the integration circle, reducing measured encircled energy at small radii. Pointing jitter broadens the effective PSF, altering the encircled-energy curve by both widening the core and potentially enhancing wings.

6.2 Defocus and focus tolerance

Defocus changes the PSF shape and typically degrades concentration. As focus error increases, encircled energy for small radii drops while radii required for a given energy fraction increase. Focus tolerance assessments often rely on how fast the encircled-energy curve moves as the focus setting is swept.

6.3 Wavelength dependence and bandwidth effects

The PSF scales with wavelength; in broadband illumination, the measured intensity distribution is a superposition across wavelengths. This can smooth out sharp features and broaden the effective PSF, changing encircled energy accumulation rates. When comparing results across systems, the spectral band and source spectrum should be reported or matched.

6.4 Manufacturing imperfections and surface errors

Surface roughness and figure errors introduce scattering and aberrations that alter both core and wings. Scattering can add a low-level background-like halo, which disproportionately affects high-percentage encircled energy radii (e.g., 90%). Figure errors can also distort the PSF, changing the curve’s curvature and possibly producing asymmetry that influences centroid-based integration.

7 Applications in imaging and illumination

7.1 Telescopes and astronomical imaging

In astronomy, encircled energy helps quantify image quality expressed by how compact stellar images remain on the detector. It can support comparisons between optical assemblies, guide alignment, and support requirements for point-source detection and photometry under realistic observing conditions.

7.2 Microscopy and endoscopic imaging

Microscopes rely on tight focusing to produce well-confined images of small features. Encircled energy provides a way to summarize how energy from a point-like specimen is distributed, supporting decisions about numerical aperture, optical correction methods, and tolerances for imaging quality in compact imaging systems.

7.3 Lithography and optical metrology

In high-resolution fabrication and measurement, controlling the spread of optical energy directly impacts pattern fidelity. Encircled energy metrics can be used to assess optical system performance, estimate blur-related tolerances, and diagnose how aberrations or illumination conditions affect the distribution near focus.

7.4 Laser beam characterization and coupling

For laser systems, encircled energy can describe beam quality and coupling efficiency into apertures. When a beam is modeled as having a spatial intensity distribution, encircled energy quantifies what fraction passes within a given aperture radius, directly linking optical beam characterization to practical alignment and coupling constraints.

8 Visualization and reporting

8.1 Typical encircled energy curves

A standard encircled-energy plot rises from near zero at very small radii and approaches unity at sufficiently large radius. The steepness near the origin reflects core quality, while the tail behavior indicates wing strength. In systems with substantial scatter, the curve may approach unity slowly, revealing the presence of extended energy tails.

8.2 Selecting radii and reference frames

Reporting commonly specifies:

  • the set of radii sampled (or the resolution of the curve),
  • the unit system (pixels, micrometers, angles),
  • and the reference frame used for centering (centroid, peak, or known geometric center).

These choices affect interpretability, particularly when the PSF is asymmetric or when cropping is used.

8.3 Reporting standards for reproducibility

Reproducibility depends on transparent documentation of:

  • normalization method (infinite-plane vs finite-window),
  • background subtraction procedure,
  • centroiding algorithm and any thresholds,
  • measurement window size and sampling rate,
  • and the relevant optical conditions (wavelength band, polarization, or illumination mode).

When these are not aligned, direct comparison of curves can be misleading.

8.4 Uncertainty quantification in published results

Uncertainty can originate from noise, background estimation, centroid variability, detector calibration, and finite sampling. A careful reporting approach may include confidence intervals on encircled-energy curves and uncertainty bounds on derived radii such as the half-energy radius, especially when used to meet engineering acceptance criteria.

9 Computational simulation workflows

9.1 Modeling PSFs from optical prescriptions

Simulation begins with an optical prescription describing lenses, mirrors, stops, and alignment. The PSF is computed using wave optics or Fourier optics methods, often based on the pupil function and system aberrations at a specified wavelength.

9.2 Aberration-driven PSF generation

Aberration terms (such as defocus, coma, astigmatism, and higher-order components) modify the phase of the pupil function and therefore the resulting intensity in the image plane. The simulated PSF intensity map becomes the input for encircled-energy calculation.

9.3 Convolution with detector and system effects

Real imaging systems include detector blur, pixel integration, jitter, and sometimes additional scattering or stray light. Simulations often incorporate these effects by convolving the optical PSF with an appropriate model of the detector or by applying sampling and noise models, producing a synthetic intensity distribution comparable to measurements.

9.4 Generating encircled energy from simulated intensity maps

Given a simulated intensity map, the encircled energy is computed by:

  • selecting a center reference (typically the intensity centroid),
  • integrating within circles of increasing radius,
  • applying the chosen normalization (full simulated extent or cropped window),
  • and producing a curve \(E(r)\) or extracting characteristic radii at fixed energy percentages.

10 Limitations and interpretation pitfalls

10.1 Dependence on centroiding and cropping

If the centroid estimation is unstable due to noise, the integration center can wander, distorting small-radius encircled energy. Similarly, cropping the image can omit energy in wings, causing the curve to saturate prematurely and potentially making systems appear better than they are relative to a full-energy normalization.

10.2 Off-axis sources and field dependence

A PSF changes across the field of view due to optical aberrations and telecentricity effects. Encircled energy measured on-axis may not represent performance at off-axis positions. Field-dependent reporting often requires sampling at multiple points and quoting the corresponding PSF center and normalization.

10.3 Non-ideal aperture shapes and obscurations

Real optical systems may include central obscurations, support structures, or non-circular apertures. These features modify the PSF and can introduce diffractive structure or redistribution of energy. While the encircled-energy definition uses a circular integration region by choice, the system’s actual pupil shape still influences the resulting curve.

10.4 Misinterpretation when energy tails are significant

When PSFs have strong wings, a large fraction of energy may lie far from the center. In such cases, metrics that emphasize only small radii or only the core can underestimate the impact of scattering on tasks such as background contamination or near-target contrast. Conversely, high-percentage radii can become sensitive to measurement window size and normalization errors.