# Ensquared Energy and Optical Centroid Efficiency in Optical Sensors: Part 1, Theory

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## Abstract

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## 1. Discrete Pixels

## 2. Energy Interception by Discrete Pixels

#### 2.1. Energy on Detector (EOD)

_{A}= 1.22 λF/#. The wavelength is denoted by λ, and the instrument is characterized by its ratio f/D = F/#. D is the aperture diameter, f is the focal distance, and F/# (sometimes also f/#) is the f-number. The pixel size is equal to the Airy disc diameter in the third illustration from the left. The two pixels on its left are larger by 20% and 40% than the diameter of the Airy disc. Both also include two point-sources positioned at the Rayleigh resolution limit. In this case, the pixel dimension acts to define the instrument resolution. The pixel on the furthest right would resolve the two point-sources at the Rayleigh’s resolution limit, but only if it were correctly centered.

_{x}d

_{y}for the pixel dimensions of 2d

_{x}by 2d

_{y}.

^{2}] or [#photons/(cm

^{2}s)] [6,7,8]. We note that the (degree of) coherence of the object point source is not of interest. The source coherence may only change the form of the psf(x,y) function to some degree, but not the theoretical development presented here. For the image centered at the origin (0,0), the psf(x,y) attains the maximum value at x = y = 0 in this EOD formulation. The denominator is the total energy in the image of a point source. In practice, the total energy only includes the summation of energy incident on the few neighboring pixels.

#### 2.2. Optical Centroiding Efficiency (OCE)

_{x}, ±Δd

_{y}) where these two quantities lie between 0 and d

_{x}, d

_{y}, respectively.

## 3. Imaging Theory When Optical and Detector Axes Are Arbitrarily Displaced

#### 3.1. Image Radiometry of a Point Source

_{P}, is most easily obtained from the power transfer equation [9,10]. This formulation considers a continuous detection process, such as the film used in the past, prior to the introduction of the digital sensors.

_{P}is the irradiance at the instrument aperture [watts/cm

^{2}] or [#photons/(cm

^{2}s)]. A

_{s}is the sensor/instrument aperture area [cm

^{2}]; τ is the instrument optical transmission factor, including any aperture obscuration or obstruction.

_{P}in Equation (2) as a product of relative power φ

_{P}and the EOD, introduced in Equation (1).

_{d}(d

_{x}, d

_{y}).

#### 3.2. Instrument Characteristic Function

_{x}by 2d

_{y}as pixel dimensions. This function is sometimes denoted as rect(x,y). It has the property that its one-dimensional Fourier transform is a sinc(x) = sinx/x. The pixel center is, in general, not aligned with the axis of the optical system; the pixel center is displaced randomly, in direction and magnitude, from the optical axis of the image-forming system. The image centroid falls on point (Δd

_{x}, Δd

_{y}). We consider that the coordinates of the detector-pixel center define the origin of the coordinate system, because the detector-pixel geometry introduces a fixed reference in the image detection and readout.

_{x}, y−Δd

_{y}), we obtain the instrument point response function, prf, or the instrument function,

_{x}, +/−10d

_{y}(see also Figure 1).

_{x}, Δd

_{y}are zero.

_{x}, Δd

_{y}are zero. Thus, recalling Equation (1) for the energy on detector, the EOD, we modify Equation (8), resulting in the normalized prf

_{n}(x,y). We denote the pixel area, 4d

_{x}d

_{y}, with A

_{d}.

_{d}(x,y) is the ensquared energy, that is, the energy enclosed within a pixel of dimensions 2d

_{x}by 2d

_{y}when the image centroid is not coincident with the pixel center. In this equation (Δd

_{x}, Δd

_{y}) are the coordinates of the image centroid. Inserting the decentering in Equation (1), we have

_{x},Δd

_{y}) with respect to the optical axis. The OCE is, then, a single number that is defined as the ratio of the energy-on-detector pixel that is misaligned by (Δd

_{x},Δd

_{y}) with respect to the optical axis, averaged over all possible misalignments, to the ensquared energy on the aligned detector pixel.

_{x},d

_{y}).

_{d}(d

_{x},d

_{y};Δd

_{x},Δd

_{y}) is given in Equation (15) and the EOD(d

_{x},d

_{y}) in Equation (1). The OCE(d

_{x},d

_{y}) is a statistical variable that depends only on the relationship between the pixel dimensions and the point spread function through EOD

_{d}(d

_{x},d

_{y};Δd

_{x},Δd

_{y}) and EOD(d

_{x},d

_{y}).

_{x},d

_{y}) is, therefore, defined as the normalized EOD(d

_{x},d

_{y}) average over a detector area, given the equal probability that the image centroid falls anywhere on the detector pixel. It may be computed when we know the psf(x,y) of the optical system and the detector pixel size, 2d

_{x}and 2d

_{y}. We are interested in the product of the OCE and the EOD because it enters the figures of merit of radiometric systems, as in Equation (7).

## 4. Modeling and Methods

^{1/2}. D denotes the aperture diameter. Both ξ and η coordinates are assumed to have non-zero values in the image plane within the range [(−d

_{x},+d

_{x}), (−d

_{y},+d

_{y})]. According to the formal analysis, the limits of integration range from minus infinity to plus infinity while the image is formed in the far field. R is the image distance. Furthermore, by the definition of the f-number, F/#, (λR/D) = λF/#, the dimensions in the image plane may be normalized by the system (λF/#)-product.

_{x},d

_{y}) and the OCE(d

_{x},d

_{y}), presented schematically in Figure 6. We set up a lens in CodeV and calculated psf(x,y) directly. We then exported psf(x,y) to MatLab, where the EOD was verified one more time to ensure the consistency of numerical and ray-trace results. The OCE was obtained with MatLab, employing the formulas here derived. We modeled the detector pixel as a square with a dimension of 2d. For the sake of due diligence, part of our work was also numerically evaluated (using Equations in Figure 6) with Mathcad. Both approaches agreed.

## 5. Effects of Pixel Size and Central Obscuration

#### 5.1. Aperture Configuration: No Central Obscuration

_{x},d

_{y}) and EOD

_{d}(d

_{x},d

_{y}) be first determined. Therefore, we first calculated the prf(x,y). Figure 8 presents the instrument point response function, prf(x,y), vs. pixel number for two orthogonal directions and two different pixel sizes, 2d = 3.33 λF/# in (a), and 2d = 13.33 λF/# in (b).

#### 5.1.1. Case 1: Small Pixel, No Central Obscuration

#### 5.1.2. Case 2: Large Pixel, No Central Obscuration

_{x},d

_{y}) was 0.967 for the 13.33 λF/#-pixel size. The psf of the optical system was convolved with the detector pixel response function (drf) to find EOD

_{d}(d

_{x},d

_{y}). Next, the OCE value was calculated by integrating the prf(x,y) over the detector area (−0.5 < x < 0.5, −0.5 < y < 0.5). We calculated a value of 0.926 for this OCE. The product of the EOD by OCE (0.967 × 0.926) was 0.895.

#### 5.2. Aperture Configuration: Circular, Incorporating Rectangular Central Obscuration

#### Case 3: Small Pixel, Round Aperture with a Central Rectangular Obscuration

## 6. Discussion

#### 6.1. Signal-Carrying Energy on Detector

#### 6.2. Small, Medium, and Large Pixels to Collect Signal-Carrying Energy

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The normalized integrated, encircled energy as a function of radius r of the enclosing circle for an ideal, diffraction-limited optical system with a circular aperture. The obscuration ratio ε is a parameter (ratio of radii), varying from 0 (no obscuration) to 0.8 in increments of 0.2. The degree of compactness of the central spot may be recognized from the average slope of these curves for radial values less than 1. About 84% of the energy is enclosed within the first dark ring of a diffraction-limited optical system without the central obscuration. All the curves also flatten to zero slope for radial distances that correspond to the zero-rings, due to the absence of radiation there.

**Figure 2.**(

**a**) Image by an optical system of a point object at infinity, with the central spot often referred to as the Airy disc; (

**b**) Rayleigh resolution distance is equal to the Airy disc radius.

**Figure 3.**A schematic illustration of how the pixel dimension relative to the size of the Airy disc affects the amount of energy that is incident on and absorbed by the detector pixel.

**Figure 4.**(

**a**) The image of a point source is in general not aligned with a pixel center. (

**b**) When in the extreme, limiting case, the image falls on the corner of four pixels, the detected signal in upper right pixel is equal to the noise in three neighboring pixels. In fact, there is no image detected against the noise, or possibly a broadly blurred image.

**Figure 5.**(

**a**) Transmission profile of a circular pupil featuring a general rectangular central obscuration. Gray indicates zero-transmission. The size of the obscuration rectangle is 0.27 D × 0.46 D, where D is the diameter of the entrance pupil. (

**b**) Point spread function, psf, of the circular pupil with the internal central obscuration shown in part (

**a**). Different profiles are obtained along the x- and y-dimensions because the rectangular obscuration is asymmetrical.

**Figure 6.**Block diagram to determine the EOD and OCE, using CodeV or some other lens design program to calculate the optical system psf(x,y).

**Figure 7.**Optical centroid efficiency vs. detector pixel size in units of [λF/#] for a perfect optical system with a circular aperture, without a central obscuration.

**Figure 8.**Two prf-s vs. pixel number shown for two orthogonal directions, x and y (in the inset denoted as red, rows, and blue, columns): (

**a**) 2d = 3.33 λF/# and (

**b**) 2d = 13.33 λF/#. Pixel coordinates along both dimensions are, in both cases, normalized to the pixel size. The general features of the curves for the small (

**left**) and large (

**right**) pixel sizes are quite similar, except that the peak for the smaller pixel (3.33 λF/#) is decreased by about 4% with respect to the peak of the large pixel. Additionally, the top of the profile for the smaller pixel size (

**left**) is more rounded and narrower than that of the large pixel (

**right**).

**Figure 9.**The OCE as a function of pixel size in [λF/#] for an optical system with a central rectangular obscuration (see Figure 5a). We observed a sharp increase in the OCE value toward the first peak to about 0.93 at about 2.05 pixels (Point B), followed by a sharp dip to about 0.82 (Point E) at about 4 pixels. Only from this, the lowest point on the curve, its shape starts to increase in a monotonical fashion.

**Figure 10.**Cross-sections of the prf as a function of two orthogonal directions, x and y, or rows and columns, for the points labeled in Figure 9. We modeled the case where the diameter of the first zero in the ideal diffraction pattern was equal to 2 pixels (in λF/# units) with the rectangular obscuration of Figure 4a. We observed that the EOD value increased with increasing detector pixel size. EOD is the value of the prf at the pixel center. Concurrently, the shape of the prf(x,y) became modified with increasing detector size from that with explicit support at the base to that with a thin base and without features there.

**Figure 11.**The OCE as a function of pixel size in [λF/#] for an optical system with and without a central rectangular obscuration. A sharp increase in the OCE value toward the first peak at about 2.05 pixels is followed by a sharp dip at about 4 pixels only for the case of the circular aperture with a central obscuration. The OCE curve for a circular aperture without a central obscuration increases in a monotonical fashion. CA denotes a circular aperture.

**Table 1.**The EOD, the OCE, and their product for small and medium pixel sizes and a circular aperture with a rectangular obscuration.

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**MDPI and ACS Style**

Strojnik, M.; Bravo-Medina, B.; Martin, R.; Wang, Y.
Ensquared Energy and Optical Centroid Efficiency in Optical Sensors: Part 1, Theory. *Photonics* **2023**, *10*, 254.
https://doi.org/10.3390/photonics10030254

**AMA Style**

Strojnik M, Bravo-Medina B, Martin R, Wang Y.
Ensquared Energy and Optical Centroid Efficiency in Optical Sensors: Part 1, Theory. *Photonics*. 2023; 10(3):254.
https://doi.org/10.3390/photonics10030254

**Chicago/Turabian Style**

Strojnik, Marija, Beethoven Bravo-Medina, Robert Martin, and Yaujen Wang.
2023. "Ensquared Energy and Optical Centroid Efficiency in Optical Sensors: Part 1, Theory" *Photonics* 10, no. 3: 254.
https://doi.org/10.3390/photonics10030254