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==== Entrance Pupil and Its Applications in Camera Imaging Testing ====

When we observe the human eye, we can see a small black hole (i.e., the pupil), through which all incident light enters the eye. Physiologically, the pupil is the opening in the center of the iris, but what we observe is not the pupil itself, but rather the image of the pupil formed by the refractive structures in front of it (such as the cornea and anterior chamber). Similarly, if we look at a camera lens from the front in a bright environment, we will see a bright spot. This spot is the image of the lens aperture (usually placed between the lens elements) formed by the elements in front of it, and is called the entrance pupil. In camera imaging testing, the entrance pupil is a very important concept with a wide range of applications. In this article, we will explore the entrance pupil together.\\
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^   Figure 1. Observing the human eye, one can see the image of the iris and pupil formed by the refractive structures in front of them. (Image source: https://commons.wikimedia.org/wiki/File:Hazel_Eye_HD.JPG)   ^
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^  Figure 2. The entrance pupil of a lens, which is the image of the aperture formed by the lens elements in front of it. (Image source: https://en.wikipedia.org/wiki/Entrance_pupil#/media/File:Apertures.jpg)  ^

**Concept of the Entrance Pupil**\\

The prototype of the modern camera is the pinhole camera, where the pinhole is the common entrance for all imaging beams, known as the aperture stop in geometrical optics. Therefore, for a luminous object point, the size of the aperture stop limits the solid angle of the imaging beam, and thus limits the luminous flux. One of the fundamental differences between modern cameras and pinhole cameras is the size of the aperture stop; only a sufficiently large aperture stop makes a camera practically useful. The position of the aperture stop in a lens is determined by the optical designer after comprehensively considering the lens structure, off-axis imaging quality, and system dimensions, and is generally located inside the lens. **In photographic camera lenses, the aperture stop is usually an adjustable aperture (referred to as the diaphragm or aperture in photographic terminology) used to control the luminous flux**.\\
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^  Figure 3. The aperture stop and the entrance pupil are in an object-image conjugate relationship, and both serve as the common entrance for all imaging rays.  ^

To determine the restriction of the imaging beam by the aperture stop, ray tracing is required, which demands extensive data and massive calculations. So, is there a more intuitive and simple way to characterize the restriction of the beam by the aperture stop? The answer is the entrance pupil. **The entrance pupil is the image of the aperture stop formed by the optical system between it and the object**, as shown in Figure 3. The aperture stop "seen" by a luminous object point is actually not the aperture stop itself, but its image. According to the principle of reversibility of light, the restriction of the incident beam by the aperture stop can be regarded as being imposed by the entrance pupil. For photographic lenses, the entrance pupil is generally a virtual image of the aperture stop, and is therefore erect and magnified. Taking Figure 3 as an example, the images of the upper and lower edges of the aperture stop formed by the lens are located at the upper and lower edges of the entrance pupil, respectively. In other words, a ray emitted from an object point and directed towards the edge of the entrance pupil will actually pass through the edge of the aperture stop. Given the position and size of the entrance pupil, the solid angle of the imaging beam and the restriction of the beam by the aperture stop can be determined without knowing the actual structure of the lens, let alone performing ray tracing and calculations.\\
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^  Figure 4. A ray emitted from an object point and directed towards the center of the entrance pupil is called the chief ray. By definition of the entrance pupil, the chief ray also passes through the center of the aperture stop.  ^

**Imagine shrinking the aperture stop to a pinhole until only rays passing through the center of the aperture stop can pass through the system to form an image; such rays are called chief rays**. The chief ray defines the propagation direction of the beam emitted from an object point during the imaging process. In addition, the intersection points of the chief ray with various optical surfaces are closely related to aberrations, which is of great significance for aberration correction. Ideally, the chief rays emitted from different object points converge at the center of the aperture stop; therefore, the chief rays (or their extensions) must also converge at the center of the entrance pupil. Corresponding to the chief ray, **the rays passing through the edge of the aperture stop (entrance pupil) are called marginal rays.**\\
The importance of the concept of the entrance pupil determines its numerous applications. Next, let us look at two common applications of the entrance pupil.\\
(It should be noted that, for ease of understanding, the concept of the entrance pupil defined above is based on the assumption of an ideal system, where all rays are paraxial and the entrance pupil is free of distortion.)\\
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^  Figure 5. Let AB be a surface element on a planar Lambertian surface. The illuminance of the inverted image A'B' of AB at the image plane is inversely proportional to the square of the f-number.^

**Application A: Definition of f-number**\\
For an imaging lens, **the f-number is defined as the ratio of the focal length to the diameter of the entrance pupil, i.e., the reciprocal of the relative aperture.** The f-number is closely related to the imaging characteristics of the lens. Here are a few examples.\\
**First, the square of the f-number is inversely proportional to the image plane illuminance.** If the object is an ideal Lambertian surface, the area of the entrance pupil is proportional to the luminous flux, while the area of the image is proportional to the square of the focal length. Obviously, as the ratio of flux to area, the illuminance is inversely proportional to the square of the f-number. Therefore, the brightness of the image we observe depends not only on the size of the entrance pupil but also on the focal length.\\
**Second, the f-number is proportional to the size of the Airy disk.** When the object distance is infinity, the size of the central bright spot (Airy disk) of the far-field diffraction pattern (Airy pattern) formed by a point object through the circular aperture stop of the lens is proportional to the f-number. Since the size of the diffraction pattern is closely related to the spatial frequency response and the cutoff spatial frequency, the f-number is tightly linked to image quality.\\
**Third, the f-number is negatively correlated with the aperture angle.** The aperture angle (the angle between the marginal rays) increases as the entrance pupil increases and as the focal length decreases, both of which correspond to a decrease in the f-number. Therefore, the aperture angle increases as the f-number decreases. Some imaging devices have requirements for the aperture angle. For example, the dichroic prism built into color television cameras, which consists of prisms, dichroic coatings, and thin air gaps, imposes a limit on the minimum f-number of the imaging lens. As another example, microlenses are often placed above the pixels of an image sensor, and the shape and position of these microlenses actually impose certain requirements on the aperture angle and f-number of the imaging beam.\\
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^  Figure 6. The angle of view is the angle subtended by the entrance window at the center of the entrance pupil.  ^

**Application B: Definition of Angle of View and Field of View**\\
**The angle of view is a very important metric, which characterizes the maximum extent of the scene that a camera can "see".** For photographic cameras, this extent is determined by the size of the image sensor. Imagine if we block the edges of the image sensor to reduce the area of the photosensitive region; the extent of the scene the camera can see would also decrease accordingly. In geometrical optics, the aperture that restricts the field of view is called the field stop, and **the image of the field stop formed by the optical system between it and the object is called the entrance window.** As shown in Figure 6, the field stop is the image sensor itself. Therefore, at the image plane, according to the object-image conjugate relationship, the entrance window is located at the object plane and determines the extent of the scene the camera can "see", which is called the field of view. The **angle subtended by the entrance window at the center of the entrance pupil is called the angle of view**. One of the traditional methods for measuring the angle of view of a camera is to use a target covered with a grid or dots, where the target is large enough to cover the camera's field of view. The physical size of the scene the camera can see can then be determined based on the extent of the target in the image (the number of grids or dots). If the distance between the camera under test and the target is also known, the angle of view can be calculated using the arctangent of the ratio of the two. It should be noted that the distance d between the camera under test and the target should be measured from the plane where the entrance pupil is located; otherwise, unnecessary errors will be introduced into the calculation results.\\
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^  Figure 7. When testing the performance at different positions on the imaging plane of a camera using collimators, the intersection of the optical axes of multiple collimators, or the center of rotation of a single collimator (or the camera), should be located at the center of the entrance pupil of the lens of the camera under test.  ^\\
In the testing of automotive cameras, it is common to encounter fixed-flange-distance prime lenses with very long design object distances, as well as cameras equipped with ultra-wide-angle (fisheye) lenses. When testing the performance of such cameras (such as spatial frequency response, distortion, angle of view, chromatic aberration, flare, etc.), collimators are typically used to form virtual images of test targets at the design object distance and at given positions in the field of view. There are two common solutions on the market: one is to use multiple collimators to simultaneously form virtual images of the target at different positions in the field of view, in which case the optical axes of all collimator lenses should converge at the center of the entrance pupil of the lens of the camera under test; the other is to use only a single collimator and rotate either the collimator or the camera under test around the center of the entrance pupil of the lens of the camera under test.\\
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^  Figure 8. For a target at a finite distance, if the center of rotation of the collimator or the camera under test is not in the same plane as the entrance pupil of the lens of the camera under test, the measured angle of view will be larger than the true value (indicated by the green dashed line) when the center of rotation is in front of the plane of the entrance pupil, and smaller when it is behind.  ^\\

If the aforementioned center of rotation is not the center of the entrance pupil, but is located in front of or behind it, two different results will occur when measuring the angle of view. When the virtual image of the target is at an infinite distance from the camera (i.e., the wave emitted by the collimator is a plane wave), as long as the light emitted by the collimator can enter the entrance pupil, it will theoretically have no effect on the measurement results. However, when the virtual image of the target is at a finite distance from the camera (i.e., the wave emitted by the collimator is a diverging spherical wave), the maximum rotation angle of the target or the camera under test is not equal to the camera's angle of view. Specifically, when the center of rotation is located in front of the entrance pupil of the lens of the camera under test, the measured angle of view will be larger than the true value (indicated by the green dashed line); conversely, when the center of rotation is located behind the entrance pupil of the lens of the camera under test, the measured angle of view will be smaller, as shown in Figure 8. Taking the [[https://rdbuy.com/product/productdetail?model_id=10443|Yanding RFT Camera Comprehensive Tester]] as an example, users can adjust the position of the camera under test according to the entrance pupil position given in the specification sheet of the lens of the camera under test, ensuring that the center of rotation of the collimator is located at the center of the entrance pupil of the lens of the camera under test, as shown in Figure 9.\\
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Figure 9. The [[https://rdbuy.com/product/productdetail?model_id=10443|Yanding RFT Camera Comprehensive Tester]] reserves adjustment space for differences in the entrance pupil positions of different cameras under test.