The Entrance pupil article describes the entrance pupil under paraxial imaging, in which the entrance pupil and the aperture stop are assumed to form a conjugate object–image pair. This idealised, fixed-pupil concept applies to ideal lenses and to telephoto (long-focal-length, narrow-angle) lenses.

In fisheye (ultra-wide-angle) lenses, however, the position of the entrance pupil is not fixed; instead, it shifts with the viewing position, as though tracking the observer like an eye. This article examines the mechanism behind this behaviour and its implications for camera image-quality measurement.

Effect of Fisheye Lens Construction on the Entrance Pupil

Figure 2 shows an example of a fisheye lens construction. In front of the aperture stop are two large-diameter lens elements, both of which are negative meniscus lenses—convex on the front surface, concave on the rear, and thin at the center. Such a front group deflects the extremely oblique rays from the edge of the object field toward the optical axis, allowing them to pass through the aperture stop. This is the fundamental reason why fisheye lenses can achieve a wide field of view. However, such a design also challenges the concept of the entrance pupil.

The entrance pupil is the image of the aperture stop formed by the lens elements in front of it. Based on this definition, the entrance pupil of a fisheye lens can be located. Figure 3 shows the results of ray tracing performed in the meridional plane for a fisheye lens, with the object at infinity (so that the incident light is parallel). At object-space chief-ray angles of 0°, 45°, and 90°, three entrance pupils at different positions are obtained (indicated by the short red line segments in the figure). At a chief-ray angle of 0°, the entrance pupil conforms to the definition given above: its center lies on the optical axis, its plane is perpendicular to the optical axis, and it is located behind the second lens element. As the chief-ray angle increases, the entrance pupil tilts progressively, consistent with the behavior shown in Figure 1: like the pupil of an eye, it turns with the direction of observation. Only when the entrance pupil tilts together with the chief ray can incident light from the scene pass through the aperture stop; conversely, if the entrance pupil always remained perpendicular to the optical axis, its projected area would be zero for rays incident at 90°, and no light could pass through the aperture stop. Therefore, from the perspective of radiative transfer, the tilting of the entrance pupil is inevitable for a fisheye lens.

In addition to tilting, the center of the entrance pupil also moves forward (toward the front of the lens) as the object-space chief-ray angle increases. The blue dashed line in Figure 4 shows the trajectory of the entrance pupil center in the meridional plane. Because a fisheye lens is rotationally symmetric about the optical axis, the set of entrance pupil centers forms a surface of revolution generated by rotating this curve around the axis. This forward movement, too, is inevitable. The interior of a fisheye lens can be compared to a well, with the first lens element at its rim: an entrance pupil buried deep inside would yield a field of view no wider than the patch of sky seen by a frog at the bottom of a well. To extend the field of view beyond 180°, the entrance pupil must be raised close to the rim—or even above it, in front of the first lens element.

Entrance Pupil Center, Field of View, and No-Parallax Point
For an ideal lens, the chief rays of different fields converge at the center of the entrance pupil. In three-dimensional space, when the chief ray angle is constant, the chief rays in all meridional planes will form the lateral surface of a right circular cone. The apex of the cone is the center of the entrance pupil, located on the optical axis, and half of the cone angle is the chief ray angle. Different chief ray angles correspond to different cones, and all cones share the same apex. Obviously, the definitions of field and field of view angle are both based on this apex. For a fisheye lens, when the chief ray angle is constant, the chief rays in all meridional planes will still form a right circular cone, and the apex of the cone will still be located on the optical axis, but at this time, the center of the entrance pupil will form a circle and will no longer be located at the apex of the cone (except when the chief ray angle is 0°). When the chief ray angle changes, the apex of the corresponding cone will move accordingly, so all cones no longer share the same apex. However, since the apex of the cone is still the intersection of the extended lines of the chief rays, we can still use this apex to define the field of view, as shown in the left image of Figure 5.

For an ideal lens, assuming an object point located on the optical axis rotates by an angle around the entrance pupil center (cone apex) in any meridional plane, then this object point must lie on the chief ray of this angle, and this angle corresponds to the respective field position. However, for the fisheye lens shown in the figure, since the cone apex is not unique, assuming an on-axis object point rotates by about 67° around the cone apex at 0° (i.e., the entrance pupil center), it actually lies on the object chief ray of about 73° (as shown in the right image of Figure 5). This is the result of the movement of the entrance pupil center. In the actual testing of fisheye lenses and cameras, if the imaging performance at different field positions is to be measured, it should be noted that simply rotating the device under test or the target around a certain cone apex is not enough; it is also necessary to move the target or the device under test along the optical axis to ensure that the center of the target is always on the chief ray of the target field.

In addition, this leads to another interesting concept—the no-parallax point. Since the entrance pupil center of an ideal lens is fixed, if there are two object points A and B in the same field but at different object distances (A is in front of B), A will occlude B. Moreover, as long as the lens or the object points A/B rotate around the entrance pupil center, this occlusion will remain unchanged, or in other words, the relative positional relationship between the foreground and background remains unchanged, as shown in the left image of Figure 6. The entrance pupil center at this time is called the no-parallax point. However, for a fisheye lens, suppose there are two object points A and B located in the 0° field, and A completely occludes B. When the lens or the two object points rotate by a certain angle around the entrance pupil center of the 0° field simultaneously, A and B will be located on chief rays of different angles, so A no longer completely occludes B, or in other words, the relative positional relationship between the foreground and background changes, as shown in the right image of Figure 6. Therefore, the fisheye lens in the figure does not have a no-parallax point. The concept of the no-parallax point is very important for panoramic image stitching. If the camera rotates around the no-parallax point during shooting, the positional relationship between the foreground and background in the image remains unchanged, making image stitching relatively simple. Otherwise, image stitching must attempt to compensate for parallax (changes in the positional relationship between the foreground and background). In addition, for an ideal imaging lens, the entrance pupil center, the no-parallax point, and the origin of the camera coordinate system in geometric calibration coincide, but for a fisheye lens, this assumption usually does not hold.

Having written this far, everyone should now have a preliminary understanding of the concept of the entrance pupil in fisheye lenses. In the actual testing of fisheye cameras and lenses, if the device under test does not have a unique no-parallax point or its position cannot be precisely located, we can start from the phenomenon and try to find the position with the minimum parallax as the equivalent no-parallax point of the device under test, so as to minimize the impact of the position change of the entrance pupil center on imaging and measurement results. Considering that the positions of the equivalent entrance pupil center or equivalent no-parallax point vary for different cameras, when using the Yanding RFT series comprehensive tester, users can finely adjust the position of the camera under test.