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 from different field points all meet at the center of the entrance pupil. In three-dimensional space, at any given object-space chief-ray angle, the chief rays in all meridional planes form the lateral surface of a right circular cone whose apex, located on the optical axis, is the center of the entrance pupil and whose half-angle is equal to the chief-ray angle. Each chief-ray angle corresponds to one such cone, and all of these cones share a common apex. The definitions of both the field and the angle of view are based on this apex, which is also the center of perspective of the lens—the point about which the camera can be rotated without introducing parallax, commonly called the no-parallax point.

For a fisheye lens, the chief rays in all meridional planes still form a right circular cone at any given chief-ray angle, and the apex still lies on the optical axis. The entrance pupil centers, however, no longer coincide with the apex (except at a chief-ray angle of 0°); instead, they form a circle around the optical axis. As the chief-ray angle changes, the apex moves along the optical axis, and the cones no longer share a common apex. Nevertheless, because the apex is still the point at which the extensions of the chief rays intersect, the angle of view can still be defined with respect to this apex, as shown in the left panel of Figure 5.

For an ideal lens, if an object point on the optical axis is rotated through a given angle about the entrance pupil center (the cone apex) within a meridional plane, the point lies on the chief ray corresponding to that angle, and that angle corresponds to a specific field position. For the fisheye lens shown in Figure 5, however, the cone apex is not unique: if an on-axis object point is rotated through about 67° about the apex corresponding to 0° (i.e., the entrance pupil center), the point actually lies on the object-space chief ray at about 73° (right panel of Figure 5). This discrepancy results from the movement of the entrance pupil center. In the testing of fisheye lenses and cameras, if the imaging performance at different field positions is to be measured, rotating the device under test or the test target about a single apex is not sufficient; the target or the device under test must also be moved along the optical axis so that the center of the target always lies on the chief ray of the intended field.

In addition, this leads to another concept—the no-parallax point. Because the entrance pupil center of an ideal lens is fixed, if two object points A and B lie in the same field but at different object distances (A in front of B), A occludes B. Moreover, as long as the lens or the object points A and B rotate about the entrance pupil center, this occlusion remains unchanged—that is, the relative positions of foreground and background remain unchanged (left panel of Figure 6). Under this condition, the entrance pupil center is called the no-parallax point.

For a fisheye lens, however, consider two object points A and B located in the 0° field, with A completely occluding B. When the lens or the two object points simultaneously rotate through a certain angle about the entrance pupil center corresponding to 0°, A and B come to lie on chief rays at different angles, so A no longer completely occludes B—that is, the relative positions of foreground and background have changed (right panel 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 essential to panoramic image stitching. If the camera rotates about the no-parallax point during shooting, the relative positions of foreground and background in the image remain unchanged, and image stitching is relatively simple. Otherwise, image stitching has to attempt to compensate for parallax (changes in the relative positions of 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; for a fisheye lens, however, this assumption generally does not hold.

At this point, readers should have a basic understanding of the entrance pupil in fisheye lenses. In the actual testing of fisheye cameras and lenses, if the device under test has no unique no-parallax point, or its position cannot be precisely located, the position with the minimum parallax can be found empirically and taken as the equivalent no-parallax point of the device under test, so as to minimize the impact of the movement of the entrance pupil center on imaging and measurement results. Given that the equivalent entrance pupil center or equivalent no-parallax point varies among cameras, the Yanding RFT series comprehensive tester allows the user to finely adjust the position of the camera under test.