The plano-convex lens is probably the most widely used type of lens in laboratories, whether for processing or testing assembly, which is the reason for its popularity. During processing, there is a flat surface that can serve as a processing reference; during testing assembly, there is a flat surface that can transfer the angular reference to the structural reference. In fact, this is also considered as much as possible in optical design, which is why plano-convex lenses are one of the most common lens forms on the shelf.

This post mainly explains the use of plano-convex lenses, specifically why the convex surface faces parallel light. The principle is relatively simple; roughly speaking, if the flat surface faces the laser, there will be back-reflected light that interferes with the laser, reducing its lifespan. From the perspective of optical theory, this can be explained in terms of aberration: when light is incident on a flat surface, the system's spherical aberration will be greater than when light is incident on the convex surface, resulting in a larger spot dispersion.

As shown in the figure above, a plano-convex lens with a diameter of 25mm and a focal length of 100mm is established, and it is inserted into parallel light for convergence. It can be seen that when the convex surface faces the parallel light, the geometric size of the converged spot is 54.732um, while when the flat surface faces the parallel light, the geometric size of the converged spot reaches 301.769um, which is nearly 6 times larger.

For a field of view only along the axis, the cause of this is naturally spherical aberration, so next we will try to find some explanations from the perspective of spherical aberration!

Our rough definition of aberration is the difference between the actual imaging and the Gaussian image, which is introduced by the optical system's aberration. Of course, there is another reason: the optical system itself is a diffraction-limited system, and the aperture cannot be infinitely large.

The cornerstone of geometric optics is Snell's law of refraction, so the difference between actual imaging and Gaussian imaging mainly lies in the approximation of angles in the formula. In the near-axis region, or in Gaussian calculations, the sine and tangent of the angle are equal to the angle itself.

This actually only holds true for small angles; as the angle increases, there will be a significant difference. This is also the reason for the introduction of geometric aberration, because in Gaussian imaging, Snell's law is linear, while in reality, it is sine-related. As can be seen from the figure below, when the angle is less than 0.5rad (about 28°), the two are still relatively close, but the larger the angle, the greater the difference!

So from the approximation of Snell's law to Gaussian imaging, it means that the smaller the incident or exit angle, the closer it is to Gaussian imaging; when the incident angle exceeds about 28°, the difference between Snell's law and Gaussian imaging becomes too large! Therefore, we will analyze the above plano-convex lens based on this 28 degrees, looking at the incident and exit angles of each surface.

The blue convergence point in the figure is the center of the sphere on the right side, and the blue converging light rays are in the direction of the sphere's normal (with slight differences). The angle between the parallel light and this converging light ray is the incident angle of this light ray on the sphere, and the complementary angle between the green light ray and this blue converging light ray is the exit angle. It can be seen that the angle on the axis is 0 degrees, and as the aperture increases, the incident angle of the light on the sphere increases, and the exit angle also increases. As the aperture increases, due to the non-linear relationship between the incident and exit angles, the light cannot converge to a single point after passing through the lens. In other words, as the aperture increases, the focal length is changing, which is spherical aberration. In ZEMAX, the angle at py=1 can be directly read, which is also the maximum incident and exit angle. Below are the incident and exit angles at each surface at PY=1 picked up using the operation numbers RAID and READ.
 

The above figure picked up eight angles, with incident angles all less than the aforementioned 28°, while the exit angle, when the flat surface is aligned with the parallel light path, has an angle of 37.59°, which is greater than 28°. Of course, when the system's focal length is extended, this 37.59° will decrease to less than 28 degrees. At this time, the system's F-number increases, the system's aberration decreases, and the spot improves. However, compared to the light path where the convex surface is aligned with the parallel light, the aberration is still poor. Below is the data when the focal length is 200mm (the above data is for a focal length of 60mm, with an effective aperture of 25mm), where the 8 angles picked up show that the original 37.59° is now 10.55°, still the largest among these 8 angles.

Both the incident and exit angles need to be considered, with a total of eight angles from both surfaces. The larger the angle, the greater the deviation of the actual refraction law from Gaussian imaging. Therefore, when analyzing, we only need to look at the largest angle. From this angle, we can conclude that parallel light should enter through the convex side and converge on the flat side.