Molded optics : design and manufacture
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With a first-order optical model defined, we can compute the image location and size of any input object. Distances are measured from the focal points, moving to the left being negative and moving to the right positive. The ratio of the image height to the object height is the magnification of the system. This would indicate that the image is located exactly at the focal point of the system. In reality, the object is at some finite distance, so the image would have some finite height and be located slightly off the focal point.
The location of the cardinal points for an optical system can be calculated using knowledge of the physical parameters of the elements contained in it. The values needed for the calculation are the radii of curvature of the optical surfaces, their locations to one another, and the refractive index of the optical materials surrounding them. We do not discuss the details of this calculation here, but refer the reader to several references.
However, if we intend to design an actual system, we need to consider whether a given ray makes it through the system or not. This leads us to the concept of pupils and stops. There are two kinds of stops, the aperture stop and the field stop, as well as two pupils, the entrance pupil and exit pupil, in each optical system. The stops, as their names imply, limit the size of the aperture and field angle of the system. In every optical system, there is some aperture that limits the size of the on-axis beam that can pass through the system.
This aperture may be the diameter of a lens element, the edge of a flange that a lens is mounted on, or an aperture placed in the system such as the iris in a digital camera to intentionally limit the beam size. The field stop sets a limit on the field of view of the system. That is, it sets an upper limit on the angular input of beams that form the captured image. In many cases, the field stop is the image capture device of the system itself. In digital cameras, where the image is captured by a detector, any light that falls outside the edges of the detector is not intentionally collected.
Thus, the detector is limiting how big a field is seen by the system, making it the field stop. The term field stop is also used to describe slightly oversized apertures placed at or near intermediate images within the system. While not strictly field stops by the definition above, since they do not actually limit the field of view, these apertures can help to control stray light.
The pupils can be considered the windows into and out of the system. The pupils are simply the images of the aperture stop, viewed through all the optics between the viewer and the stop itself. Looking into the back of a camera lens, we can see an effective aperture from which all the light appears to come. This effective aperture is known as the exit pupil.
Since all beams pass through the aperture stop, all beams appear to pass through its image, the exit pupil. Similarly, looking into the front of the camera lens, we see the entrance pupil, through which all beams appear to enter the system. If the aperture stop is in front of all the optical elements, the entrance pupil is located at the same position as the aperture stop since there are no elements for the stop to be imaged through. Similarly, if the aperture stop is behind all the optical elements, the exit pupil and aperture stop are coincident.
This ratio is important in that it relates to the amount of light captured by the system, through the solid angle, as will be discussed in Chapter 3. By image quality, we mean how well the image represents the true characteristics of the object, such as its fine detail. To predict the image quality provided by an optical system, we turn to ray tracing. This law relates the direction of a ray after an optical surface to the direction of the ray before it through the ratio of the refractive indices of the materials on each side of the surface.
It should be noted that the incident ray, the refracted ray, and the surface normal all lie in a common plane. In this special case, we see that the magnitude of angle of incidence and refraction actually reflection are equal to each other. The reflected ray lies on the opposite side of the normal to the surface, at the same angle as the incident ray.
As above, the incident ray, reflected refracted ray, and surface normal all lie in the same plane. We previously stated that materials with higher refractive index bend light more than materials with lower refractive index. Therefore, we see that the higher-index material bends the ray more from its original direction than the lower-index material does.
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The input ray, parallel to the axis, strikes the lens at a height of 1. At this location, the ray makes an angle of Using the refracted angle, the direction of the normal, and the distance to the planar surface, we determine that the ray strikes the rear surface of the lens at a height of 1. Solving, we determine a ray height of zero occurs at a distance of 4.
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We have now traced a single ray through our one-element optical system. Ray tracing calculations were previously performed manually, using log tables or basic calculators, depending on the era. Needless to say, tracing even a single ray through a multielement system took considerable effort. With modern computers, thousands of rays can be traced per second, greatly simplifying the calculations required of the optical designer. If we were to trace multiple rays in this input beam, entering the lens at varying heights, we would find that all the rays do not pass through the same axial point, as would be predicted by first-order optics.
This is due to the presence of aberrations, which are not included in our first-order model and are the subject of the next section. We can consider aberrations to fall into one of two general categories: those that result in a point not being imaged to a point, and those that result in a point being imaged to a point in the incorrect location. Examples of the first type are spherical aberration, coma, and astigmatism, while examples of the second type are Petzval curvature and distortion.
The amount of each aberration also varies as a function of the entrance pupil diameter or the field angle of the system. We now briefly discuss each of the aberrations mentioned, along with methods of controlling them. The rays striking the outer portion of the lens cross the axis closer to the rear surface than rays striking the lens near its center.
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For this lens, the rays near the edge of the lens are refracted more sharply than is needed for them to come to a focus position coincident with the rays in the center, a condition referred to as undercorrected spherical aberration. The amount of spherical aberration depends upon the third power of the pupil diameter, but is independent of the field angle.
Thus, operating a system at twice its original pupil diameter will result in an eightfold increase in spherical aberration, with the aberration being a constant value over the field of view. Conversely, reducing the aperture stop size and thus the pupil size , known as stopping down the system, will reduce the spherical aberration. Figure 1. By increasing the radius of the front surface, the angles of incidence of the rays are decreased, reducing the spherical aberration contribution of the surface.
Adding power to the rear surface maintains the focal length of the element, but increases the angles of incidence on it and its spherical aberration contribution. However, the magnitude of the increase at the rear surface is not as great as the decrease in the front surface contribution, resulting in an overall reduction in the spherical aberration of the element. Splitting the lens allows longer radii of curvature for each of the surfaces, reducing the angles of incidence of the rays, and decreasing the associated surface spherical aberration contributions.