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The refracting or reflecting surfaces in an optical system are not infinite in size but limited, generally, to a round shape. This finite transverse extension limits the beam of light passing through them. Let us consider a centered optical system. If the light beam entering this system comes from a point object on the optical axis, very likely only one of the surfaces will limit the transverse extension of the beam, as shown in Figure 1.22. This limiting surface is called the stop of the system. If the stop is a diaphragm, we may think of it as a dummy refracting surface whose refractive indi-ces are the same before and after the surface (diaphragm). The system stop may be at any surface. It need not be in the middle or at one end of the system; some optical

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Virtual object

Virtual image

FIGURE 1.21 Formation of a virtual image with a virtual object.

surfaces are located before the stop and some others after it. If the stop is observed from the entrance of the system, it will be observed through the surfaces that precede it, changing its apparent size and position. This observed image of the stop is called the entrance pupil. If the stop is observed from the back of the system, it will be observed through the surfaces that are after it, changing again its apparent size and position. This observed image of the stop is called the exit pupil.

As shown in Figure 1.22, of all meridional rays going from a point off-axis on the object plane, to the point on the image plane, only one passes through the center of the stop. This ray is the principal ray, defined as the ray that passes through the off-axis point object and the center of the stop. The intersection of the extension of the segment of the principal ray in the object space with the optical axis is the center of the entrance pupil. Similarly, the intersection of the extension of the segment of the principal ray in the image space with the optical axis is the center of the exit pupil.

An image of the stop can also be observed from any medium in the optical sys-tem, not only from the object or image media. As shown in Figure 1.23, the real (or virtual) image of the stop is located at the point where the principal ray (or its exten-sion) crosses the optical axis. This image of the stop is the pupil of that surface or medium.

All quantities referring to the principal ray are represented with a bar on top of the symbol; for example, y is the paraxial height of the principal ray and ū is its paraxial angle with respect to the optical axis. By definition, the value of y is equal to zero at the stop. All quantities referring to the axial rays (meridional rays from a point object on the axis) are written without the bar.

The meridional ray heights at the pupil for the medium j are represented by Y_pj for the marginal rays or y_pj for the paraxial rays. The meridional ray heights at the entrance and exit pupils are represented by Yentr and Yexit for the marginal rays and yentr and yexit for the paraxial rays.

Exit pupil

position Principal

ray

Entrance pupil position

Stop Field

stop

FIGURE 1.22 Definitions of principal ray, entrance pupil, and exit pupil.

24 Handbook of Optical Design

Summarizing, the stop is the aperture that limits the amount of light entering the optical system and its images are the pupils. The field stop, on the other hand, is located on the image plane and limits the image lateral extension, as shown in Fig-ure 1.22.

If the light beam entering the system comes from an off-axis object point, as shown in Figure 1.24, several surfaces may limit the transverse extension of the beam, producing an apparent aperture with a nearly elliptical shape. Then, the sys-tem is said to have vignetting. The vignetting effect appears only when the angle of incidence of the beam exceeds a certain limit. It is frequently desirable to avoid vignetting in a centered optical system, as shown in Figure 1.25, to avoid excessive decreasing of the illuminance of the image at the edge of the field and to have a better control of the image analysis during the design stage. Sometimes, however, vignett-ing is introduced on purpose, to eliminate some difficult to correct aberrations.

The tangential and sagittal planes, defined previously in Section 1.3, may now be more formally defined. The tangential plane is a meridional plane that contains the principal ray (also off-axis point object). The sagittal plane is a plane perpendicular

Principal ray

Principal ray

Principal ray extension

Pupil for

medium j Pupil for medium j−1

Surface j Surface j+1

Medium j−1 Medium j Medium j+1

Y_j Y_j+1

FIGURE 1.23 Location of the pupil of a surface in an optical system.

apertureFree

Optical axis

Lens 1 Lens 2 Lens 3

Diaphragms

FIGURE 1.24 Vignetting in a lens.

to the tangential plane, which contains the principal ray. As we may notice, there is a single common tangential plane for all media between two consecutive optical surfaces in a centered optical system. However, there is a sagittal plane for each medium, because the principal ray is refracted at each surface.

In order to trace the principal ray through an optical system, we must know its direction in the object medium. This direction must be such that the principal ray passes through the center of the stop.

Let us consider a small circle centered on the entrance pupil whose diameter is very small compared with the distance from an off-axis point light source to the optical system. Then, all light rays from the point light source that enter the pupil through this circle will be nearly parallel and close to the principal ray. These rays are real (non-paraxial) and we will refer to them as parabasal rays (from the Greek, near the fundamental or principal).

1.6.1 telecentric systems

A frontal telecentric system is one that has its entrance pupil placed at infinity.

Since the stop (diaphragm) is at the back focal plane, the object must be at a finite distance to avoid forming the image on the focal plane. Let us consider the optical system in Figure 1.26a where the principal ray is parallel to the optical axis, since the entrance pupil is at infinity. A small defocusing by a small change in the distance from the object to the system does not introduce any change in the magnification of the image. This property makes these systems useful for measuring systems where a small defocusing does not introduce any errors.

A rear telecentric system has its exit pupil at infinity as in Figure 1.26b. The stop is at the front focal plane. The object may be at any distance from the system. In these systems, a small defocusing by changing the distance from the optical system to the observing screen does not change the image size.

An optical system may be simultaneously frontal and rear telecentric, with both the object and the image at finite distances from the system. In this case, the stop is in the middle of the system, at the back focal plane of the part of the system preced-ing the stop and at the front focal plane of the part of the system after the stop, as in Figure 1.26c.

assemblyLens Free aperture

Lens 1 Lens 2 Lens 3

Optical axis

Diaphragms

FIGURE 1.25 Stop size to avoid vignetting in a lens for a given off-axis angle.

26 Handbook of Optical Design