Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo

For imaging to occur, the rays starting at a point A must converge to a point B regardless of angle. The total ray-transfer matrix is given by

(b) When imaging occurs, we have

Hence, the magnification A is given by

A¼ y2 ¼ 1 d2 : y1 f

EXAMPLE 8.12 Two planes in an optical system are referred to as conjugate planes if the intensity distribution on one plane is the image of the intensity distribution on the other plane (imaging means the two distributions are the same except for magnification or demagnification). (a) What is the ray-transfer matrix between two conjugate planes? (b) How are the angles y1 and y2 related on two conjugate planes?

Solution: (a) In the previous problem, the ray transfer matrix M was calculated in general. Under the imaging condition, M becomes

where my and my equal ð1 d2=f Þ, ð1 d1=f Þ, respectively.

Rear Focal Plane

Figure 8.19. The rear focal point and the rear focal plane of a negative lens.

Front Focal

Point

Front Focal Plane

Figure 8.20. The front focal point and the front focal plane of a negative lens.

EXAMPLE 8.14 The focal points and focal planes of a negative lens can be similarly defined. This is shown in Figures 8.19 and 8.20.

Note that the front and rear focal points and focal planes are the reverse of the front and rear focal points and focal planes of a positive lens. The ray-transfer matrix between the two planes can be shown to be also given by Eq. (8.6-9).

8.7THICK LENSES

A thick lens can be analyzed just like a thin lens by defining two principal planes. Consider the front focal point of a thick lens as shown in Figure 8.21. By definition, the rays emerging from this point exit the lens as a parallel bundle. An incident ray on the lens intersect with the existing ray from the lens projected backwards at a

Figure 8.21. (a) P1: first principal plane, (b) P2: second principal plane.

certain point. The plane passing through this point and perpendicular to the optical axis is called the first principal plane.

By definition, a parallel bundle of rays passing through a lens converge to the second focal point of the lens. An incident ray on the lens projected forward into the lens intersect the ray emerging towards the second focal point of the lens at a certain point. The plane passing through this point and perpendicular to the optical axis is called the second principle plane.

The ray-transfer matrix between the two principle planes of a thick lens can be shown to be the same as the regular ray-transfer matrix, namely,

Consequently, the focal points, the focal length, and the imaging distances are as shown in Figure 8.22. The focal length is defined as the distance of a principle plane from the corresponding focal point. The two focal lengths are the same.

Figure 8.22. Object, image, principal and focal planes, focal lengths, and object and image distances in a thick lens.

132 GEOMETRICAL OPTICS

EXAMPLE 8.15 Show that the distances d1 and d2 between two conjugate planes satisfy the lens law.

Solution: The ray-transfer matrix between the two planes is given by

This is the same ray-transfer matrix discussed in Example 8.11. Hence, the imaging condition is given by

1 ¼ 1 þ 1 f d1 d2

8.8 ENTRANCE AND EXIT PUPILS OF AN OPTICAL SYSTEM

An optical system typically contains a number of physical apertures. One of these apertures causes the most severe limitation for the system. It will be called the effective physical aperture.

Entrance and

Exit Pupils

(a)

Entrance Pupil

Exit Pupil

(b)

Exit Pupil

(c)

Figure 8.23. (a) Entrance pupil ¼ exit pupil, (b) Exit pupil ¼ physical pupil, (c) Entrance pupil ¼ exit pupil.

The entrance pupil of the system is the image of the effective physical aperture when looking from the object space. The exit pupil of the system is the image of the effective physical aperture when looking from the image space.

In order to clarify these concepts, consider Figure 8.23. In part (a), the effective physical aperture is on the plane of the lens. In this case, the entrance and exit pupils coincide, and are the same as the effective physical aperture. In part (b), the effective physical aperture is the same as the exit pupil. Its image seen from the object side is a virtual image, which is the entrance pupil. In part (c), the effective physical aperture is the same as the entrance pupil. Its image seen from the image space is a virtual image, which is the exit pupil.

In considering diffraction effects, it is the exit pupil that effectively limits the wave field passing through the optical system. As far as diffraction is concerned, the lens system has the exit pupil as its aperture, and Fraunhofer diffraction can be used to study diffraction effects on the image plane. For this purpose, the distance on the optical axis is the distance from the exit pupil to the image plane.

Figure 9.1. The three parts of a thin lens.

negative radius of curvature. A thin lens has the property that a ray enters and exits the lens at about the same (x,y) coordinates.

The three parameters discussed about can be combined in a single parameter called f, the focal length, by

f is positive or negative, depending on R1 and R2. For example, a double-convex lens has positive f whereas a double-concave lens has negative f. In the following discussion, f will be assumed to be positive.

The phase transformation of a perfect lens can be written as

where t0 is the maximum thickness of the lens. Since ejkt0 gives a constant phase change, it can be neglected. Equation (9.2-2) will be derived below. The finite extent of the lens aperture can be taken into account by defining a pupil function P(x,y):

In order to drive Eq. (9.2-2), the lens is considered in three parts as shown in Figure 9.1. The thickness ðx; yÞ at a location (x,y) can be shown to be [Goodman, 2004]

where t0 is the total thickness of the lens. This expression is simplified by using paraxial approximation stated by