Basic Geometry of an Optical System

The model of a pinhole camera is an oversimplifi cation of an imaging system. A pinhole camera forms an image of an object at any distance while a real optical system forms a sharp image only within a certain dis- tance range. Fortunately, the geometry even for complex optical systems can still be modeled with a small modifi cation of perspective projection

186                                                                                               7 Image Formation

 

 

x

 

Figure 7.7: Optical imaging with an optical system modeled by its principal points P₁ and P₂ and focal points F₁ and F₂. The system forms an image that is a distance d' behind F₂ from an object that is the distance d in front of F₁.

 

as illustrated in Figs. 7.6 and 7.7. The focal plane has to be replaced by two principal planes. The two principal planes meet the optical axis at the principal points. A ray directed towards the fi rst principal point appears — after passing through the system — to originate from the second principal point without angular deviation (Fig. 7.6). The distance between the two principal planes thus models the axial extension of the optical system.

As illustrated in Fig. 7.6, rays between the two principal planes are always parallel and parallel rays entering the optical system from left and right meet at the second and fi rst focal point, respectively. For practical purposes, the following defi nitions also are useful: The eff ective focal length is the distance from the principal point to the corresponding focal point. The front focal length and back focal length are the distances from the fi rst and last surface of the optical system to the fi rst and second focal point, respectively.

The relation between the object distance and the image distance be- comes very simple if they are measured from the focal points (Fig. 7.7),

 

dd' = f 2.                                                  (7.19)

This is the Newtonian form of the image equation. The possibly better known Gaussian form uses the distances as to the principal points:

1           1       1

d' + f + d + f = f                                            (7.20)

7.4.2 Lateral and Axial Magnifi cation

The lateral magnifi cation m_l of an optical system is given by the ratio of the image size, x, to the object size, X:

ml = x₁     f   d'

X₁ = d =

f.                                      (7.21)

7.4 Real Imaging                                                                           187

A less well-known quantity is the axial magnifi cation that relates the positions of the image plane and object plane to each other. Thus the axial magnifi cation gives the magnifi cation along the optical axis. If we shift a point in the object space along the optical axis, how large is the shift of the image plane? In contrast to the lateral magnifi cation, the axial magnifi cation is not constant with the position along the optical axis. Therefore the axial magnifi cation is only defi ned in the limit of small changes. We use slightly modifi ed object and image positions d + ∆ X₃ and d'− ∆ x₃ and introduce them into Eq. (7.19). Then a fi rst-order Taylor expansion in ∆ X₃ and ∆ x₃ (assuming that ∆ X₃, d and ∆ x₃, d') yields

∆ x₃

∆ X₃ ≈

d'                                                (7.22)

d

and the axial magnifi cation m_a is given by

m    d'   f 2       d'2              2

a ≈ d = d2 = f 2 = ml.                                        (7.23)

 

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