Читать книгу Optical Engineering Science - Stephen Rolt - Страница 86

Long before the advent of powerful computer ray tracing models, there was a powerful incentive to develop simple rules of thumb to guide the optical design process. This was particularly true for the complex task of ameliorating system aberrations. Working in the nineteenth century, Ernst Abbe set out the Abbe sine condition, which directly relates the object and image space numerical apertures for a ‘perfect’, unaberrated system. Essentially, the Abbe sine condition articulates a specific requirement for a system to be free of spherical aberration and coma, i.e. aplanatic. The Abbe sine condition is expressed for an infinitesimal object and image height and its justification is illustrated in Figure 4.18.

In the representation in Figure 4.18 we trace a ray from the object to a point, P, located on a reference sphere whose centre lies on axis at the axial position of the object and whose vertex lies at the entrance pupil. At the same time, we also trace a marginal ray from the object location to the entrance pupil. The conjugate point to P, designated, P′, is located nominally at the exit pupil and on a sphere whose centre lies at the paraxial image location. For there to be perfect imaging, then the OPD associated with the passage of the marginal ray must be zero. Furthermore, the OPD of the ray from object to image must also be zero. It is also further assumed that the relative OPD of the object to image ray when compared to the marginal ray is zero on passage from points P to P′. This assumption is justified for an infinitesimal object height. Therefore, it is possible to compute the total object to image OPD by simply summing the path differences relative to the marginal ray between the object and point P and between the image and point P′. For there to be perfect imaging this difference must, of course be zero.

Figure 4.18 Abbe sine condition.

(4.46)

n is the refractive index in object space and n′ is the refractive index in image space.

Equation 4.46 is one formulation of the Abbe sine condition which, nominally, applies for all values of θ and θ′, including paraxial angles. If we represent the relevant paraxial angles in object and image space as θ_p and θ_p' then the Abbe sine condition may be rewritten as:

(4.47)

One specific scenario occurs where the object or image lies at the infinite conjugate. For example, one might imagine an object located on axis at the first focal point. In this case, the height of any ray within the collimated beam in image space is directly proportional to the numerical aperture associated with the input ray.

Figure 4.19 illustrates the application of the Abbe sine condition for a specific example. As highlighted previously, the sine condition effectively seeks out the aplanatic condition in an optical system. In this example, a meniscus lens is to be designed to fulfil the aplanatic condition. However, its conjugate parameter is adjusted around the ideal value and the spherical aberration and coma plotted as a function of the conjugate parameter. In addition, the departure from the Abbe sine condition is also plotted in the same way. All data is derived from detailed ray tracing and values thus derived are presented as relative values to fit reasonably into the graphical presentation. It is clear that elimination of spherical aberration and coma corresponds closely to the fulfilment of the Abbe sine condition.

The form of the Abbe sine condition set out in Eq. (4.46) is interesting. It may be compared directly to the Helmholtz equation which has a similar form. However, instead of a relationship based on the sine of the angle, the Helmholtz equation is defined by a relationship based on the tangent of the angle:

It is quite apparent that the two equations present something of a contradiction. The Helmholtz equation sets the condition for perfect imaging in an ideal system for all pairs of conjugates. However, the Abbe sine condition relates to aberration free imaging for a specific conjugate pair. This presents us with an important conclusion. It is clear that aberration free imaging for a specific conjugate (Abbe) fundamentally denies the possibility for perfect imaging across all conjugates (Helmholtz). Therefore, an optical system can only be designed to deliver aberration free imaging for one specific conjugate pair.

Figure 4.19 Fulfilment of Abbe sine condition for aplanatic meniscus lens.