Читать книгу Optical Engineering Science - Stephen Rolt - Страница 57

In the preceding section, we considered the impact of optical imperfections on the transverse aberration and the construction of ray fans. Unfortunately, this treatment, whilst providing a simple introduction, does not lead to a coherent, generalised description of aberration. At this point, we introduce the concept of optical path difference (OPD). For a perfect imaging system, with no aberration, if all rays converge onto the paraxial focus, then all ray paths must have the same optical path length from object to image. This is simply a statement of Fermat's principle. We now consider an aberrated system where we accurately (not relying on the paraxial approximation) trace all rays through the system from object to image. However, at the last surface, we (hypothetically) force all rays to converge onto the paraxial focus. For all rays, we compute the optical path from object to image. The OPD is the difference between the integrated optical path of a specified ray with respect to the optical path of the chief ray. Of course, if there were no aberration present, the OPD would be zero. Thus, the OPD represents a quantitative description of the violation of Fermat's principle.

The general concept is shown in Figure 3.5. Rays are accurately traced from the object through the system, emerging into image space. That is to say, ray tracing proceeds until the last optical surface, mirror or lens etc. Following the preceding discussion, at some point, we force all rays to converge upon the paraxial focus. However, the convention for computing OPD is that all rays are traced back to a spherical surface centred on the paraxial focus and which lies at the exit pupil of the system. Of course, it must be emphasised that the real rays do not actually follow this path. In the generic system illustrated, the real ray is traced to point P located in object space and the optical path length computed. Thereafter, instead of tracing the real ray into the object space, a dummy ray is traced, as shown by the dotted line. This dummy ray is traced from point P to point Q that lies on the reference surface – a sphere located at the exit pupil and centred on the paraxial focus. The optical path length of this segment is then added to the total.

Figure 3.5 Illustration of optical path difference.

After calculating the optical path length for the dummy ray OPQ, we need to calculate the OPD with respect to the chief ray. The chief ray path is calculated from the object to its intersection with the reference sphere at the pupil, represented, in this instance, by the path OR. In calculating the OPD, the convention is that the OPD is the chief ray optical path (OR) minus the dummy ray optical path (OPQ). Note the sign convention.

Having established an additional way of describing aberrations in terms of the violation of Fermat's principle, the question is what is the particular significance and utility of this approach? The answer is that, when expressed in terms of the OPD, aberrations are additive through a system. As a consequence of this, this treatment provides an extremely powerful general description of aberrations and, in particular, third order aberrations. Broadly, aberrations can be computed for individual system elements, such as surfaces, mirrors, or lenses and applied additively to the system as a whole. This generality and flexibility is not provided by a consideration of transverse aberrations.

There is a correspondence between transverse aberration and OPD. This is illustrated in Figure 3.6. At this point, we introduce a concept that is related to that of OPD, namely wavefront error(WFE). We must remember that, according to the wave description, the rays we trace through the system represent normals to the relevant wavefront. The wavefront itself originates from a single object point and represents a surface of equal phase. As such, the wavefront represents a surface of equal optical path length. For an aberrated optical system, the surface normals (rays) do not converge on a single point. In Figure 3.6, this surface is shown as a solid line. A hypothetical spherical surface, shown as a dashed line, is now added to represent rays converging on the paraxial focus. This surface intersects the real surface at the chief ray position. The distance between these two surfaces is the WFE.

In terms of the sign convention, the wavefront error, WFE, is given by:

The sign convention is important, as it now concurs with the definition of OPD. As the wavefronts form surfaces of constant optical path length, there is a direct correspondence between OPD and WFE. A positive OPD indicates the optical path of the ray at the reference sphere is less than that of the chief ray. Therefore, this ray has to travel a small positive distance to ‘catch up’ with the chief ray to maintain phase equality. Hence, the WFE is also positive.

Figure 3.6 Wavefront representation of aberration.

Figure 3.7 Simplified wavefront and ray geometry.

Both OPD and WFE quantify the violation of Fermat's principle in the same way. OPD is generally used to describe the path length difference of a specific ray. WFE tends to be used when describing OPD variation across an assembly of rays, specifically across a pupil. The concept of WFE enables us to establish the relationship between OPD and transverse aberration in that it helps define the link between wave (phase and path length) geometry and ray geometry. This is shown in Figure 3.7. It is clear that the transverse aberration is related to the angular difference between the wavefront and reference sphere surfaces.

We now describe the WFE, Φ, as a function of the reference sphere (paraxial ray) angle, θ. The radius of the reference sphere (distance to the paraxial focus) is denoted by f. This allows us to calculate the difference in angle, Δθ, between the real and paraxial rays. This is simply equal to the difference in local slope between the two surfaces.

(3.9)

n is the medium refractive index.

In this analysis, the WFE represents the difference between the real and reference surfaces with the positive axial direction represented by the propagation direction (from object to image). In this convention, the WFE has the opposite sign to the OPD. The transverse aberration, t, can be derived from simple trigonometry.

(3.10)

If θ describes the angle the ray makes to the chief ray, then Eq. (3.10) may be reformed in terms of the numerical aperture, NA. The numerical aperture is equal to nsinθ, and Eq. (3.11) may be recast as:

(3.11)

So, the transverse aberration may be represented by the first differential of the WFE with respect to the numerical aperture. In terms of third order aberration theory, the numerical aperture of an individual ray is directly proportional to the normalised pupil function, p. If the overall system, or marginal ray, numerical aperture is NA₀, then the individual ray numerical aperture is simply NA₀p. The transverse aberration is then given by:

(3.12)

Equation (3.12) provides a simple direct relationship between OPD and transverse aberration. Of course, we know that, for third order aberration, the transverse aberration is proportional to the third power of the pupil function, p. If this is the case, then it is apparent, from Eq. (3.12), that the OPD is proportional to the fourth power of the pupil function. So, for third order aberration, the transverse aberration shows a third power dependence upon the pupil function whereas the OPD shows a fourth power dependence.

Applying these arguments to the analysis of the simple on-axis example illustrated earlier, with the object placed at the infinite conjugate, then the WFE can be represented by the following equation:

(3.13)

p is the normalised pupil function.

Figure 3.8 shows a plot of the OPD against the normalised pupil function; such a plot is referred to as an OPD fan.

Despite the fact that this simple aberration has a quartic dependence on the pupil function, it is still referred to as third order aberration after the transverse aberration dependence. As with the optimisation of transverse aberration, the OPD can be balanced by applying defocus to offset the aberration. We saw earlier that a simple defocus produces a linear term in the transverse aberration. Referring to Eq. (3.12), it is clear that defocus may be represented by a quadratic term. Equation (3.14) describes the OPD when some defocus has been added to the initial aberration.

(3.14)

An OPD fan with aberration plus balancing defocus is shown in Figure 3.9.

In this instance, the plot has a characteristic ‘W’ shape, with the curve in the vicinity of the origin dominated by the quadratic defocus term. As with the case for transverse aberration, the defocus can be optimised to produce the minimum possible OPD value when taken as a root mean squared value over the circular pupil. Again, using a weighting factor that is proportional to the pupil function, p, (to take account of the circular geometry), the mean squared OPD is given by:

(3.15)

Figure 3.8 Quartic OPD fan.

Figure 3.9 OPD fan with balancing defocus.

The above expression has a minimum where α = −¾. To understand the magnitude of this defocus, it is useful first to convert the new OPD expression into a transverse aberration using Eq. (3.12).

(3.16)

From Eq. (3.16), it can be seen that the optimum defocus is 3/8 of the distance between the paraxial and marginal ray foci. This value is different to that derived for the optimisation of the transverse aberration itself. It should be understood that the optimisation of the transverse aberration and the OPD, although having the same ultimate purpose in minimising the aberration, nonetheless produce different results. Indeed, in the optimisation of optical designs, one is faced with a choice of minimising either the geometrical spot size (transverse aberration) or OPD in the form of rms WFE. The rationale behind this selection will be considered in later chapters when we examine measures of image quality, as applied to optical design.

The balanced defocus, as illustrated in Eq. (3.15) does significantly reduce the rms OPD. In fact, it reduces the OPD by a factor of four. Resultant rms values are set out in Eq. (3.17).

(3.17)