Evaluation

Paraxial analysis

Although the lens setup routine contains a paraxial ray trace, a separate paraxial ray trace routine is used to compute data for display to the user. At a minimum, the paraxial ray heights and slopes of the axial and chief ray are shown for each surface, in each color, in each configuration.

The equations used for paraxial ray tracing were described in the previous section. Although such equations become exact only for "true" paraxial rays that are infinitesimally displaced from the optical axis, it is customary to consider paraxial ray data to describe "formal" paraxial rays that refract at the tangent planes to surfaces, as shown in Fig. 2 below. Here, the ray ABC is a paraxial ray that provides a first-order approximation to the exact ray ADE. Not only does the paraxial ray refract at the (imaginary) tangent plane BVP, but also it bends a different amount from the exact ray.

In addition to the computation of ray heights and slopes for the axial and chief ray, various paraxial constants that characterize the overall system are computed. The particular values computed depend on whether the system is focal (finite image distance) or afocal (image at infinity). For focal systems, the quantities of interest are (at a minimum) the focal length efl, the f/number FN, the paraxial (Lagrange) invariant PIV, and the transverse magnification m. It is desirable to compute such quantities in a way that does not depend on the position of the final image surface. Let the object height be h, the entrance pupil radius by r, the axial ray data in object and image space be  and , the chief ray data in object and image space be  and , and the refractive indices be n and .

The above-mentioned paraxial constants are then given by

In addition to the paraxial constants, most programs display the locations of the entrance and exit pupils, which are easily determined using chief-ray data. Surprisingly, most optical design programs do not explicitly show the locations of the principal planes. In addition, although most programs have the capability to display "yybar" plots, few have integrated this method into the main data entry routine.

Aberrations

Although most optical design is based on exact ray data, virtually all programs have the capability to compute and display first-order chromatic aberrations and third-order monochromatic (Seidel) aberrations. Many programs can compute fifth-order aberrations as well. The form in which aberrations are displayed depends on the program and the type of system under study, but as a general rule, for focal systems aberrations are displayed as equivalent ray displacements in the paraxial image plane.

In the case of the chromatic aberrations, the primary and secondary chromatic aberration of the axial and chief rays are computed. In a system for which three wavelengths are defined, the primary aberration is usually taken between the two outer wavelengths, and the secondary aberration between the central and short wavelengths.

The Seidel aberrations are computed according to the usual aberration polynomial. If we let be the displacement of a ray from the chief ray, then

For a relative field height h and normalized entrance pupil coordinates r and , the third-order terms are

The interpretation of the coefficients is generally as follows, but several optical design programs display tangential coma, rather than the sagittal coma indicated in the table.

The fifth-order terms are

These equations express the fifth-order aberration in terms of the Buchdahl coefficients. In systems for which the third-order aberrations are corrected, the following identities exist:

Some programs display only the aberrations that have corresponding third-order coefficients, omitting oblique spherical aberration and elliptical coma.

The formulas needed to calculate the chromatic and third-order aberrations are given in the U.S. Military Handbook of Optical Design. The formulas for calculating the fifth-order aberrations are given in Buchdahl's book[i].

Aberration coefficients are useful in optical design because they characterize the system in terms of its symmetries, because they allow the overall performance to be expressed as a sum of surface contributions, and because they are calculated quickly. On the negative side, aberration coefficients are not valid for systems that have tilted and decentered elements, and for systems that cover an appreciable field of view, the accuracy of aberration coefficients in predicting performance is usually inadequate. Moreover, for systems that include unusual elements like diffractive surfaces and gradient index materials, the computation of aberration coefficients is cumbersome at best.

Ray tracing

Exact ray tracing is the foundation of an optical design program, serving as a base for both evaluation and optimization. From the programmer's standpoint, the exact ray trace routines must be accurate and efficient. From the user's viewpoint, the data produced by the ray trace routines must be accurate and comprehensible. Misunderstanding the meaning of ray trace results can be the source of costly errors in design.

To trace rays in an optical design program, it is necessary to understand how exact rays are specified. Although the details may vary from one program to the next, many programs define a ray by a two-step process. In the first step, an object point is specified. Once this has been done, all rays are assumed to originate from this point until a new object point is specified. The rays themselves are then specified by aperture coordinates and wavelength.

Exact ray starting data is usually normalized to the object and pupil coordinates specified by the axial and chief rays. That is, the aperture coordinates of a ray are specified as a fractional number, with 0.0 representing a point on the vertex of the entrance pupil, and 1.0 representing the edge of the pupil. Field angles or object heights are similarly described, with 0.0 being a point on the axis, and 1.0 being a point at the edge of the field of view.

Although the above normalization is useful when the object plane is at infinity, it is not so good when the object is at a finite distance and the numerical aperture in object space is appreciable. Then, fractional aperture coordinates should be chosen proportional to the direction cosines of rays leaving an object point. There are two reasons for this. One is that it allows an object point to be considered a point source, so that the amount of energy is proportional to the "area" on the entrance pupil. The other is that for systems without pupil aberrations, the fractional coordinates on the second principal surface should be the same as those on the first principal surface. Notwithstanding these requirements, many optical design programs do not define fractional coordinates proportional to direction cosines.

It is sometimes a point of confusion that the aperture and field of view of a system are specified by paraxial quantities, when the actual performance is determined by exact rays. In fact, the paraxial specifications merely establish a normalization for exact ray data. For example, in a real system the field of view is determined not by the angle of the paraxial chief ray, but by the angle at which exact rays blocked by actual apertures just fail to pass through the system. Using an iterative procedure, it is not to hard to find this angle, but because of the nonlinear behavior of Snell's law,it does not provide a convenient reference point.

There are two types of exact rays: ordinary or Lagrangian rays, and iterated or Hamiltonian rays. The designation of rays as Lagrangian or Hamiltonian comes from the analogy to the equations of motion of a particle in classical mechanics. Here we use the more common designation as ordinary or iterated rays, An ordinary ray is a ray that starts from a known object point in a known direction. An iterated ray also starts from a known object point, but its direction is not known at the start. Instead, it is known that the ray passes through some known (non-conjugate) point inside the system, and the initial ray direction is determined by an iterative procedure.

Iterated rays have several applications in optical design programs. For example, whenever a new object point is specified, it is common to trace an iterated ray through the center of the aperture stop (or some other point) to serve as a reference ray, or to trace several iterated rays through the edges of limiting apertures to serve as reference rays. In fact, many programs use the term reference ray to mean iterated ray (although in others, reference rays are ordinary rays). Iterated rays are traced using differentially displaced rays to compute corrections to the initial ray directions. Because of this, they are traced slower than ordinary rays. On the other hand, they carry more information in the form of the differentials, which is useful for computing ancillary data like field sags.

Reference rays are used as base rays in the interpretation of ordinary ray data. For example, the term ray displacement often refers to the difference in coordinates on the image surface between a ray and its corresponding reference ray. Similarly, the optical path difference of a ray may compare its phase length to that of the corresponding reference ray. The qualifications expressed in the preceding sentences indicate that the definitions are not universal. Indeed, although the terms ray displacement and optical path difference are very commonly used in optical design, they are not precisely defined, nor can they be. Let us consider, for example, the optical path difference.

Imagine a monochromatic wavefront from a specified object point that passes through an optical system. Fig. 3 shows the wavefront PE emerging in image space, where it is labeled "actual wavefront". Because of aberrations, an ordinary ray perpendicular to the actual wavefront will not intersect the final image surface at the ideal image point I, but at some other point Q. The optical path difference may be defined as the optical path measured along the actual ray between the actual wavefront and a reference sphere centered on the ideal image point.

Unfortunately, the ideal image point is not precisely defined. In the figure, it is shown as the intersection of the reference ray with the image surface, but the reference ray itself may not be precisely defined. Is it the ray through the center of the aperture stop, or perhaps the ray through the center of the actual vignetted aperture? These two definitions will result in different reference rays, and correspondingly different values for the optical path difference. In fact, in many practical applications neither of the definitions is used, and the actual ideal image point is defined to be the one that minimizes the variance of the optical path difference (and hence maximizes the peak intensity of the diffraction image).

Moreover, the figure shows that even if the ideal image point is precisely defined, the value of the optical path difference depends on the point E where the actual wavefront intersects the reference sphere. For the particular point shown, the optical path difference is the optical length along the ordinary ray from the object point to the point T, less the optical length along the reference ray from the object point to the point I. As the radius of the reference sphere is increased, the point T merges with the point S, where a perpendicular from the ideal image point intersects the ordinary ray.

The above somewhat extended discussion is meant to demonstrate that even "well-known" optical terms are not always precisely defined. Not surprisingly, various optical design programs in common use produce different values for such quantities. There has been little effort to standardize the definitions of many terms, possibly because one cannot legislate physics. In any case, it is important for the user of an optical design program to understand precisely what the program is computing.

Virtually all optical design programs can trace single rays and display the ray heights and direction cosines on each surface. Other data, such as the path length, angles of incidence and refraction, and direction of the normal vector, are also commonly computed. Another type of ray-data display that is nearly universal is the ray-intercept curve, which shows ray displacement on the final image surface vs (fractional) pupil coordinates. A variation plots optical path difference vs pupil coordinates.

In addition to the uncertainty concerning the definition of ray displacement and optical path difference, there are different methods for handling the pupil coordinates. Some programs use entrance pupil coordinates, while others use exit pupil coordinates. In most cases, there is not a significant difference, but in the case of systems containing cylindrical lenses for example, there are major differences.

Another consideration relating to ray-intercept curves is the way in which vignetting is handled. This is coupled to the way that the program handles apertures. As mentioned before, apertures have a special status in many optical design programs. Rays can be blocked by apertures, but this must be handled as a special case by the program, because there is nothing inherent in the ray-trace equations that prevents a blocked ray from being traced, in contrast to a ray that misses a surface or undergoes total-internal reflection.

Even though a surface may have a blocking aperture, it may be desirable to let the ray trace proceed anyway. As mentioned before, blocking rays in optimization can produce instabilities that prevent convergence to a solution, even though all the rays in the final solution are contained within the allowed apertures. Another situation where blocking can be a problem concerns central obstructions. In such systems, the reference ray may be blocked by an obstruction, so its data are not available to compute the displacement or optical path difference of an ordinary ray (which is not blocked). The programmer must anticipate such situations and build in the proper code to handle them.

In the case of ray-intercept curves, it is not unusual for programs to display data for rays that are actually blocked by apertures. The user is expected to know which rays get through, and ignore the others, a somewhat unreasonable expectation. The justification for allowing it is that the designer can see what would happen to the rays if the apertures were increased.

In addition to ray-intercept curves, optical design programs usually display field sag plots showing the locations of the tangential and sagittal foci as a function of field angle, and distortion curves. In the case of distortion, there is the question of what to choose as a reference height. It is generally easiest to refer distortion to the paraxial chief ray height in the final image surface, but in many cases it is more meaningful to refer it to the centroid height of a bundle of exact rays from the same object point. Again, it is important for the user to know what the program is computing.

Spot diagram analysis

Spot diagrams provide the basis for realistic modeling of optical systems in an optical design program. In contrast to simple ray-trace evaluation, which shows data from one or a few rays, spot diagrams average data from hundreds or thousands of rays to evaluate the image of a point source. Notwithstanding this, it should be understood that the principal purpose of an optical design program is to design a system, not to simulate its performance. It is generally up to the designer to understand whether or not the evaluation model of a system is adequate to characterize its real performance, and the prudent designer will view unexpected results with suspicion.

From a programmer's point of view, the most difficult task in spot-diagram analysis is to accurately locate the aperture of the system. For systems that have rotational symmetry, this is not difficult, but for off-axis systems with vignetted apertures it can be a challenging exercise. However, the results of image evaluation routines are often critically dependent on effects that occur near the edges of apertures, so particular care must be paid to this problem in writing optical design software. Like many other aspects of an optical design program, there is a tradeoff between efficiency and accuracy.

A spot diagram is an assemblage of data describing the image-space coordinates of a large number of rays traced from a single object point. The data may be either monochromatic or polychromatic. Each ray is assigned a weight proportional to the fractional energy that it carries. Usually, the data saved for each ray include its xyz coordinates on the image surface, the direction cosines klm, and the optical path length or optical path difference from the reference ray. The ray coordinates are treated statistically to calculate root-mean-square spot sizes. The optical path lengths yield a measure of the wavefront quality, expressed through its variance and peak-to-valley error.

To obtain a spot diagram, the entrance pupil must be divided into cells, usually of equal area. Although for many purposes the arrangement of the cells does not matter, for some computations (e.g. transfer functions), it is advantageous to have the cells arranged on a rectangular grid. To make the computations have the proper symmetry, the grid should be symmetrical about the x and y axes. The size of the grid cells determines the total number of rays in the spot diagram.

In computing spot diagrams, the same considerations concerning the reference point appear as for ray fans. That is, it is possible to define ray displacements with respect to the chief-ray, the paraxial ray height, or the centroid of the spot diagram. However, for spot diagrams it is most common to use the centroid as the reference point, both because many image evaluation computations require this definition, and also because the value for the centroid is readily available from the computed ray data.

The displacements of rays on a plane shifted in the z-direction from the nominal image plane by an amount z is given by

If there are n rays, the coordinates of the  centroid of the spot diagram are

where W is a normalizing constant that ensures that the total energy in the image adds up to 100%, and

The mean-square spot size can then be written as

Usually, the root-mean-square (rms) spot size, which is the square root of this quantity, is reported. Since the MSS is a quadratic form, it can be written explicitly as a function of the focus shift by

where

Differentiating this expression for the MSS with respect to focus shift, then setting the derivative to zero, determines the focus shift at which the rms spot size has its minimum value:

Although the above equations determine the rms spot size in two dimensions, similar one-dimensional equations can be written for x and y separately, allowing the ready computation of the tangential and sagittal foci from spot diagram data. In addition, it is straightforward to carry out the preceding type of analysis using optical path data, which leads to the determination of the center of the reference sphere that minimizes the variance of the wavefront.

Beyond the computation of the statistical rms spot size and the wavefront variance, most optical design programs include a variety of image evaluation routines that are based on spot diagram data. It is useful to characterize them as belonging to geometrical optics or physical optics, according to whether they are based on ray displacements or wavefronts, although of course all are based on the results of geometrical ray tracing.

Geometrical optics

Most optical design programs provide routines for computing radial energy diagrams and knife-edge scans. To compute a radial energy diagram, the spot diagram data is sorted according to increasing ray displacement from the centroid of the spot. The fractional energy is then plotted as a function of spot radius. The knife-edge scan involves a similar computation, except that the spot diagram data is sorted according to x or y coordinates, instead of total ray displacement.

Another type of geometrical image evaluation based on spot-diagram data is the so-called geometrical optical transfer function (GOTF). This function can be developed as the limiting case, as the wavelength approaches zero, of the actual diffraction MTF, or alternately in a more heuristic way as the Fourier transform of a line spread function found directly from spot diagram ray displacements (see, for example, Smith's book[ii]). From a programming standpoint, computation of the GOTF involves multiplying the ray displacements by 2 times the spatial frequency under consideration, forming cosine and sine terms, and summing over all the rays in the spot diagram. The computation is quick, flexible, and if there are no more than a few waves of aberration, accurate. The results of the GOTF computation are typically shown as either plots of the magnitude of the GOTF as a function of frequency, or alternately in the form of what is called a "through-focus" MTF, in which the GOTF at a chosen frequency is plotted as a function of focus shift from the nominal image surface.

Physical optics

The principal physical optics calculations based on spot diagram data are the modulation transfer function, sometimes called the "diffraction" MTF, and the point spread function (PSF). Both are based on the wavefront derived from the optical path length data in the spot diagram. There are various ways to compute the MTF and PSF, and not all programs use the same method. The PSF, for example, can be computed from the pupil function using the Fast Fourier transform algorithm, or alternately using direct evaluation of the Fraunhofer diffraction integral. The MTF can be computed either as the Fourier transform of the PSF, or alternately, using the convolution of the pupil function.[iii] The decision as to which method to use involves speed, accuracy, flexibility, and ease of coding.

In physical-optics based image evaluation, accuracy can be a problem of substantial magnitude. In many optical design programs, diffraction-based computations are only accurate for systems in which diffraction plays an important role in limiting performance. Systems that are limited primarily by geometrical aberrations are difficult to evaluate using physical optics, because the wavefront changes so much across the pupil that it is impossible to sample it sufficiently using a reasonable number of rays. If the actual wavefront in the exit pupil is compared to a reference sphere, the resultant fringe spacing defines the size required for the spot diagram grid, since there must be several sample points per fringe to obtain accurate diffraction calculations. To obtain a small grid spacing, one can either trace many rays, or trace fewer rays but interpolate the resulting data to obtain intermediate data.

Diffraction calculations are necessarily restricted to one wavelength. To obtain polychromatic diffraction-results it is necessary to repeat the calculations in each color, adding the results while keeping track of the phase shifts caused by chromatic aberration.

Simulation

There is increased interest in using optical design programs to simulate the performance of actual systems. The goal is often to be able to calculate radiometric throughput of a system used in conjunction with a real extended source. It is difficult to provide software to do this with much generality, because brute force methods are very inefficient and hard to specify, while elegant methods tend to have restricted scope, and demand good judgment by the person modeling the physical situation. Nevertheless, with the speed of computers increasing as fast as it has recently, there is bound to be an increasing use of optical design software for evaluating real systems.