Eigenmode Analysis: Understanding Tolerance-Induced Aberrations

Introduction

Optical system performance is increasingly limited by sensitivity to manufacturing and assembly tolerances, making it essential for designers to understand not just how far the “performance” of a system drifts from nominal, but more specifically, which aberrations drive that degradation and where those effects appear in the field. Traditional tolerance analysis quantifies the performance loss – typically in terms of increased RMS wavefront error or decreased MTF - but often fails to reveal the underlying mechanisms responsible for the performance loss, leaving engineers without the insight needed to diagnose or mitigate the true sources of error.

Eigenmode analysis fills this gap.

Instead of reporting changes in overall quality metrics such as RMS wavefront error, eigenmode analysis examines the changes in individual aberrations and reports a set of orthogonal aberration pattern changes, ranked by their relative impact on performance. These “tolerance eigenmodes” form an orthonormal basis for the set of all tolerance-induced aberrations. This tells the designer not only how severely the performance will be degraded, but in what manner the degradation will occur. Furthermore, the eigenmodes can be rank-ordered in terms of severity, so the designer knows which eigenmodes are the most important to be addressed.

Examples of Eigenmodes

To give an idea of the kinds of aberration pattern changes that are typically associated with tolerances, we show examples of common eigenmodes below, beginning with the modes of the microscope objective shown in Figure 1. We emphasize that these are examples only: tolerance-induced eigenmodes vary from system to system and depend on the tolerances assigned.

Figure 1. Microscope objective.

Because there are usually more ways to drive a system out of focus than to create any other problem, field-uniform defocus, shown in Figure 2, is the dominant tolerance-induced aberration mode of most optical systems.

Figure 2. Field-uniform defocus.

The green circles in the figure represent defocus. Contributions from all aberration types are plotted in this figure, but because this mode is strongly dominated by defocus, the other aberrations appear only as small dots at the center of each defocus circle.

For this system, the second eigenmode is field-constant coma, shown in Figure 2.

Figure 3. Field-constant coma.

The mode shown in Figure 3 is the dominant eigenmode (after defocus) of many narrow-field optical systems. In fact, many optical engineers simply assume that this will be the dominant mode of their system, and plan on using a “compensator adjustment” such an intentional decentration of an element to address this type of error. Unfortunately, the assumption that this will be the dominant mode is often incorrect. Guessing incorrectly as to the nature of the dominant eigenmode can lead to the selection of an inappropriate compensator for the system.

The third mode for this system is field-constant coma oriented horizontally rather than vertically. Conceptually this is the same mode as #2, but the software reports it as a separate eigenmode.

The next mode for this system is the astigmatic pattern shown in Figure 4. The astigmatism is zero at the center of the field and grows linearly toward the edge of the field, although the orientation of the astigmatism is asymmetric in the field.

Figure 4. Field-linear, field-asymmetric astigmatism.

This pattern is predicted to exist by Nodal Aberration Theory^1,2 but the CODE V Modal Analysis tool simply recognizes this pattern in the behavior of the optical system without the use of any aberration theory.

This eigenmode frequently results from a misalignment of an optical system and is in many cases significantly more important than the Field-Uniform Coma pattern of Figure 3. Because the aberration of this mode vanishes at the center of the field, it is necessary to measure the aberrations at several peripheral field points to identify the mode. Measuring only at the on-axis field point would give no hint as to the existence of this mode, much less its magnitude or orientation.

As was the case with field-uniform coma, the pattern shown in Figure 4 shows a particular orientation, and a similar mode (#5 in this case) exists with the pattern rotated 90 degrees.

The next mode for this system is the field-constant astigmatism pattern shown in Figures 5 and 6.

The patterns shown in Figures 5 and 6 are mathematically orthogonal even though they are rotated by 45° rather than 90°. This is for the same reason that the two Zernike polynomials for astigmatism are rotated by 45° rather than 90°.

Figure 5. Field-constant astigmatism, 45° orientation.

Figure 6. Field-constant astigmatism, horizontal orientation.

It is also worth noting that although Figures 5 and 6 are both related to astigmatism, they are mathematically orthogonal to the field-linear, field-asymmetric pattern shown in Figure 4. This is because neither pattern can be improved by addition of any amount of the of the other pattern.

The Field-Uniform Astigmatism pattern shown in Figures 5 and 6 is typically not caused by a misalignment, but by astigmatic errors of the optical surfaces. The most effective way to reduce this mode is to either tighten the surface figure tolerances of the system or specify that the surfaces are to be “clocked” by appropriate amounts after the system behavior has been determined.

The next mode for this system is field-linear defocus i.e., image plane tilt, shown in Figure 7.

Figure 7, field-linear defocus, horizontal orientation.

In this case the dashed green circles represent defocus in the negative direction, and solid circles represent positive defocus. The linear behavior from left to right is evident in the figure. Even though each field point suffers only defocus, this pattern is mathematically orthogonal to field-uniform defocus (Figure 2), because it cannot be improved by adding or subtracting any amount of field-uniform defocus. As may be expected, this mode is paired with another mode with the image tilt running in the vertical direction.

Figure 8 shows the next mode for this system, which is field-constant longitudinal chromatic aberration.

Figure 8. Field-Uniform Axial Color.

There is typically no good compensator for this aberration short of respacing the entire system or by sorting the fabricated elements into pairs that happen to have compensating dispersion errors. Because these two approaches are labor-intensive, it is often more cost-effective simply to tighten the dispersion tolerances on the optical elements.

Compensator Selection

Knowing the eigenmodes and rank-ordering them according to severity provides an ideal way to select compensators for the system. In fact, the CODE V Tolerance-Induced Modal Analysis tool considers not only the effectiveness of prospective compensators, but also the effect of errors in the motions of those compensators, i.e., the precision to which they can be adjusted. Figure 9 demonstrates the effectiveness and sensitivity to implementation error for several possible compensators for defocus error.

Figure 9. Compensator Selection.

Each color-coded curve in the plot shows the performance of a different tentative compensator. In this case, longitudinal adjustments of both “DLT” type and “DSZ” type are considered. A “DLT” type is a change to a spacer, meaning that all elements after the spacer are shifted according to the thickness change of the spacer in question. A “DSZ” type adjustment represents the longitudinal adjustment of a single element; this can be thought of as a change to a spacer before the element followed by a corresponding negative change to the spacer following the element. Often, the choice between a “DLT” type or a “DSZ” type motion is determined by the nature of the mechanical design of the housing. In this example, we are assuming that the housing design has not yet been made, so we are exploring the efficacy of both types of compensator.

The vertical axis of the figure depicts the performance of the system relative to a set of user-entered metrics, with lower values being preferred over higher values. The figure is scaled so that the top edge of the plot corresponds to starting (pre-compensation) performance of the system.

The horizontal axis of the figure represents the error in implementing the compensator motion, relative to user-entered estimates of how precisely each compensator can be implemented. (Users can indicate different precision values for different compensators.) The horizontal center of the figure represents a normalized error of zero, i.e., each compensator is implemented perfectly. The right and left edges of the figure represent the cases in which the error in implementing each compensator is 100% of the user-entered precision for that compensator.

The figure demonstrates that the performance metric can be reduced to approximately one-quarter of the pre-compensation value using any one of the 14 tentative compensators shown, provided that the compensator is implemented with zero error.

On the other hand, if we assume that each compensator suffers an error equal to its user-entered precision value, then the performance metrics for the red compensator (“DLT S19”) and the blue compensator (“DSZ S18..19”) are much worse than for the other compensators. This indicates to the designer that these tentative compensators should not be chosen for focus control.

Of course, it may happen that for reasons of mechanical convenience, it may be necessary to use one of the more sensitive compensators. In this example, DLT S19 represents adjustment of the distance to the focal plane, and this is the common method of adjusting focus in a microscope. In the event that the compensator has been “preselected” for reasons of mechanical convenience, the figure can be read differently: in order to achieve the same level of performance with DLT S19 as with the other compensators, the DLT S19 needs to be adjusted to a relative precision of 0.1 rather than 1.0. That is, the adjustment must be made 10x more precisely than the other compensators.

Orthogonality is important

We have emphasized that the eigenmodes of a system are orthogonal, i.e., no eigenmode can by “improved” by adding or subtracting any amount of any other eigenmode. Because no mode can be used to balance any other mode, every mode can be thought of as a detrimental feature that should be eliminated or reduced.

Modal analysis identifies the different ways in which an optical system will respond to stimuli. In the analysis, the stimuli are the tolerances themselves, and the goal is to understand not only the extent of MTF degradation but also how aberration patterns change under perturbation. These changes define the failure modes of the system.

Example 2 — Wide-Angle Lens

As a second example, we consider the wide-angle system, shown in Figure 10, designed to cover a rectangular field.

Figure 10: Wide-Angle System

For the system shown, the eigenmodes are different. As with most systems, the dominant mode is defocus, followed by (in this case) image plane tilt. These modes are compensated for in a straightforward manner by adjusting the location and tilt of the image plane. The dominant modes after correction of defocus and image tilt are (in order): field-linear, field-asymmetric astigmatism (shown in Figure 11 below), rotationally symmetric astigmatism, field-uniform astigmatism, and field-uniform coma.

Figure 11. Dominant mode after compensation of focus and image tilt.

As mentioned earlier, characterization of the orientation and severity of this mode requires that the aberrations be measured at several points around the periphery of the field. This is in contrast to the case of field-constant coma, which only requires measurement at the on-axis field point and which, for this system, is about four times less severe than the dominant mode. Knowledge of the order in which the modes will appear is crucial for planning the compensation steps and the associated metrology.

Conclusion

Eigenmode analysis helps optical engineers plan metrology more effectively by showing where measurements should be taken across the field. It also provides a clear ranking of system sensitivities, allowing compensators to be chosen and sequenced based on the eigenvalues that matter most. Aberrations that cannot be corrected—such as axial color tied to Abbe value error—should have their tolerances tightened instead. Running eigenmode analysis early in the design and mechanical planning process ensures these insights guide decisions before costly changes are made. Finally, because the eigenmodes are orthogonal, each represents an independent degradation mechanism that must be understood and managed on its own.

References

[1] K.P. Thompson,“Aberration fields in tilted and decentered optical systems, Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

[2] K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry”, J. Opt. Soc.Am A 22, p1389 – 1401 (2005).

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