Predicting Wavefront Error in Optical Coatings

Written by the Alluxa Engineering Team

An analytical approach to predicting coating-contributed surface error when measurement is limited or inadequate.

Introduction

Modern optical systems increasingly demand tighter tolerances, higher throughput, and greater consistency across larger apertures. As these performance requirements grow, so does the need to understand—and confidently control—the factors that degrade optical wavefront quality. Among these factors, Transmitted Wavefront Error (TWE) remains one of the most critical, yet most frequently misunderstood, contributors to system level performance.

TWE quantifies how much an optical element distorts the wavefront of a light wave as it propagates through the component. In an ideal system, a plane wave should emerge from a filter or window without any change in phase front. In practice, however, real optical components introduce non-ideal phase variations or errors from sources such as coating thickness gradients, substrate shape, internal material inhomogeneity, or the interaction of multiple surfaces. These variations/errors can reduce image fidelity, shift focal positions, and introduce aberrations into the optical system.

Circumstances can render it infeasible to empirically measure TWE. A combination of coating properties, wavelength limitations, and instrument constraints can interfere with measurement; still, demonstrating compliance remains essential. To address this challenge, Alluxa’s engineering team has developed an analytical framework for predicting TWE when direct interferometric measurement is impractical or when cost and equipment limitations prohibit full characterization. This predictive modeling approach—based on spectral uniformity—is showcased as a way to estimate coating-contributed TWE with high confidence when direct measurement is not an option.

TWE Discussion

Transmitted Wavefront Error is the industry metric used to quantify the distortion imparted onto a light wave as it passes through an optical component, such as a filter, lens, or window. In an idealized optical system, a plane wavefront entering a flat component should emerge as an unaltered plane wavefront.

Figure 1a. Light beam wavefront transmitted and reflected by a perfectly flat plane parallel transparent substrate.

Figure 1b. Light beam wavefront transmitted and reflected by the same glass substrate after bending introduces curvature to the surfaces.

However, in practice, essentially all wavefronts deviate from this ideal shape. TWE measures the difference between the emerging wavefront and the ideal reference wavefront.

As a metric, TWE does not typically include all distortions experienced by a wavefront traveling through an optical component. Piston, tilt and power (Figure 2) are typically not included in the final TWE value. The reasons for this exclusion are:

Piston simply represents a uniform shift in the phase of the entire wavefront along the axis of propagation. Although this can sometimes be important, in most standalone imaging or filtering applications, piston can be ignored because it does not change the shape of the wavefront or the quality of the image; it simply shifts the absolute phase, which neither the human eye nor standard sensors can detect.
Tilt occurs when the wavefront arrives at an angle relative to the optical axis. It can be caused by thickness variation in a part, such as a physical wedge in a substrate, or by misalignment within the system. Tilt does not introduce blur and can often be corrected in system alignment by adjusting a mirror or realigning a sensor.
Power is a rotationally symmetric curvature of the wavefront, causing a plane wave to converge or diverge, effectively acting like a weak lens. Although this aberration can be critical in fixed-focus systems, in systems with adjustable focus, this can typically be compensated and is therefore excluded from the total contributed wavefront distortion.

Figure 2. Total wavefront distortion is the sum of piston, tilt and power components in addition to irregularity. TWE always includes irregularity and sometimes power but rarely piston or tilt.

While distortions on reflection —referred to as Reflective Wavefront Error (RWE)—can also be important, the focus here is on TWE. RWE is distinctly different from TWE. For example, reflection from a slightly bent, plane parallel substrate (Figure 1b) will show curvature in the reflected wavefront, while the transmitted wave remains nearly unaffected aside from a small tilt.

One common misconception when specifying wavefront error is the assumption that surface flatness directly correlates to TWE. However, the individual flatness contributions from the two surfaces alone cannot reliably predict transmitted wavefront error. Surface geometries of an optical flat may either cancel or compound distortions. If the aspect of one surface matches that of the other surface, TWE can be minimal (Figure 3b). If they are similar but noncomplementary, TWE can double (Figure 3c).

Figure 3. Transmitted wavefront distortion for a) flat substrate, b) substrate non-flat in the same way on both sides, c) substrate with the same non-flat top and bottom surfaces as in b) but with the bottom surface flipped.

Determining TWE from Wavelength Uniformity

TWE is determined solely by the change in the transmitted wavefront, i.e., the relative changes in phase at different positions across the clear aperture of the optical component. Interferometry is the standard method for measuring TWE, as it directly maps wavefront phase. This method is not always practical, though, since interferometric measurements can be compromised by ghost reflections, alignment errors, or where the optic is opaque at the interferometer’s source wavelength.

For cases where interferometry is impractical, TWE can be determined analytically. The analysis can be done using wavelength uniformity measurements over the working clear aperture of the filter. A change in coating thickness, i.e. non-uniformity, leads to a proportional change in optical phase thickness. Considering this relationship, wavelength shifts can be converted directly into TWE (Figure 4).

Figure 4. Phase shifts in an incident wavefront after transmission through a filter with variations in coating thickness across the surface. Wavelength dependent features appear at longer wavelengths in areas with thicker coating.

This method relates coating thickness and spectral position to changes in phase across the part. The approach relies on some relatively easy-to-satisfy requirements driven by the coating process itself.

The first requirement is the dominance of the coating thickness as the principal source of wavefront distortion. Wavelength variations across the clear aperture should be driven solely by coating thickness, and influence from the substrate or other factors should be negligible. The second is that the layer thicknesses should scale proportionally across the part. To a first order approximation, all layers in the coating stack should increase or decrease in thickness together. Third, the thickness changes of each layer should be small relative to the layer thickness. This ensures any shift is only a slight change in phase and, consequently, can be determined to sufficient accuracy using a first order approximation. Finally, the thickness variations should occur gradually across the surface. This reduces the total number of measurements required to characterize the entirety of the part and allows for each measurement to be made over a reasonable area.

These requirements are typically satisfied by precision PVD coating processes such as evaporation and sputtering. During the coating process, slight variations in thickness are inevitable due to the variation in deposition thickness throughout a coating chamber. As coating thickness increases, spectral features shift to longer wavelengths; when thickness decreases, they shift shorter.

The transmission curve of a thin-film filter is shaped by interference between layers and is extremely sensitive to layer thickness. Subnanometer variations can noticeably distort the spectral response curve. To enable the use of spectral performance as a measurement of TWE, the spectral curve should closely match the expected theoretical curve, with minimal distortion. For the analysis to be effective, the spectral non-uniformity should manifest in a first order shift in overall wavelength position; more complicated variations in the layer-to-layer thickness impacting the final spectral curve would render this proposed method inaccurate. For high-layer count coatings with precise wavelength features, if the spectral curve retains its shape and matches the theoretical model, this provides confidence that the thickness change is small relative to the layer thickness. Then, the wavelength shifts can be modeled as a constant percentage change in coating thickness.

Once the integrity of the spectral shape is determined, the process to calculate TWE can be summarized as follows. A spectrophotometer or a laser/detector system measures a specific or unique spectral feature (such as a 50% edge or peak wavelength) at multiple points across the aperture. These measurements are plotted in the form of wavelength shifts as shown in Figure 5.