There are many fabrication and mounting errors to consider when tolerancing an optical system.  ZEMAX’s tolerancing capabilities can model a number of different tolerances, including tolerance on radius, thickness, tilts and decenters of surfaces or elements, surface irregularity, and much more.  Each of these is supported via their own tolerance operand in ZEMAX.

Sometimes, it is desirable to tolerance for machined optical glass errors, such as the spatial inhomogeneity of the refractive index of a material.  As there is not a built-in tolerance operand for inhomogeneity, this article is designed to cover the alterative and superior approach to tolerancing for such fabrication errors.

Accurately modeling the inhomogeneity of a material is not as “standard” as you might expect.  Think of it this way:  Inhomogeneity basically means NOT homogeneous.  In other words, there are an infinite number of solutions to modeling a non-homogenous medium.  Which one do you pick, and how do you know for certain that this assumption is accurately representing your particular blank of glass?

If we know the exact profile of how the index varies over the aperture and thickness of the lens, then we can mathematically model the varying refractive index using ZEMAX’s GRIN capabilities.  However, it is not often that a manufacturer will provide the index profile for you.

Instead, the inhomogeneity of a glass is often expressed as a grade, which corresponds to some maximum refractive index variation of the cut glass.  For example, the ISO 10110 part 4 homogeneity grades range from 0 to 5, while SCHOTT denotes their grades using the symbols below (S0, S1, and H1 through H5) [1]:



Table courtesy of SCHOTT glass.  Please see Reference [1].

Given a single number or value representing the inhomogeneity of a material, it is impossible to exactly predict the index profile of the glass. 

Therefore, the most accurate and superior approach to modeling the inhomogeneity of a material can be performed via the statistical results of Monte Carlo Tolerance Analysis using tolerances on surface irregularity. 

For example, the TEZI tolerance operand in ZEMAX is used to analyze random irregular deviations of small amplitude on a surface that is either type Standard or Even Aspheric.  Within the Monte Carlo Analysis, the specified surface is converted to a Zernike Standard Sag surface, and each polynomial term is assigned a coefficient randomly chosen between zero and one.  The resulting coefficients are normalized to yield the exact specified RMS tolerance.
 
So how are these related?  Basically, the surface irregularity model will “warp” the wavefront much like the non-homogeneity of a specific material.  Index variation within a piece of glass will lead to a deformation of the wavefront passing through it [1].  From SCHOTT’s technical information article on homogeneity (also reference [1]), the deformation to the wavefront as a function of inhomogeneity is given by:

Δs = d • Δn

where Δs is the wavefront deviation, d is the thickness of the glass, and Δn is the peak to valley refractive index variation in the glass.  In ZEMAX terms, we might write this as:

ΔOPL = t • Δn

where ΔOPL is the change in optical path length, t is the thickness of the glass, and Δn is the peak to valley refractive index variation in the glass.  For example, if we considered a plane wave incident on a 25mm thick plane parallel plate of glass of +/- 5*10-6 index varation, the maximum ΔOPL would be equal to:

ΔOPL = 25mm • 10 • 10-6 = 250nm

Though the TEZI operand is a direct tolerance for the exact RMS error of the surface in lens units, we can approximate the inhomogeneity spec as sag error.  Most conservatively, we can estimate the RMS error by multiplying the center thickness (assuming this is your largest thickness over the aperture of the lens) of your element by the inhomogeneity tolerance.

In the example given on the previous page, we determined that the maximum ΔOPL was 250nm for a 25mm PPP of H2 grade.  If we use the TEZI tolerance operand in the Tolerance Data Editor (TDE), the maximum tolerance can be set to 2.5E-4 (provided that the Lens Units are in millimeters).  The minimum tolerance for the TEZI operand is automatically set to the negative of the max value.





The number of Zernike terms used for the analysis may be between 0 and 231.  Generally speaking, if fewer terms are used, the irregularity will be of low frequency, with fewer “bumps” across the surface.  The maximum number of terms should be chosen accordingly. 

In a number of Monte Carlo Runs, we can gather a significant amount of statistical data relating to the change in RMS Wavefront Error due to the inhomogeneity of the glass.  The more Monte Carlo tolerance runs that are performed, the better the statistical average of performance degradation (change in criteria) will be.

Since our nominal system was simply a planar wavefront passing through a perfectly planar slab of glass, the wavefront error was initially zero, and thus the criteria is equivalent to the change in criteria in the tolerance output.  On average, the inhomogeneity of the glass resulted in nearly a quarter wave RMS wavefront error.  We can evaluate each wavefront profile by individually opening each of the saved Monte Carlo files and reviewing the Wavefront Map.  Nominally, our wavefront had no variation across the pupil:



Yet after the randomly chosen Zernike coefficients from the tolerance analysis, the wavefront is clearly aberrated, or “warped.”



Each Monte Carlo trial will have a slightly different representation of the inhomogeneity of your glass.  Therefore, a statistical listing of the entire Monte Carlo set is essential for estimating the probable effects the inhomogeneity has on your system performance.

When tolerancing for various manufacturing/assembly defects of an optical system, it is sometimes important to consider effects of inhomogeneity, or irregular variations of refractive index in the glass.

Although ZEMAX does not have a specific tolerance operand for inhomogeneity, a statistical method via the TEZI operand can be used instead, which tolerances for irregular deviations of small amplitude on a surface.

The Monte Carlo tolerance analysis is used to randomize the inhomogeneity (irregularity) and provide you with accurate, statistical results of how this irregularity is affecting the performance of your optical system.

References

1.  Schott AG (July 2004).  TIE – 26: Homogeneity of optical glass.  In Technical Information – Optics For Devices (Section 2.2).  Retrieved from http://www.us.schott.com/optics_devices/english/download/tie-26_homogeneity_of_optical_glass_us.pdf

2.  ZEMAX Optical Design Program User’s Guide, ZEMAX Development Corporation