**Wavefront Issues to Address for ANSI US Standard: 6/25/06**

**1. Slope requirement for a surface or wavefront:**

**Slope specifications are currently written as follows:**

**A slope value is calculated for every pixel in the data acquisition array**

**based on the following relationship:**

**Y-slope = pixel above minus pixel below divide by two.**

**X-slope = pixel left minus pixel right divide by two.**

**Slope Mag = SQRT [(Y-slope)**

2

** + (X-slope)**

2

**]**

**The result is in fringes/pixel or nm/pixel etc. The value is further**

**converted to fringes/mm or nm/mm by multiplying this result by**

**pixel/mm.**

**This method for defining slope is adequate for smooth wavefront**

**functions where the change in pixel values is small. Also, when the part**

**diameter is small, the pixel/mm value can get large which in turn gives**

**large values for slope in terms of nm/mm even for small changes in pixel**

**values due to erroneous data acquisition "noise", rendering the**

**specification technique ineffective.**

**2. Smoothing:**

**Common methods for smoothing data are convolutions or polynomial fitting.**

**This is a common practice for surface and wavefront measurement applications to**

**suppress the influence of data acquisition "noise" sources.**

**Convolution smoothing - Every pixel in the data acquisition array is averaged by**

**itself and all of it's adjacent neighboring pixels. Therefore 9 pixels would be**

**averaged except for edge pixels. This would be defined as a 3 by 3 averaging**

**kernel, which represents the 3 row and 3 columns of pixels.**

**Convolution kernels can be increased to 5 by 5, 7 by 7, etc. etc, at the cost of**

**suppressing higher frequency spatial features in the data acquisition array.**

**Polynomial Fitting -- The use of a polynomial such as the Zernike polynomial**

**can be used to report the result of a surface or wavefront as a function of a least**

**squares fit of the acquired data. This will sufficiently suppress all spatial content**

**that is beyond the degree of the polynomial used.**

**3. Datapoint Statistical Analysis:**

**The peak-to-valley (PV) or slope specifications are highly influenced by the value**

**of all pixels in the data acquisition array because the result reported is the**

**maximum value obtained within the array, which can involve thousands of**

**individual pixel datapoints. It is common to report the number ofdata points**

**acquired in the data acquisition array that fall outside of some threshold limit, and**

**further statistically evaluate the number ofpixel datapoints that fall within the**

**threshold limit as a percentage.**

**Erroneous noise sources that exist in data acquisition can have a detrimental**

**effect on the value reported for PV and slope to name a few. A percentage of all**

**datapoints that meet a defined threshold value can be assigned to the specification**

**of single pixel dependent tolerances.**

**4. Universal Units Definition:**

**Historically, wavefront and surface units are reported in waves or fringes. This**

**method for defining results is ambiguous unless the wavelength is defined as**

**well. The metric of nanometer does not rely on the definition of the wavelength.**