The Peak-to-Valley wavefront error is a measure of the distance from the highest to the lowest point on the test wavefront relative to the reference wavefront. According to Optical Shop Testing by Daniel Malacara; "The P-V error must be regarded with some skepticism because it is calculated from the worst two interferometric data points out of possible thousands. It might make the system under test appear worse than it really is." Note that a mirror with a true P-V wavefront error of .25 wave, as verified by interferometry, will in any case meet or exceed the RMS wavefront and Strehl ratio criterion described below. Because P-V uses two points without regard to location on the mirrors surface, two mirrors may have the same P-V error but will be very different in overall quality. The difference in quality will be evident in the RMS and Strehl values. As an example, two mirrors may have the same P-V value due to a zone on the mirrors surface. Both of the mirrors have a high zone. The first mirror has its high zone near the center while the second has its high zone near the edge. The first mirror will have better RMS and Strehl values because the surface area covered by the high zone will be smaller in the center than at the edge.

To obtain the RMS wavefront error, a large number of interferometric data points are measured over the entire area of the test wavefront. As explained in Optical Shop Testing; "The RMS error is a statistic that is calculated from all of the measured data and gives a better indication of the overall system performance." Due to its statistical nature, professional optical shops consider the RMS wavefront error to be the most useful measure of optical quality. By common convention an optic with RMS wavefront error of 0.0712 wave or less is considered diffraction limited.

The Strehl ratio is another statistical measure of optical performance calculated from the interferometric test data. The Strehl ratio is the ratio of intensity of an aberrated wavefront to an unaberrated wavefront. In other words, the use of the Strehl ratio is a fundamental description of the amount of intensity reduction due to wavefront errors. A common convention is to consider an optic with a Strehl ratio of 0.8 or higher to be diffraction limited. The Strehl ratio and RMS wavefront error are mathematically related. It can be shown that a Strehl ratio of 0.8 corresponds to an RMS wavefront error of 0.0714 wave or approximately 1/14 wave.

Why Interferometry is Important

During the figuring process, the surface of your mirror is polished from a sphere to a paraboloid using an iterative process of testing and polishing to achieve the desired results. During this process we interpret the appearance of fringes in the Ronchi test and shadows in the Foucault test. Once the optician feels that the optic will pass the scrutiny of the interferometer, this test is performed. Due to the qualitative nature of the Ronchi and Foucault tests, and to the subjectivity in their interpretation, it is not infrequent that the interferometer will reject a mirror, sending it back for touch-up polishing. The interferometer is the final impartial go/no-go point in the quality control chain. It is a completely objective and accurate assessment of the quality of the optic under test.

Although it is possible to produce diffraction limited mirrors using methods such as the Ronchi and Foucault test, it is impossible to accurately verify and the quality of the entire wavefront to a fraction of a wave without interferometry. The Ronchi and Foucault tests are excellent evaluation tools during mirror fabrication as they show the general shape of the wavefront as well as localized and high frequency errors extremely well. These tests can show localized errors as small as 1/100 wave. This makes them indispensable tools during mirror figuring. However, unlike interferometry, they do not provide a means of accurately quantifying the whole wavefront because they only measure a few points. Interferometry on the other hand, is an extremely strict statistical analysis that assures the customer of a truly diffraction limited optic over its entire wavefront.

Optical Mechanics OMI Newtonian Mirror Wavefront Quality Specifications

The wavefront quality specification for Optical Mechanics OMI mirrors is 0.0712 wave RMS at 550 nm with a Strehl ratio of 0.8 or higher. The P-V wavefront error is not used in the specification, however it is reported in the interferometric test report provided with each mirror and is typically less than .25 wave. Each OMi mirror must meet or exceed these specifications before it is certified for shipment to the customer. Every mirror is delivered with its interferometric test report and a Certificate of Optical Quality.

Visual Interpretation of Your Interferogram

A full, statistically valid analysis of your interferometric test data requires computer fringe analysis. It is, however, possible to draw some general conclusions about a mirror from the visual appearance of the interference fringes. In particular, the third order Seidel aberrations; spherical aberration, astigmatism and coma, if present in significant amounts, can be seen in the interference fringes. Some examples follow:

The spherical aberration in the interferogram shown in Figure 1 is indicated by the slight gull-wing shape of the fringes. The gull-wing is most notable in the fringe just above the central obstruction. You can estimate the magnitude of the spherical aberration by stretching a string from one end of the fringe to the other. The fringe spacing in an interferogram is 1/2 wave in a single pass test and 1/4 wave in a double pass test. In this case, we are testing a Pro-Spec mirror in double-pass auto-collimation and thus the fringe spacing is 1/4 wave. In the image in Figure 1, a dark line representing the string has been drawn across the above mentioned fringe. The test shows that the fringe deviates above the string by approximately 1/3 of the fringe spacing on the left side of the mirror, and below the string by approximately 1/3 of the fringe spacing on the right side of the mirror. The P-V error along this fringe in waves is the sum of these deviations divided by 4 (double-pass), that is (1/3 + 1/3)/4 or approximately 1/6 wave.		Figure 1. - Spherical Aberration
In the interferogram shown in Figure 2 notice that in the lower left quadrant, the fringes curve slightly downward, while in the upper left quadrant they curve slightly upward. This indicates astigmatism. In this case, the string test indicates just under 1/4 wave of astigmatism. There is also a small amount of coma (see Figure 3) in this interferogram. Coma indicates an error in the alignment between the mirror and the auto-collimating test flat. To a certain extent, coma and astigmatism are coupled together in the sense that both can be induced by alignment error. Therefore it is very important that the alignment of the mirror and flat be adjusted such that the coma term is less than 1/4 wave before performing the final interferometric certification. Otherwise the mirror will exhibit an alignment induced astigmatism term that is not inherent to the mirror's wavefront.		Figure 2. - Astigmatism
The serpentine, or S-shaped, appearance of the fringes in the interferograms shown in Figure 3 indicate coma. This mirror has just under 1/4 wave of coma which, as mentioned above, is induced by misalignment between the mirror and the test flat. In other words, the coma is not inherent to the mirror's wavefront. The coma term is a very useful metric during the optical alignment process due to its direct correlation to the accuracy of the alignment. If the coma term is low, the alignment is good and the interferometric results will give an accurate representation of the mirror's wavefront quality. Since the residual coma in the final interferometric certification is due to residual misalignment and not to errors of the mirror's wavefront, the coma term is set to zero in the interferometric test report.		Figure 3. - Coma

Cosmetic Surface Quality Specification

Optics can have cosmetic defects such as small scratches and digs (pits) in the surface. Military specification, MIL-0-13830A, sets the standards for defining these surface blemishes. Two numbers specify the Scratch-Dig designation for an optic; the first number defines the allowable maximum scratch width and the second number defines the allowable maximum dig diameter. A scratch is defined as any marking or tearing of a polished optical surface. The maximum allowed scratch width is determined by comparison with traceable standard scratches. Designations for scratch widths are:

#10 scratch = 10 microns
#20 scratch = 20 microns
#40 scratch = 40 microns
#60 scratch = 60 microns
#80 scratch = 80 microns

If scratches with the maximum allowable width are present, their combined lengths cannot exceed one quarter of the part diameter.

A dig is defined as a pit on the polished optical surface. Digs numbers are defined by the dig diameter measured in hundredths of a millimeter. The diameter of an irregular shaped dig is defined as ½ x (length + width). Designations for dig diameters are:

#5 dig = 0.05 mm
#10 dig = 0.10 mm
#20 dig = 0.20 mm
#40 dig = 0.40 mm
#50 dig = 0.50 mm

The designation 80-50 represents a commonly acceptable cosmetic standard. 60-40 represents an acceptable standard for most scientific research applications. 10-5 represents a precise standard for very demanding laser applications. The Optical Mechanics OMI mirror Scratch-Dig ratio specification is 40-20.

For additional information please feel free to contact us via telephone or E-mail at:

Tel. (319) 351-3960
Fax (319) 351-3943

jcmulherin@opticalmechanics.com (James Mulherin)

About the Author

James Mulherin is the president of Optical Mechanics and a master optician. James manages the optics shop at Optical Mechanics and is a leading expert in the design and fabrication of astronomical optics.