With apologies to Gertrude Stein, this month we address the question of whether the image degradation resulting from a quarter wave of wavefront deformation is the same regardless of the type of aberration causing the deformation. Let us consider the effect of a quarter wave of deformation arising from different types of aberration on the performance of a system as gauged by the modulation transfer function (MTF; see **oe**magazine, March 2004, p. 40).

A widely used form of the MTF for this purpose is one in which we normalize the spatial frequency scale to make the cutoff frequency V0 equal to one. The cutoff frequency represents the absolute diffraction limit on the spatial frequency that an optical system can pass, and is defined as:

V_{0} = 2 NA/λ

= 1/λ(*f*/#)

where NA is the numerical aperture (NA = n sin U) of the lens, *f*/# is the *f*-number, and λ is the wavelength.

*Normalized MTF for system with λ/4 wavefront error (top) shows that the source of wavefront distortion is as important as the amount. The curve showing transverse third-order spherical aberration represents a system focused midway between the marginal and paraxial foci; the curve showing transverse zonal spherical aberration represents a system with the marginal spherical corrected and focused 75% of the way from the paraxial to the zonal focus. The curve showing coma represents the average of the sagittal and tangential MTFs; the tangential coma is 1.5λ/NA. Bottom plot identical to the top except for λ/2 wavefront error shows significant performance deviation depending on the source of the error.*

Consider a plot of normalized MTFs for an optical system with aberrations that produce a quarter wave of peak-to-peak (or peak-to-valley) wavefront deformation (see figure). All of the curves shown are at the focus, or reference position, which minimizes the peak-to-peak wavefront deformation; all curves correspond to deformations of exactly a quarter wave under these conditions.

We can see that the image degradation resulting from the simpler, low-order aberrations such as defocus, third-order coma, or spherical aberration is quite similar. When we look at the higher-order deformations (e.g., the green line), however, we see a noticeable difference, especially at the lower frequencies. The lesson is simple: The higher the order of the aberration, the greater the difference in MTF between the various aberrations; for lower orders, however, a quarter wave is a quarter wave.

Another useful measure of image quality is the Strehl ratio, which is the illumination at the center of an aberrated point-spread function relative to the illumination for a perfect lens. A Strehl ratio of 0.8 corresponds to a quarter wave of defocusing and is most reliable when the image quality is quite good. The Strehl ratio also turns out to be equal to the volume under the 3-D MTF plot, normalized by the volume under the MTF plot for a perfect lens. Some workers approximate this by looking at the area under the MTF curve, weighting the high frequency areas a bit more than the low. If we apply this test to our plots in the figure, it's apparent that the Strehl ratio is not too different for any of these aberrations. So, while our rose may begin to look just a little like a dahlia, at this point it still smells like a rose.

If we increase the aberrations so that they produce a half wave of wavefront deformation, however, the differences become much more apparent. For larger amounts of aberration, then, the type does count. For a quarter wave or less of error, the wavefront deformation looks like a very reliable metric with which to evaluate the quality of the image formed by an optical system, and, indeed, a quarter wave is a quarter wave is a quarter wave. **oe**

*Further Reading*

*W. Smith, Modern Optical Engineering, Third Edition, SPIE Press, McGraw-Hill, New York, NY (2000).*

**Warren Smith**

*Warren Smith is chief scientist and consultant for Kaiser Electro-Optics in Carlsbad, CA. *