# RMS Error and the Strehl Ratio

#### A short explanation of how you can go from one to the other

It's often pointed out that RMS (root mean squared) wavefront error is a better characterization of optical quality than peak-to-valley wavefront error. The latter only tells you the difference between the "highest" and "lowest" parts of the wavefront, while the former tells you how much the "height" varies across the entire wavefront.

The Strehl ratio is another expression of optical quality. Under perfect conditions, a telescope's objective focuses light from a distant star to a spot called the Airy disc. This disc has some definite, positive size, not because the telescope isn't perfectly made, but because of the wave nature of light. (See my essay, "Diffraction," for more on this.) The brightest part of the Airy disc is at its center, naturally enough. The more light the optics bring to the center of the Airy disc, the sharper the image is. (Think of how it's easier to draw a crisp picture with a sharp pencil than a thick lump of charcoal.) The Strehl ratio is the amount of light that the optics put in the center of the Airy disc, divided by the amount of light that would have been put there, if the optics were perfect. Therefore, a Strehl ratio of 1.0 signifies perfect optics.

Even though they seem like two entirely different ways to think about optical quality, they are actually quite closely related when the optical quality is high. Specifically, for any given set of optics, the RMS wavefront error w and Strehl ratio r are related by

r = 1 – 4 pi2 w2

provided that w is sufficiently low (small error) and r is sufficiently high. A common criterion for this is that w < 1/14 and r > 0.80; as it happens, these are called the Marechal and Strehl criteria for diffraction-limited optics, respectively. And if you plug in those two values, you find that the relationship above approximately holds for them. Where on earth does that relationship come from?

To explain that, let's consider what RMS really means. It means that if you take the square of the wavefront error, average it over the entire wavefront, then the square root of that average is the RMS error. (Hence, "root mean square.") If we denote the wavefront error at any point on a wavefront A by u, then the square of the error is u2, and

w2 = (IntA u2 dA) / m(A)

where Int signifies the integral operation, and m(A) is the area (more precisely, solid angle) covered by the wavefront. On the other hand, the Strehl ratio is determined by how much in phase the wavefront arrives at the focal point. The relative amplitude Z at the center of the Airy disc (compared to perfect optics) is obtained by integrating cos u over the entire wavefront. That is,

Z = (IntA cos u dA) / m(A)

So far, the two formulae look similar, but we seem stuck. However, whenever the optical quality is "sufficiently high," the error u should be relatively small, and in that case, we can use the approximation

cos u = 1 – u2 / 2

In that case, the expression for Z becomes

Z = (IntA (1 – u2 / 2) dA) / m(A)

Breaking the integration into parts, we then get

Z = (m(A) – (1/2) IntA u2 dA) / m(A) = 1 – (IntA u2 dA) / 2 m(A)

Notice that the fraction on the right is basically our expression for w2, so we can finally write

Z = 1 – w2 / 2

Now, in order to use the approximation for cos u, the variable u must be expressed in radians. That means that w too must be expressed in radians. But wavefront errors are generally not expressed in terms of radians, but in wavelengths. Since a wavelength is equal to 2 pi radians, the proper expression of the previous formula for wavelengths (instead of radians) is

Z = 1 – 2 pi2 w2

We're not quite done yet. The Strehl ratio is not the ratio of amplitudes, but the ratio of intensities, and intensity is the square of amplitude. So

r = Z2 = (1 – 2 pi2 w2)2 = 1 – 4 pi2 w2 + 4 pi4 w4

When optical quality is high, w is small, and the last term is also small. (For example, for the Marechal criterion w = 1/14, the last term is 0.01.) We can therefore omit it without much error, which leaves us with

r = 1 – 4 pi2 w2

Q.E.D.