Image Processing Reference

In-Depth Information

Effects of Turbulence on a Wavefront

19

that occurs when the image screen is shifted from the ideal position. Some common

terms for aberrations are defocus, coma, astigmatism, and distortion, as identified

by Seidel in 1856. (Mahajan 1991; Smith 2000).

Optical aberrations are described within mathematical formalisms, the most

common being Zernike polynomials (Born and Wolf 1999). Zernike polynomials

provide a set of expressions that can be combined to describe any aberration. Each

polynomial form is independent from the others, forming an orthonormal basis set

of expressions that is a complete description of the observed aberration. Several

Zernike polynomials can be combined to produce a composite aberration, and as a

result all wavefronts can be completely represented. Conversely, a complex wave-

front can be decomposed into its individual Zernike polynomials.

Zernike polynomials contain a hierarchy of complexity. The simplest or low-

est-order Zernike polynomial is a uniform phase change across a wavefront, known

as piston. The next-lowest-order aberration comes as a pair known as tip and tilt,

which are angular changes in the wavefront. These three Zernike polynomials do

not change the shape of wavefront locally, but rather change the overall position or

angle of propagation of the wavefront. Tip and tilt are commonly encountered

when changing the angle of a mirror to an object and observing the positional

change in the reflection.

Lower-order Zernike polynomials can also be thought of as smoothly varying

in phase across a wavefront. As the Zernike order increases, the phase variations

over the wavefront increase on smaller spatial scales. Such higher-order modes can

be visualized as a bending or warping of the wavefront into shapes more like those

of potato chips; that is, a collection of curves in the surface of the wavefront. Table

2.1 shows a series of Zernike polynomials and Fig. 2.8 shows the corresponding

shapes of these polynomials.

Zernike polynomials provide a notation for describing an aberration, allowing

very complex aberrations to be constructed from a superposition of specific poly-

nomials of varying amplitude. The notation for aberrations is not standardized, so it

is important to be consistent in how one identifies the Zernike polynomials. Fig-

ure 2.10 shows one form of the notation.

Table 2.1
Zernike Polynomals.

|m|/n

1

2

3

4

0

rsin
θ

2rcos
θ

1.73(2r
2

r
2

2.236(6r
2

6r
2
+1)

1

1)

2r)sin
θ

2.83(3r
2

2r)cos
θ

r
2
cos2
θ

2.4r
2
sin2
θ

r
4

3r
2
)sin
θ

2

3.162(4r
4

3r
2
)cos
θ