**Rotationally Invariant Polynomials:**

These polynomials must have m 0 .

The set of rotationally invariant polynomials that exist in the current set of 36 Zernike

terms described in ISO 10110-5, Appendix A.3, first edition are as follows:

Z

2,0

is

Z

3

r , 2r

2

1

Z

4,0

is

Z

8

r , 6r

4

6r

2

1

Z

6,0

is

Z

15

20r

6

30r

4

12r

2

1

Z

8,0

is

*Z*

24

*70r*

8

*140r*

6

*90r*

4

*20r*

2

*1*

Z

10,0

is Z

35

r , 252r

10

630r

8

560r

6

210r

4

30r

2

1

These polynomials have no azimuthal dependence and describe the wavefront aspheric

approximation f

wri

in section 3.2.6. Additional rotationally invariant terms can be

added to more accurately approximate the wavefront.

**Rotationally Varying Polynomials:**

These polynomials must have m 0 .

Rotationally varying polynomials have a radial function as well as a sine or cosine

function which dictates the order of the azimuthally varying function.

These polynomials come in pairs, having an equal value for n and the positive/negative

equal integer walue for m.

For example:

Z

2,2

is astigmatism about 0 degrees.

r

2

cos

Z

2, 2

is astigmatism about 45 degrees.

*r*

2

*sin*

A vector summation along with the arc tangent expression will give the magnitude and

the directio of the resultant pair of matching polynomials.

Directional Angle 1 m arctan Z

n , m

Z

n ,m

*Magnitude 2*

*Z*

n , m

2

*Z*

n , m

2

The Zernike Polynomial Term is written:

Z

n , m

where:

n = radial order

m = azimuthal order

n is an integer > 0.

n - |m| must be a non-negative even integer including 0.

If

n = 8; then m can equal 8, 6, 4, 2, -2, -4, -6, -8 only.

Definitions:

Z

n , m

is the coefficient value for the expression:

R

nm

r

sin m

for m is a negative number

R

nm

r

cos m

for m is a positive number

And further: R

nm

r is the radial function:

R

nm

r

s 0

l

1

s

n s

s l s n l s r

n 2s

where l n m

2

sin m a and cos m a are the azimuthal functions

Example:

Z

8, 4

(pronounced, Z eight minus four) would be the magnitude of the Zernike

coefficient for the following equation.

28r

8

42r

6

15r

4

sin 4

Which is an 8

th

order radial function and 4

th

order azimuthal function.