Aberrations and the Benefit of Their Correction in Confocal Microscopy - Adaptive Optics for Biological Imaging - page 53

If we denote the axial position of the pupil plane of the lens with z 0 , the refractive index distribution of
the unaberrated case with n x y z

, and the aberrated case with n x y z , we may deine the aberration

0 ( ,

, ) ( , , )

function ψ( x , y ) in the pupil plane of the lens by

P x ( , y , )

0

2 π

λ

( )

d

∫

ψ ( , ) x y = n x y z ( , , ) − n x y z w ( , , ) (4.2)

0

F

It should be noted that the refractive index distributions describe the whole optical system including
the lenses. herefore, the aberration function ψ( x , y ), which is measured in the following experiments,
takes into account all types of aberrations relative to an ideal optical system. For a perfect system, ψ( x , y )
would be constant within the pupil plane of the lens.

4.3 Zernike Mode Analysis of the Aberrated Wavefront

To describe the aberration present in the wavefront, it is convenient to use a Zernike mode representa-
tion, which is a decomposition of the aberration function ψ( x , y ) into Zernike polynomials that may be
expressed as follows:

∞
∑ 1

ψ θ ( , ) r = M Z r

i

( , ) θ (4.3)

i

i =

where the modal coeicients M i describe the strength of each Zernike polynomial within the wavefront.
Since the Zernike polynomials Z i ( r ,θ) (Born and Wolf 1983; Noll 1976)* are a set of orthogonal functions
deined over the unit circle, we may write

1 2 π

1

∫ ∫

M = ψ θ ( , ) r Z r ( , ) d d θ r θ r (4.4)

i i

π

0 0

he set of Zernike coeicients M i fully describes the aberration function ψ( r ,θ). Some Zernike modes
correspond to the classical terms of abberations; for instance, astigmatism (modes 5 and 6) or coma
(modes 7 and 8). In our data analysis, we characterized the wavefront by the Zernike modes 2 through 22.
Several deinitions with diferent normalization factors and mapping schemes for the indices exist. In
this chapter, we use the following deinition (Neil et al. 2000):

m < 0 :

:
:

2 R

n m

−

( )sin( r − m θ )

Z r

n m

( , ) θ =

m
m

=
>

0 0
0

(4.5)

2 R ( r )cos( m )

n m

θ

( n m − )/ 2

( − 1 ) (

s

n s − )!

∑

R r

n m

( ) = n + 1 r

n − 2 s

s n !(( + m )/ 2 − s ))!(( n m − )/ 2 − s )!

s = 0

where the indices m and n , restricted to the conditions n ≥ | m | and n − | m |, are even. he rules to map
the double indices ( n , m ) to the single index i are i starts at 1 for n = 0 and rises irst with n , then all
allowed values of m are ordered with rising magnitude where the positive values come irst. he irst 22
polynomials Z r

n m ( , θ are listed in Table 4.1 , and a few modes are plotted in Figure 4.2 . Note that some

* Note that the normalization and indexing are diferent between the deinitions in Born and Wolf (1983) and Noll (1976).