13. Wavefield aberrations#
As we have seen in Eq. (psf4f)
the OTF is the Fourier transform of the complex-valued pupil function. In case of a uniformly filled pupil and constant wavefield phase this corresponds to an Airy profile PSF. But what if the pupil is only partially filled or if the phase is not constant? In this case we speak about non-optimal imaging. In general, partly filled pupils are not considered as abberation, but a non-constant phase is. In this case we speak of wavefield aberrations that lead to a deterioration of the performance of an optical imaging system.
13.1. General wavefield aberrations#
The PSF of an ideal imaging system is derived using the transmission function for a perfect (paraxial) lens as
with
13.1.1. Defocus aberration#
Consider an imaging system as shown in Fig. Fig. 13.1 that normally focuses a distance

Fig. 13.1 Imaging geometry of a lens in case of defocud imaging by a distance
When the image is taken a distance
Hence, the transfer function
Using the approximation
which, when compared to the conventional definition for an aberration
Equation (13.2) shows that defocus is associated with a parabolic phase profile over the pupil plane. This is not entirely surprising since it is such a parabolic phase profile that leads to the focusing process itself.
One might wonder whether defocus is actually a true aberration. You could just as well place the detector at the image plane located at
13.2. Image quality metrics#
There are a variety of metrics that can be used to quantify the quality of an imaging system. Here the most import ones that can be captured in a single parameter are discussed.
13.2.1. Strehl ratio#
The Strehl ratio
This is shown graphically in Figure Fig. 13.2(a).

Fig. 13.2 (a) Definition of Strehl ratio in the image plane. (b) Definition of the Strehl ratio in the Fourier space.#
Since
The on-axis PSF without abberations is
In the presence of aberrations it always follows that
For relatively small aberrations we can make the approximation
and the wavefront variance
the Strehl ratio can be written out as
This means that any spatially varying aberration leads to a reduction of the PSF height. In another definition of the PSF one knows that for incoherent imaging
Whereas the previous definition of the Strehl ratio was defined as the ratio of the height of the PSF, in this definition the Strehl ratio can be interpreted as the ratio of integrated frequencies as shown graphically in Figure Fig. 13.2(b). Similarly, the Strehl ratio for the coherent case also can be defined although this is less common and will not be discussed here.
13.2.2. Maréchal criterion#
At zero phase all parts of the wavefront interfere constructively at the focal point leading to optimal interference and the heighest PSF. When the phase deviates from zero, constructive interference of different parts of the wavefront is less than ideal. When the aberrations increase parts of the wavefront start to destructively interfere when
which is known as the Rayleigh criterion. For the particular case of defocus aberration and the condition
and hence puts a lower limit on the Strehl ratio according to
Obviously, the precise type of the aberration determines the exact Strehl ratio but Maréchal found that the Strehl ratio limit is always close to 0.8 for any aberration.
13.3. Zernike polynomials#
Instead of using an analytical expression for the wavefront aberration an alternative way is to decompose the wavefront into fundamental modes. One of the most common modal decompositions is in Zernike modes, invented by Dutch physicist Frits Zernike (nobel prize 1953). Zernike modes have properties such as orthogonality and symmetry that make the very suitable for describing arbitry wavefields. The general wavefield decompositions in Zernike modes is given by

Fig. 13.3 The unit circle with the definitions of
The Zernike modes are defined as
with
The Zernike polynomials are defined by
Note that there also exist a Noll index starting at 1 and that Zernike modes are also stated without normalization. Here we keep to the stated definition. Below is a table with the first 10 Zernike modes in polar and cartesian coordinates. For cartesian coordinates it is required that
Figure Fig. 13.4 shows the plots of the Zernikes modes represented above. Increasing rows represent increasing radial order. From left to right are the modes with varying azimuthal order.

Fig. 13.4 Plot of the first ten Zernike modes with Noll indexing.#
The great strength of a Zernike decomposition is that various properties of the wavefield can be determined quite easily without having to apply the calculations on the wavefield itself. For example, the mean of the wavefield is
Equation (13.6) shows that all Zernike polynomials, besides the piston, have zero mean. The piston wavefield is constant over the unit circle, hence, the mean of the wavefield is equal to coefficient of the piston. If the piston coefficient is zero, the entire wavefield has zero mean. The average of the square of the wavefield over the unit circle is
Since
Again, the strength of the Zernike polynomials comes from the fact, for example, the Strehl ratio of an aberrated wavefield described by Zernikes is easily calculated from the
Consequently, any wavefield on a pupil can be decomposed in a number of Zernike modes that have coefficients
In describing the quality of optical imaging, the Zernike polynomials can be used to describe the aberrations in the pupil. In the developed theory this would give a space-invariant PSF irrespective of the image point. However, note that this is based on the paraxial approximation. For points far away from the optical axis the paraxial approximation breaks down and the the total aberration can be described, but then with a Zernike decomposition that depends on the image coordinate.