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Strehl ratio,
wavefront power series expansion &
Zernike polynomials expansion
in small aberrated optical systems
By Sheng Yuan
OPTI 521
Fall 2006
Introduction
The wave aberration function(OPD), W(x,y), is defined
as the distance, in optical path length, from the
reference sphere to the wavefront in the exit pupil
measured along the ray as a function of the transverse
coordinates (x,y) of the ray intersection with a
reference sphere centered on the ideal image point. It
is not the wavefront itself but it is the departure of
the wavefront from the reference spherical wavefront
(OPD)
Wave aberration function
Exit
Pupil
y
x
z
Wave
Aberration
W(x,y)
Aberrated
Wavefront
Actual Ray
(normal to Aberrated Wavefront)
Reference
Spherical
Wavefront
Image
Plane
Strehl Ratio
• Strehl ratio is a very important figure of merit
in system with small aberration, i.e.,
astronomy system where aberration is almost
always “well” corrected, thus a good
understand of the relationship between Strehl
ratio and aberration variance is absolutely
necessary.
Defination of Strehl Ratio
• For small aberrations, the Strehl ratio is defined as
the ratio of the intensity at the Gaussian image point
(the origin of the reference sphere is the point of
maximum intensity in the observation plane) in the
presence of aberration, divided by the intensity that
would be obtained if no aberration were present.
How to calculate Strehl ratio?
How to calcuate wavefront variance?
Power series expansion of
Aberration function
What is the problem with power series
expansion?
How can we solve this coupling
problem?
• If we can expand the aberration function
(OPD) in a form that each term is orthogonal
to one another!!
• Zernike Polynomial in the orthogonal choice!
Why Use Zernike Polynomials?
What is the unique properties of
Zernike Polynomials?
How Zernike Polynomials looks like?
Zernike Polynomials expansion of
Aberration function (OPD)
How the variance of the aberration
function looks like now?
Is Zernike Polynomials Superior
than Power Series Expansion?
• Why don’t we use Zernike Polynomials
always?
• Why don’t we abandon the classical power
series expansion?
Comparison of both expansion
• Zernike Polynomials can only be useful in
circular pupil!!
• Power series expansion is an expansion of
function, have nth to do with the shape of
pupil, thus it is always useful!!
Reference