Synopsis of a published paper “Strehl ratio for primary aberrations:
some analytical results for circular and annular pupils”
By Sheng Yuan
Oct 28, 2006
Abstract
In this synopsis paper, I am going to introduce prof. Virendra N. Mahajan’s paper “Strehl
ratio for primary aberrations: some analytical results for circular and annular pupils 1 ”.I
am going to present the key result of this paper and I will also infer who would want to
use it and which type of application it will be used.
1. Introduction:
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, S  I (0) / I (0)0 , here  is phase
aberration.
In this paper, the author obtains simple analytical expressions for the Strehl ratio of
images formed by imaging systems with circular or annular pupils aberrated by primary
aberrations(third order). Both classical (Seidel aberration) and balanced (orthogonal,
Zernike) primary aberrations are considered. Numerical results are obtained from these
expressions for up to five waves of a classical aberration. These result are compared with
the approximate results obtained by using the Marechal formula ( S1  (1 
 2
)2 ) to
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determine its range of validity. It is shown that, in general, a long as the Strehl ratio is
greater than 0.6, the Marechal formula gives results with an error of less than 10%.
The author also checked the famous Rayleigh quarter-wave rule--that the aberration
tolerance for a Strehl ratio of 0.80 is a quarter-wave. The Strehl ratio is calculated for a
primary aberration when its aberration coefficient, peak absolute value, and peak-to-peak
value, are each a quarter-wave. It is shown that only a quarter-wave of spherical
aberration and a quarter-wave peak value of balanced spherical aberration give a Strehl
ratio of 0.80. A quarter-wave of coma and astigmatism do not give a Strehl ratio of 0.80.
The author thus concluded that the Rayleigh quarter-wave rule does not provide a
quantitative, but only a qualitative, measure of aberration tolerance.
The author also considered non-optimally balanced aberrations. It is shown that, unless
the Strehl ratio is quite high, an optimally balanced aberration (in the sense of minimum
variance) does not give the maximum possible Strehl ratio. As an example, spherical
aberration is discussed in detail in this paper. A certain amount of spherical aberration in
balanced with an appropriate amount of defocus to minimize its variance across the pupil.
However, it is found that minimum aberration variance does not always lead to a
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maximum of Strehl ratio. The author find out for the circular pupil chase, only for less
than 2.3 waves of spherical aberration, the maximum Strehl ratio is obtained with
balanced spherical aberration(with defocus). Otherwise, minimum variance does not
corresponding to minimum Strehl ratio. In the case of coma, the situation is even worse,
the author find out only for very small coma aberrations(less than 0.7 waves) does
balanced coma give the maximum Strehl ratio; otherwise a non-optimal value of tilt gives
the maximum Strehl ratio.
2. Key result:
From the definition of Strehl ratio,
S=
1
2
2
2
1
 e
i (  , )
0
 d  d
(1)
0
By expanding the complex exponential of equation (1) into a power series and keep the
first 2 terms only for small aberration, the author derived the following approximated
expression for Strehl ratio (Marechal formula),
 2
)2 ,   is the standard deviation of phase aberration
(2)
2
Now neglecting the   4 term in formula (2), the author provided another famous
approximated expressions for Strehl ratio,
S2 1    2 ,
(3)
S1
(1 
If the pupil is annular, then the definition of Strehl ratio changes to,
1
S= 2
 (1   2 )2
2
2
1
  e
i (  , )
 d  d
(4)
0
Using formula(1)-(4), the author calculated the primary aberrations and their
corresponding standard deviations and the Strehl ratios for both circular and annular pupil,
as listed blow in Table 1.Both classical and balanced aberration(in the sense of minimum
variance) are considered. The corresponding orthogonal (Zernike) aberrations are given
in Ref 6 .Note that the form of an orthogonal aberration is identical with that of a
balanced aberration, except in the case of a rotationally symmetric aberration, in which
case it differs by a constant (independent of pupil coordinate) term. Since the variance or
the Strehl ratio for an aberration does not depend on a constant aberration term, the
study in this paper of a balanced aberration is equivalent to that of an orthogonal
aberration.
Using the expressions given in Table 1, the author then go ahead compared formula (1)
and (2), numerically calculated how the Strehl ratio varies with the coefficient of primary
aberrations up to 5 waves. The variance of the percent difference 100(S-S 1 )/S for circular
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pupil is shown blow in Fig 1 as an example. (To see all other calculations, please refer to
the paper itself)
Fig1. Percent error 100(S-S 1 )/S as a function of S for circular pupil
From Fig1, it is easy to see for S  0.8 ; the error is negligible and practically insensitive
to the type of aberration being considered. The error is less than 10% for S  0.6 . Thus
the author concluded that as long as    0.67 , or the standard deviation of the wave

aberration  w  ( / 2 )  
, S1 gives Strehl-ratio results with less than 10% error to
9.4
S.
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The author then go ahead examined the famous Rayleigh quarter-wave criterion, namely,
that the quality of an aberrated image will be good if the absolute value of the aberration
at any point on the pupil is less than a quarter-wave. Table 2 shows the values of the
Strehl ratio for primary aberrations of absolute peak value of a quarter-wave.
Table 2 Strehl ratio for primary aberrations of absolute peak value of a quarter-wave
From this table it is easy to see, that except for spherical aberration (the example
provided by Rayleigh) and balanced spherical aberration, a Strehl ratio of 0.8 is not
obtained for a quarter-wave of peak absolute value (or peak to peak value) of a primary
aberration. Thus the Rayleigh quarter-wave rule does not provide a quantitative measure
of the quality of an aberrated image. At best, it provides an indication of a reasonable
good image.
Finally, the author examined the Strehl ratio for non-optimally balanced aberration. He
found out for large aberrations, there is no simple relationship between the Strehl ratio
and the aberration variance. For the case of Spherical aberration, the author calculated the
following result as shown in Fig 2 blow.
From fig 2, it is easy to see, minimum aberratio variance does not necessary means
minimum Strehl ratio.
3. Discussion:
Prof Virendra N. Mahajan published a series of papers on Strehl ratio, classical aberration
expansion Vs. Zernike polynomial in aberration balancing 18 . This paper is pretty much a
basis of what he is going to discuss in his other papers, because in this paper, different
models for calculating Strehl ratio is examined , and the detail relationship between
aberration variance and Strehl ratio is provided. From here, we are able to discuss how
we can make use of Zernike polynomial to minimize the aberration variance thus benefit
in aberration balancing.
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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. This paper provided a very useful and complete reference for this kind of
application.
Fig 2.
Reference:
1. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and
annular pupils,” J. Opt.
Soc. Am. 72, 1258–1266 (1982), Errata, 10, 2092 (1993); “Strehl ratio for primary aberrations in
terms of their
aberration variance,” J. Opt. Soc. Am. 73, 860–861 (1983).
2. V. N. Mahajan, “Symmetry properties of aberrated point-spread functions,” J. Opt. Soc. Am.
A11, 1993–2003 (1994).
3. V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. 2, 833–846
(1985).
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4. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt.
Soc. Am. 71, 75–85(1981); 71, 1408 (1981); 1, 685 (1984); “Zernike annular polynomials and
optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994).
5. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction,
obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
6. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular
pupils,” Appl. Opt. 33,8121–8124 (1994); and “Zernike polynomials and optical aberrations,” Appl.
Opt. 34, 8060–8062 (1995).
7. V. N. Mahajan, “Zernike-Gauss polynomials for optical systems with Gaussian pupils,” Appl.
Opt. 34, 8057–8059(1995).
8. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, SPIE Press,
Bellingham,Washington (2001).Proc. of SPIE Vol. 5173 17
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