Indian Journal of Pure & Applied Physics
Vol. 51, December 2013, pp. 837-843
Zernike polynomiales for optical systems with rectangular and
square apertures of area equal to π
Ali Hadi Al-Hamdani* & Sundus Y Hasan**
*University of Technology, Laser & Optoelectonics Engineering Department, Baghdad, Iraq
**Physics Department, Education College for Girls, Kufa University, Najaf 54001, Iraq
*E-mails: [email protected], [email protected], [email protected]
Received 9 June 2013; revised 29 August 2013; accepted 11 September 2013
The property of orthogonality of Zernike circle polynomials and their representation of balanced spherical aberrations
made them in a widespread use for wavefront analysis. In the present paper, we derived closed form polynomials that are
orthonormal over horizontal and vertical rectangular pupils of an area equal to π just the same as the area of Zernike circular
pupil. Then the polynomials suitable for a square aperture with the same area are found, using the circular polynomials as
the basis functions. The polynomials are given in cartesian and polar coordinates. The values of standard deviation for
balanced and unbalanced primary aberration are also calculated for the concerned apertures and compared with that of
circular aperture.
Keywords: Zernike polynomials, Optical systems, Wavefront analysis, Spherical aberrations
1 Introduction
Most of optical systems have circular pupils, there
are some with noncircular pupils like annular,
elliptical,
rectangular,
square,,..etc.
Zernike
polynomials, suggested first by Zernike in his paper
of phase contrast method for testing circular mirrors1,
are suitable only for the unit circular aperture. A lot of
work has been done to extend these polynomials to
apertures other than circular2-5, to have benefit of their
properties
of
orthogonalization
over
the
corresponding aperture and representing balanced
classical aberration that yielding minimum variance
across the aperture.
In the present work, rectangular and square pupils
are of interest. An example of a system having these
pupils is the high-power laser beams4,6. These
apertures have been studied before to the fourth order
for rectangular and eighth order for square aperture by
Mahajan and Dai4. They took these apertures as a unit
figures inscribed inside a unit circle of radius equal to
one.
Orthogonal square polynomials have also obtained
by Bray7 who chose a square inscribed outside the
circle. In the present work, the rectangular and square
apertures are taken to be of an area equal to π which is
equal to the area of Zernike circle aperture of radius
equal to 1.
2 Zernike Polynomials for Noncircular Pupils
Zernike circle polynomials Zj(ȡ,ș) may be written
in the form8:
­° N m R m cos(mθ ) for m ≥ 0
Z ( r ,θ ) = ® n m n m
°̄ − N n Rn sin( mθ ) for m < 0
…(1)
where n and m are positive integers, |m|≤n, n−|m| is
even.
Rnm ( r ) = ¦ s = 0
( n −|m|)/ 2
(−1) s (n − s )!
r n−2 s
ª n+ | m | º ª n− | m | º
s !«
− s » !«
− s »!
¬ 2
¼ ¬ 2
¼
…(2)
And the normalization factor:
N nm =
2(n + 1)
1 + δ m ,0
…(3)
The index n represents the radial degree or the
order of the polynomial, since it represents the highest
power of r in the polynomial, and m is called the
azimuthal frequency. These polynomials can be easily
converted from polar coordinates to Cartesian
coordinates, and so the latter one can be used as a
INDIAN J PURE & APPL PHYS, VOL 51, DECEMBER 2013
838
basis for finding Cartesian Zernike polynomials for
rectangular and square apertures.
In Cartesian coordinates (x,y), the aberration
function W(x,y) for a noncircular pupil may be
expanded in terms of polynomials Fj(x,y) that are
orthonormal over the corresponding pupil9:
Fj =
Fj′
§1
2·
¨ ³ ³ A Fj ¸
A
©
¹
…(11)
3 Present Work
3.1 Rectangular aperture
W(x,y)=Σaj Fj(x,y)
… (4)
where aj is the aberration coefficient of the
polynomial Fj(x,y). The orthonormality of the
polynomials is represented by:
1
³ ³ A Fj ( x, y ) Fj ′ ( x, y )dxdy = δ jj ′
A
…(5)
and the coefficient aj is given by:
aj =
1
³ ³ A W ( x, y ) Z j ( x, y )dxdy
A
…(6)
where the double integration is over the area of the
aperture.
It is evident that the value of coefficient is
independent of the number of expansion polynomials;
this is the result of the property of orthogonality. So,
the mean value of each polynomial is zero except the
first one is unity, and hence, the mean value and mean
square value of the aberration function10 are:
¢W(x,y)²=F1
…(7)
¢[W(x,y)]2²=Σj=1Fj2
…(8)
In the present paper, the rectangular pupil is of area
equal to π. As shown in Fig. 2, the half widths of the
rectangle along the x and y axes are 1 and π/4,
respectively for horizontal rectangular aperture
[Fig. 1(a)], while it is π/4 and 1 for vertical
rectangular aperture [Fig. 1(b)]. So, by using Zernike
polynomials for circular aperture and Gram Schmidt
orthgonalization method, and by programming
equations 10 and 11with MATLAB code using the
limits of integration shown in Fig. 1, the orthonormal
rectangular polynomials for horizontal and vertical
apertures have been obtained, up to order 5 and
presented in Tables 1 and 2, respectively in Cartesian
coordinates. Then these polynomials were converted
to polar coordinates as presented in Tables 3 and 4,
respectively.
3.2 Square aperture
The square aperture of area equal to π, is shown in
Fig. 2. Where the half widths along x and y are equal
to
/2. The orthonormal polynomials in Cartesian
coordinates for this aperture up to order five are
presented in Tables 5 and 6 which show these
polynomials in polar coordinates.
3.3 Standard deviation
The Seidel aberrations may be represented in
Cartesian coordinates by:
and the variance is:
V = ¢[W(x,y)]2²−¢W(x,y)²2=Σj=2Fj2
…(9)
The orthogonal polynomials Fj(x,y) can be obtained
from the circle polynomials Zj(x,y) using Gram–
Schmidt orthogonalization process11. Using an
abbreviated notation, where we omit the argument
(x,y) of the polynomials, we may write:
W(x,y)=W0+W11x+W20(x2+y2)+W40(x2+y2)2
+W31[(x3+y2x)+W22x2 ]
… (12)
F1=Z1=1
Fj′+1 = Z j +1 − ¦ i =1
j
³ ³ A Z j Z j +1
³ ³ A Z i Z i +1
while the normalization
Zi
…(10)
Fig. 1 — Rectangular aperture of an area equal ʌ, (a) Horizontal,
(b) Vertical
AL-HAMDANI & HASAN: ZERNIKE POLYNOMIALES FOR OPTICAL SYSTEMS
where the terms of this equation represent the piston,
tilt, focus, astigmatism, and spherical aberration,
respectively3.
The obtained Zernike polynomials are representing
balanced classical aberrations, which make the
Table 1 — Orthonormal polynomials for horizontal rectangular
pupil of area equal to π and half widths along x and y axes are
1 and π/4,respectively, in Cartesian coordinates
839
variance minimum. As stated before, standard
deviation (SD) is the square root of the variance
[see Eq. (9)]. In present work, the values of S D of the
primary (Seidel) balanced and unbalanced aberrations
have been calculated for the concerned apertures, and
compared with that of circular aperture in Table 7.
Table 3 — Orthonormal polynomials for vertical rectangular
pupil of area equal to π and half widths along x and y axes are
π/4 and 1 respectively, in Cartesian coordinates
No.
Polynomial
No.
Polynomial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
1.73x
2.20y
−1.54+2.86r2
3.82xy
1.75x2−4.61y2+0.37
(−4.32+5.36r2)x
(−4.56+6.48r2)y
1.86−9.21x2−8.69y2+7.26r4
3.53yx2−12y3+3.26y
3.79x3−7.69xy2+0.68x
xy(12.4(x2−y2)−2.85)
10.1x4−7.16x2−17.2y4+11.5y2−0.19−7.13x2y2
xy(7.69x2+20.2y2−12.1)
4.23x4−8.47x2y2+28.1y4+0.69−1.89x2−12y2
17.9x4y−35.8x2y3+3.58y5−2.56y−2.09yx2+9.49y3
11.5x5−23x3y2−8.15x3+32xy2−34.5xy4−1.16x
6.87y−28.8x2y−21.7y3+22.2x4y+26.9x2y3+18.7y5
7.55x−28.1x3−11.2xy2+21.5x5+20.1x3y2−1.58xy4
−3.1yx2+73.8y3+3.8x4y+1.04x2y3−108y5−9.45y
9.66x5−18.1x3y2+46.2xy4+1.56x−7.02x3−13.4xy2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
17
19
20
21
1
2.2x
1.73y
−1.54+2.86r2
3.82xy
4.61x2−1.75y2−0.37
−4.56x+6.48(x3+y2x)
−4.32y+5.36(x2y+y3)
1.86−8.69x2−9.21y2+7.26r4
7.69x2y−3.79y3+0.680y
12.0x3−3.53y2x−3.26x
12.4xy(x2−y2)+2.85xy
17.286x4+(−11.49+7.13y2)x2+0.19−10.1y4+7.16y^2
20.2x3y+7.69xy3−12.1xy
28.1x4+(−12.04−8.467y2)x2−1.89y2+4.23y4+0.69
17x4y+(11.2y−33.9y3)x2+3.07y3+3.39y5−2.6268y
8.4x5+(−22/125−16.8y2)x3+(−25.2y4−3.46+27.8y2)x
30.8x4y+(8y3−21.4y)x2+6.73y+20y5−23.7y3
8.36x−42.6x3−11xy2+43.2x5+39.1x3y2−3.76xy4
31.4x4y+(2.23y3−17.7y)x2−22.2y3−3.55y−20.5y5
66.6x5−16.6x3y2+15xy4+5.13x+39.9x3−6.45xy2
Table 2 — Orthonormal polynomials for horizontal rectangular pupil of area equal to π and half widths along x and y axes
are 1 andπ/4 respectively, in polar coordinates
No.
Polynomial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
1.73rcos(q)
2.2rsin(q)
−1.54+2.86r2
1.91r2sin(2q)
r2(−1.43−3.18cos(2q))+0.37
(−4.32+5.36r2)rcos(q)
(−4.56+6.48r2)rsin(q)
1.86−0.26r2cos(2q)−8.95r2+7.25r4
3.88r3sin(3q)+(3.26−12r2)rsin(q)
5.74r3cos(3q)+(0.68−3.79r2)rcos(q)
3.1r4sin(4q)−1.43r2sin(2q)
−0.19−9.33r2cos(2q)+13.64r4cos(2q)+2.17r2−3.56r4
−1.56r4sin(4q)+(−6.05+10.1r2)r2sin(2q)
0.68+22.02r4cos(4q)+5.05r2cos(2q)−72.43r4cos(2q)−17.05r2+94.4r4
3.58r5sin(5q)−2.56rsin(q)−2.9r3sin(3q)+6.595r3sin(q)
11.5r5cos(3q)−10.04r3cos(3q)+1.9r3cos(q)−1.16rcos(q)
6.87rsin(q)−1.88r3sin(3q)−23.48r3sin(q)+0.88r5*sin(5q)+17.82r5sin(q)
7.55rcos(q)−15.43r3cos(q)+4.23r3cos(3q)+4.11r5cos(q)−4.29r5cos(3q)−0.01r5cos(5q)
54.58r3sin(q)−19.23r3sin(3q)−66.9r5sin(q)+34.48r5sin(3q)−6.58r5sin(5q)+1.04r5−9.45rsin(q)
69.42r5cos(q)+4.54r5cos(3q)+4.62r5cos(5q)+1.56rcos(q)−10.37r3cos3(q)−10.05r3cos(q)
INDIAN J PURE & APPL PHYS, VOL 51, DECEMBER 2013
840
Table 4 — Orthonormal polynomials for vertical rectangular pupil of area equal to π and half widths along x and y axes
are π/4 and 1 respectively, in polar coordinates
No.
Polynomial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
2.2rcos(q)
1.73rsin(q)
−1.54+2.86r2
1.91r2sin(2q)
−0.37+(1.43+3.38cos(2q))r2
(6.48r2−4.56)rcos(q)
(5.36r2−4.16)rsin(q)
1.86+7.26r4+(0.26cos(2q)−8.95)r2
5.74r3sin(3q)+(−3.79r2+0.68)rsin(q)
−3.88r3cos(3q)+(0.35r2−3.26)rcos(q)
3.1r4sin(4q)+1.43r2sin(2q)
0.19+(3.55+13.65cos(2q)r4+(−2.17−9.33cos(2q))r2
(1.56sin(4q)+3.85sin(2q))r4−12.1r2sin(2q)
0.694+(13.11+11.94cos(2q)+7.14cos(4q))r4+(−6.96−5.07cos(2q))r2
(−10.17sin(3q)+54.29/16sin(5q)+3.39sin(q))r5+3.07sin(q)+2.03sin(3q)r3−2.62rsin(q)
(−16.8cos(q)−8.4cos(3q))r5+(20.81cos(q)+6.99cos(3q))r3−3.66cos(q)r
(−8sin(3q)+2.18cos(5q)+20sin(q))r5+(−23.7sin(q)+0.58sin(3q))r3+6.73rsin(q)
8.36rcos(q)−18.9r3cos(q)+7.9r3cos(3q)+8.12r5cos(q)−7.78r5cos(3q)+0.03r5cos(5q)
−326.177r5sin(5q)+12.44r5sin(3q)−40r5sin(q)−14.33r3sin(q)−9.98r3sin(3q)−3.55rsin(q)
6.66r5cos(5q)+19.77r5cos(3q)+11.59r3cos(3q)+2.81r5cos(2q)+46.31r5cos(q)−11.59r3cos(q)+5.13rcos(q)
Table 5 — Orthonormal polynomials for square pupil of area
equal to π and half widths along x and y axes are π1/2/2, in
Cartesian coordinates
Fig. 2 — Square aperture of area=π
To illustrate how these values are evaluated, an
example for calculating S D for balanced and
unbalanced astigmatism aberration for vertical
rectangular aperture will be taken:
The 6th polynomial is S6
S6=4.61 x2−1.75 y2 – 0.37= 6.36 x2−1.75 r2 – 0.37
(where r2=x2+y2) =6.36 x2−1.75/2.86 S4 +c S1
where c is constant, S1 and S4 are the1st and 4th
Zernike polynomials for the vertical rectangular
aperture.
S6/6.36= x2-1.75/(2.86*6.36) S4 +c1 S1
and this means that the astigmatism aberration is
balanced with focus error and the value of S D of
No.
Polynomial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
1.95x
1.95y
−1.59+3.02r2
3.82xy
3.03(x2−y2)
−4.58x+6.24xr2
−4.58y+6.24yr2
1.90−9.36r2+7.8r4
5.5x2y−7.06y3+1.9y
7.11x3−5.53y2x−1.91x
13.1xy(x2−y2)
14.9(x4−y4)−10(x2−y2)
(13.2r2−12.4)xy
13x4+4.39(−1.44−2.16y2)x2−5.97y2+12.6y4+0.71
18.1x4y−36.2x2y3+3.62y5−2.92y+4.81x2y+6.23y3
11.1x5−22.2x3y2−3.84x3+33y2x−33.3xy4−2.9x
7.21y−29.2x2y−24.6y3+32.1x4y+16.8x2y3+22.7y5
7.9x−3.72x3−6.77xy2+34.8x5+16.4x3y2−8.4xy4
−11.6x2y+37y3+14.2x4y+3.75x2y3−42.9y5−5.22y
30x5−21.4x3y2+24.8xy4+3.07x−20.6x3−6.59y2x
balanced astigmatism aberration is equal to
(σba=1/6.36=0.157). While The S D of astigmatism
aberration is found as follows:
x2= S6/6.36 + 1.75/(2.86*6.36) S4 –c1 S1
AL-HAMDANI & HASAN: ZERNIKE POLYNOMIALES FOR OPTICAL SYSTEMS
841
Table 6 — Orthonormal polynomials for square pupil of area equal to π and half widths along x and y axes
are π1/2/2, in polar coordinates
No.
Polynomial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
1.95rcos(q)
1.95rsin(q)
−1.59+3.02r2
1.91r2sin(2q)
3.03r2cos(2q)
2.67r3sin(3q)−0.84rsin(q)
2.67r3cos(3q)+0.84rcos(q)
1.9−9.36r2+7.8r4
(4.03cos(q)−1.67cos(3q))r3−4.91cos(q)r
(−1.67sin(3q)+9.11sin(q))r3−4.91sin(q)r
3.28r^4sin(4q)
(15r2−10.1)r2(cos(2q))
(−6.19+6.56r2)r2sin(2q)
0.71+(10.21−0.01cos(2q)+6.14cos(4q))r4+(−6.18+0.01cos(2q))r2
3.63r5sin(5q)−2.98rsin(q)−0.362r3sin(3q)+0.596r3sin(q)
(−22.28cos(q)−11.14cos(3q))r5+(23.78cos(q)+9.23cos(3q))r3−2.88cos(q)r
(24.56+4.7cos(2q)+6.59cos(q))sin(q)r5+(−27−2.7cos(2q))sin(q)r3+7.15sin(q)r
7.9rcos(q)−6r3cos(q)−0.76r3cos3q+6.78r5cosq+3.23r5cos3q+1.15r5cos5q
18.5r3sin(q)−2.03r5sin(5q)+16.3r5sin(3q)−37.77r5sin(q)−5.22rsin(q)
59.5r5cos(q)+16.7r5cos(3q)+4.7r5cos(5q)−19.1r3cos(q)+3.51r3cos(3q)+3.07rcos(q)
Table 7 — Comparison of standard deviation (σ) of the
balanced and unbalanced aberrations for circular, rectangular
and square pupils, all of an area equal to π.
(t-tilt, f-focus, ba-balanced astigmatism, a-astigmatism,
bc-balanced comma, c-comma, bs-balanced spherical,
s-spherical)
Standard
deviation
Circle
(Ref.4)
Horizontal
rectangular
Vertical
rectangular
Square
σt
σf
σba
σa
σbc
σc
σbs
σs
0.5
0.29
0.2
0.25
0.118
0.354
0.07
0.3
0.58
0.35
0.157
0.298
0.187
0.502
0.138
0.396
0.46
0.35
0.157
0.184
0.154
0.355
0.138
0.421
0.51
0.33
0.165
0.234
0.16
0.409
0.128
0.418
Now, the S D is equal to the square root of the
addition of the squares of the coefficients of Zernike
polynomials except the first (Eq. 9):
σa={(1/6.36)2+[1.75/(2.86*6.36)]2}0.5=0.184
and so on.
4 Discussion
To have a good comparison between Zernike
polynomials of different aperture shapes, it is better to
have the same areas. So, the apertures have been
taken to be of area equal to π. Zernike polynomials
will lose their beneficial properties when they are
used for other aperture than circle, but by finding the
orthonormal polynomials for these apertures, the
benefits of Zernike polynomials on these apertures are
returned, like the orthogonality which makes the
coefficient of the first polynomial represents the mean
value, while the other coefficients represent the
standard deviation of the corresponding polynomials,
and therefore, the variance represents the sum of the
squares of Zernike coefficients except the first. In
addition to that, the orthogonality makes the value of
coefficients does not depend on the number of
polynomials used, which makes it appropriate to
determine the first and third order aberrations or
higher.
The results in Tables (2, 4, 6) showed that each
polynomial is written with only one kind of
trigonometric function (sin or cos),which means that
there is a biaxial symmetry. And, in contrast with the
circular aperture where the polynomials can be
written in separable form of (r and q), in rectangular
aperture only the first six polynomials are separable
and for square aperture, in addition to the first six
there are the (9, 12, 13 and 14). So, the polynomial
numbering with two indices (n and m, see Eq. 1) lost
significance and they must be numbered with one
index, j.
Fig. 3 — Graphical representation of first 19 orthonormal square Zernike polynomials in 2D and 3D
842
INDIAN J PURE & APPL PHYS, VOL 51, DECEMBER 2013
AL-HAMDANI & HASAN: ZERNIKE POLYNOMIALES FOR OPTICAL SYSTEMS
Figure 3 shows graphical representation for
orthonormal square Zernike polynomials in 2D and
3D. Note, the spherical symmetry is obvious in 4th
polynomial (S4) and 9th polynomial (S9). And it is
clear that the complexity in figures is increase as the
number of polynomial increase, where the degree of
polynomial increase.
5 Conclusions
In present work, several conclusions can be drawn
as follows:
1 Common Zernike polynomials (Z P) are suitable
only for circular aperture of area equal to π or of
unit radius; these polynomials lost their important
properties when the shape of the aperture is
changed.
2 Z P for square and rectangular (vertical and
horizontal) apertures, like that of circular
aperture, are having biaxial symmetry, this is
clear from the ability of writing the polynomials
in terms of only one kind of trigonometric
functions (sin or cosine).
3 Circular Z P were written with two indices
(n and m), this cannot be done for square and
rectangular apertures because the polynomials
could not be written as separable functions of
(r and q) as that in circular aperture, so they were
written with one index j.
4 In rectangular aperture only the fourth polynomial
have spherical symmetry (r dependent only),
while for square aperture, in addition to the fourth
polynomial, the ninth polynomial is also of
spherical symmetry (see S4 and S9 in Fig. 3).
5
6
7
8
843
Tilt standard deviation (S D) for square aperture
is nearly equal to that of circular aperture, and it
is nearly equal to the mean of the two values of
vertical and horizontal vertical apertures, all of
the same area (π).
S D values of focus error, spherical, and balanced
spherical for circular aperture are smaller than
those of square and rectangular apertures. This is
because of the circular symmetry of the circular
aperture.
S D values of comma and balanced comma is
smaller for circular aperture than that for the other
apertures.
S D for balanced astigmatism values for square
and rectangular apertures are smaller than that of
circular aperture.
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