## Abstract

The problem of determining the orthonormal polynomials for hexagonal pupils by the Gram–Schmidt orthogonalization of Zernike circle polynomials is revisited, and closed-form expressions for the hexagonal polynomials are given. We show how the orthonormal coefficients are related to the corresponding Zernike coefficients for a hexagonal pupil and emphasize that it is the former that should be used for any quantitative wavefront analysis for such a pupil.

© 2006 Optical Society of America

### References

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1. http://scikits.com/KFacts.html.
2. R. Upton and B. Ellerbroek, Opt. Lett. 29, 2840 (2004).
[CrossRef]
3. C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987). From the statement on p. 348 it can be inferred that the real orthonormal polynomials are given by (1/2)(pn,k+pn,−k)/(1/2)||pn,k|| and (1/2i)(pn,k−pn,−k)/(1/2)||pn,k||. However, wrong answers are obtained in some cases, e.g., when k=0 or pn,k is real. For example, 2 does not apply in some cases, such as p2,0 and p4,0. Similarly, p3,3 and p3,−3 are real and unequal, and p3,3+p3,−3 does not yield the correct form of the polynomial. Instead, p3,3/||p3,3|| yields our polynomial H10 and p3,−3/||p3,−3|| yields our polynomial H9. There are other mistakes as well. For example, p4,4 and ||p4,4||2 should equal z4 and 319/3150, respectively.
[CrossRef]
4. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
5. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
[CrossRef]
6. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE, 2004).
7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).
8. V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
[CrossRef]

#### 2003 (1)

V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
[CrossRef]

#### 1987 (1)

C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987). From the statement on p. 348 it can be inferred that the real orthonormal polynomials are given by (1/2)(pn,k+pn,−k)/(1/2)||pn,k|| and (1/2i)(pn,k−pn,−k)/(1/2)||pn,k||. However, wrong answers are obtained in some cases, e.g., when k=0 or pn,k is real. For example, 2 does not apply in some cases, such as p2,0 and p4,0. Similarly, p3,3 and p3,−3 are real and unequal, and p3,3+p3,−3 does not yield the correct form of the polynomial. Instead, p3,3/||p3,3|| yields our polynomial H10 and p3,−3/||p3,−3|| yields our polynomial H9. There are other mistakes as well. For example, p4,4 and ||p4,4||2 should equal z4 and 319/3150, respectively.
[CrossRef]

#### Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

#### Dunkl, C. F.

C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987). From the statement on p. 348 it can be inferred that the real orthonormal polynomials are given by (1/2)(pn,k+pn,−k)/(1/2)||pn,k|| and (1/2i)(pn,k−pn,−k)/(1/2)||pn,k||. However, wrong answers are obtained in some cases, e.g., when k=0 or pn,k is real. For example, 2 does not apply in some cases, such as p2,0 and p4,0. Similarly, p3,3 and p3,−3 are real and unequal, and p3,3+p3,−3 does not yield the correct form of the polynomial. Instead, p3,3/||p3,3|| yields our polynomial H10 and p3,−3/||p3,−3|| yields our polynomial H9. There are other mistakes as well. For example, p4,4 and ||p4,4||2 should equal z4 and 319/3150, respectively.
[CrossRef]

#### Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

#### Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

#### Mahajan, V. N.

V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE, 2004).

#### Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

#### Proc. SPIE (1)

V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
[CrossRef]

#### SIAM J. Appl. Math. (1)

C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987). From the statement on p. 348 it can be inferred that the real orthonormal polynomials are given by (1/2)(pn,k+pn,−k)/(1/2)||pn,k|| and (1/2i)(pn,k−pn,−k)/(1/2)||pn,k||. However, wrong answers are obtained in some cases, e.g., when k=0 or pn,k is real. For example, 2 does not apply in some cases, such as p2,0 and p4,0. Similarly, p3,3 and p3,−3 are real and unequal, and p3,3+p3,−3 does not yield the correct form of the polynomial. Instead, p3,3/||p3,3|| yields our polynomial H10 and p3,−3/||p3,−3|| yields our polynomial H9. There are other mistakes as well. For example, p4,4 and ||p4,4||2 should equal z4 and 319/3150, respectively.
[CrossRef]

#### Other (4)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE, 2004).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

http://scikits.com/KFacts.html.

### Cited By

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### Figures (1)

Fig. 1

Coordinate system for a hexagonal pupil represented by a unit hexagon inscribed inside a unit circle.

### Tables (2)

Table 1 Orthonormal Zernike Circle Polynomials $Z j ( x , y )$ , Hexagonal Polynomials $H j ( x , y )$ in Cartesian Coordinates

Table 2 Orthonormal Zernike Circle Polynomials in Polar Coordinates and Hexagonal Polynomials in Cartesian and Polar Coordinates, where $( x , y ) = ρ ( cos θ , sin θ )$ .

### Equations (28)

$W ( x , y ) = ∑ j a j H j ( x , y ) ,$
$A − 1 ∫ hexagon H j ( x , y ) H j ′ ( x , y ) d x d y = δ j j ″ ,$
$a j = 1 A ∫ hexagon W ( x , y ) H j ( x , y ) d x d y ,$
$σ 2 = ∑ j a j 2 , j ≠ 1 ,$
$G 1 = Z 1 = 1 ,$
$G j + 1 = ∑ k = 1 j c j + 1 , k H k + Z j + 1 ,$
$H j + 1 = G j + 1 ‖ G j + 1 ‖ = G j + 1 ( 1 A ∫ hexagon G j + 1 2 d x d y ) 1 ∕ 2 ,$
$c j + 1 , k = − 1 A ∫ hexagon Z j + 1 H k d x d y .$
$H l ( x , y ) = ∑ i = 1 l M l i Z i ( x , y ) , M l l = 1 ∥ G l ∥ .$
$∫ x 2 + y 2 ⩽ 1 Z j ( x , y ) Z j ′ ( x , y ) d x d y ∫ x 2 + y 2 ⩽ 1 d x d y = δ j j ′ .$
$G 2 = c 21 H 1 + Z 2 = Z 2 = 2 x ,$
$H 2 = 2 x ( 1 A ∫ hexagon 4 x 2 d x d y ) 1 ∕ 2 = 6 ∕ 5 Z 2 .$
$H 3 = 6 ∕ 5 Z 3 ,$
$c 41 = 1 ∕ 3 , c 42 = 0 = c 43 ,$
$G 4 = ( 1 ∕ 3 ) Z 1 + Z 4 = 3 ( 2 ρ 2 − 5 ∕ 6 ) ,$
$H 4 = 3 ( 2 ρ 2 − 5 ∕ 6 ) [ 1 A ∫ hexagon 3 ( 2 ρ 2 − 5 ∕ 6 ) 2 d x d y ] 1 ∕ 2 = 3 ( 2 ρ 2 − 5 ∕ 6 ) 43 ∕ 60 = 5 ∕ 43 Z 1 + 2 15 ∕ 43 Z 4 .$
$M 11 = 1 , M 22 = 6 ∕ 5 = M 33 , M 41 = 5 ∕ 43 ,$
$M 44 = 2 15 ∕ 43 ,$
$M 55 = 10 ∕ 7 = M 66 , M 73 = 16 14 ∕ 11055 = M 82 ,$
$M 77 = 10 35 ∕ 2211 = M 88 , M 99 = ( 2 ∕ 3 ) 5 ,$
$M 10 , 10 = 2 35 ∕ 103 , M 11 , 1 = 521 1072205 ,$
$M 11 , 4 = 88 15 ∕ 214441 , M 11 , 11 = 14 43 ∕ 4987 .$
$W ̂ ( x , y ) = ∑ j = 1 11 b j Z j ( x , y ) ,$
$W ̂ ( x , y ) = ∑ j = 1 11 a j ∑ i = 1 j M i j Z i ( x , y ) = ∑ j = 1 11 ∑ i = j 11 a i M i j Z j ( x , y ) .$
$b j = ∑ i = j 11 a i M i j .$
$b 1 = a 1 M 11 + a 4 M 41 + a 11 M 11 , 1 , b 2 = a 2 M 22 + a 8 M 82 ,$
$b 3 = a 3 M 33 + a 7 M 73 , b 4 = a 4 M 44 + a 11 M 11 , 4 ,$
$b j = a j M j j , j = 5 , 6 , … , 11 .$