The Zernike polynomials are an orthogonal basis thatis a product of radial and azimuthal functions definedon a unit circle.1 The rotational symmetry of opticalsystems with circular pupils allows their phase aberra-tions to be well represented by the Zernike basis. Inoptical systems that have hexagonal and other non-circular primary mirror segments the Zernike poly-nomials are no longer orthogonal. As a result, thecoeff icients of the basis functions can no longer be com-bined in quadrature to calculate rms wave-front error.Since most interferometric reduction software and op-tical modeling software use Zernike polynomials to de-scribe phase aberrations, it may be useful to develop aprocess for generating associated orthogonal bases onapertures of arbitrary shape.
Reference 2 provides a good introduction to theGram–Schmidt orthogonalization (GSO) techniqueas applied to optical engineering. However, thetreatment provided considers only circular aperturesand is not extended to apertures of general shape.Reference 3 discusses the generation and aberrationbalancing of Zernike-type basis functions for circu-lar apertures with annular portions that are alsosectioned in polar angles. It makes reference togenerating Zernike-type phase functions by means ofa digital GSO technique, without further elaboration.References 4 and 5 present the derivation of ortho-gonal basis functions over annular pupils of arbitraryobscuration ratio.
In this Letter we describe how the GSO methodcan be extended to apertures of arbitrary shape. Thetreatment presented constructs each basis functionover the arbitrary aperture as a linear combination ofcircular Zernike polynomials. The output of the GSOis a matrix C that contains the expansion coefficientsdefining these combinations. Function number n ofthe new basis is defined in terms of the first n Zernikepolynomials. An example is provided that generatesZernike polynomial-type basis vectors that are or-thonormal over a hexagonal aperture. The resultsof the GSO analysis applied to the f irst 15 Zernikepolynomials are presented.
The GSO is presented in a number of texts on linearalgebra6 and wave theory.7 The task is to generate
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an orthonormal basis on an aperture of arbitraryshape from the circular Zernike basis �U1 . . .Un�.Orthogonality is defined with respect to the innerproduct,
� f ,g� �
Rdrf �r�g�r�H �r�R
drH �r�, (1)
where H is a zero-one-valued function definingthe aperture. In this Letter the basis �V1 . . .Vn�is defined over a hexagon and is referred to as thehex-Zernike polynomials. The vectors contained inthe space �U1 . . .Un� are referred to as circular-Zernikepolynomials.
The GSO is represented by
V 0n � Un 1
n21Xm�1
D 0nmVm . (2)
The vectors �V 01 . . .V 0
n� are the set of orthogonal, butnot orthonormal, hexagonal Zernike vectors. Notethat vector V 0
n is formed by summing Zernike polyno-mial n with the hex-Zernike polynomials �V1 . . .Vn21�,which have lower order than n. The relationshipbetween �V1 . . .Vn� and �V 0
1 . . .V 0n� is
Vn �V 0
n
kV 0nk
. (3)
D 0nm are calculated with Eq. (2). For an orthogonal ba-
sis set
0 � �V 0n,Vm� � �Un,Vm� 1
n21Xl�1
D 0nl�Vl,Vm� ,
D 0nm � 2�Un,Vm� (4)
for each m , n. This essentially completes theGSO. However, it is useful to define �V1 . . .Vn�solely in terms of the circular-Zernike polynomials�U1 . . .Un�. In defining �V1 . . .Vn� in this way, matrixC can be formed, which is approximately equal to
the identity for apertures that are hexagonal andhigher-order polygons.
Consider the expansion in terms of Vi �U1 . . .Un�
Vi �nX
m�1CimUm . (5)
Vector Uj can also be written in terms of �V1 . . .Vn�.Hence
Uj �nX
l�1
BjlVl �nX
l�1
�Uj ,Vl�Vl , (6)
where the second equality follows because �V1 . . .Vn�are an orthonormal basis on the hexagonal aperture.Equation (6) can be substituted into Eq. (5) to form
Vi �nX
m�1Cim
MXl�1
BmlVl �NXl�1
√NX
m�1CimBml
!Vl . (7)
It follows that the coefficients Cim and Bml are relatedby matrix equation C � B21.
As a test case, the first nine hex-Zernike polynomi-als are generated from the circular-Zernike polynomi-als whose coefficients are equal to the rms wave-fronterror over the pupil.8 These polynomials are incommon use in commercially available optical designpackages.9,10
The C matrix transforming �U1 . . .U9� onto�V1 . . .V9� is shown in Table 1. For example, byuse of Eq. (5) and the coeff icients contained in the lastrow of Table 1, the hex-Zernike polynomial V9 is
V9 �9X
m�1C9mUm , (8)
V9 � 0.0049U1 1 0.0204U3 1 0.0069U4
1 0.0112U5 1 0.0214U7 1 1.138U9 . (9)
Note that C is strongly diagonal and approximatelyequal to the identity matrix.
Table 1. C Matrix Containing the Transformation from ���U1 . . .U9��� to ���V1 . . .V9���
Wave-front maps of �U1 . . .U3�, �V1 . . .V3�, and�jU1 2 V1j, jU2 2 V2j, jU3 2 V3j�, which are trun-cated by the hexagonal aperture, are shown inFig. 1; wave-front maps of �U4 . . .U6�, �V4 . . .V6�,and �jU4 2 V4j, jU5 2 V5j, jU6 2 V6j� are shown inFig. 2; and wave-front maps of �U7 . . .U9�, �V7 . . .V9�,and �jU7 2 V7j, jU8 2 V8j, jU9 2 V9j� are shown in
Fig. 1. Wave-front maps �U1 . . .U3�, �V1 . . .V3�, and �jU1 2V1j, jU2 2 V2j, jU3 2 V3j� truncated by the hexagonal aper-ture. The scale of the data is 66 waves.
Fig. 2. Wave-front maps �U4 . . .U6�, �V4 . . .V6�, and �jU4 2V4j, jU5 2 V5j, jU6 2 V6j� truncated by the hexagonal aper-ture. The scale of the data is 66 waves.
Fig. 3. Wave-front maps �U7 . . .U9�, �V7 . . .V9�, and �jU7 2V7j, jU8 2 V8j, jU9 2 V9j� truncated by the hexagonal aper-ture. The scale of the data is 66 waves.
Fig. 4. Plot of the rms residual arising from takingthe difference between Un and Vn for a given Zernikenumber n.
Fig. 3. Figure 4 is a plot of the rms residual arisingfrom the difference between Un and Vn for a givenorder n. The plot indicates a general trend thatthe rms difference between Un and Vn increaseswith Zernike number n. Zernike numbers 5, 6, and9 depart from this general trend. The departurefor Zernike number 9 is greatest ��33%� becauseof the 120± triangular periodicity of the Zernikenumber, which is trefoil. The departures for Zernikenumbers 5 and 6, which both correspond to astigma-tism, arise due to the similarity of astigmatism totrefoil.
The GSO presented in this Letter can also be usedto generate explicit polynomial representations forthe hex-Zernike polynomials. The formulation ofthis technique is not limited to hexagonal aperturesor Zernike polynomials. It is simple to extend it toapertures of arbitrary shape or a different choice ofbasis functions.
1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Perg-amon, New York, 1993), pp. 464–468.
2. D. Malacara and S. L. DeVore, in Optical Shop Testing,D. Malacara, ed. (Wiley, New York, 1992), pp. 480–484.
3. W. Swantner and W. Chow, Appl. Opt. 33, 1832 (1994).4. V. N. Mahajan, J. Opt. Soc. Am. 71, 75 (1981).5. V. N. Mahajan, Optical Imaging and Aberrations
Part II. Wave Diffraction Optics (SPIE Press,Bellingham, Wash., 2001), Sec. 3.4.
6. G. Strang, Linear Algebra and Its Applications (Har-court Brace Jovanovitch, San Diego, Calif., 1988),pp. 166–174.
7. D. G. Dudley, Mathematical Foundations for Electro-magnetic Theory (Institute of Electrical and Electron-ics Engineers, Piscataway, N.J., 1994), pp. 16–18.
8. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).9. The Zemax optical design program is developed by the
Zemax development corporation, www.zemax.com.10. The CODE V optical design program is developed by
