Abstract

In this research, Zernike polynomials for a unit annular elliptical aperture (ellipse inscribed by a unit circle of unit radius obscured by elliptical obscuration) have been studied in Cartesian coordinates and in polar coordinates. These polynomials have been shown to form a complete basis orthogonal on a unit annular ellipse aperture, and they represent balanced classical aberrations just as Zernike circular polynomials in a unit circle.

© 2012 Optical Society of America

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References

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  1. F. Zernike, “Diffraction theory of the knife-edge test and its improved form: the phase-contrast method,” Physica 1, 689–704 (1934).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999), pp. 905–910.
  3. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  4. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994).
    [CrossRef]
  5. D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87–95 (2001).
    [CrossRef]
  6. V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. 31, 2462–2464 (2006).
    [CrossRef]
  7. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [CrossRef]
  8. A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. 2, 07012 (2007).
    [CrossRef]
  9. S. Ling and M. Dongmei, “A new method for comparing Zernike circular polynomials with Zernike annular polynomials in annular pupils” in International Conference on Computer, Mechatronics, Control and Electronic Engineering (CMCE, 2010).
  10. R. Navarro, R. Rivera, and J. Aporta, “Represented the wavefronts in free-form transmission pupils with complex Zernike polynomials,” J. Optom. 4, 41–48 (2012).
  11. J. C. Wyant, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 1–53 (1992).
  12. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [CrossRef]
  13. D. Malacara, Optical Shop Testing, 2nd ed. (John Wiley, 1992).
  14. E. Suli and D. F. Mayers, An Introduction to Numerical Analysis (Cambridge, 2003).

2012 (1)

R. Navarro, R. Rivera, and J. Aporta, “Represented the wavefronts in free-form transmission pupils with complex Zernike polynomials,” J. Optom. 4, 41–48 (2012).

2007 (2)

N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

2006 (1)

2001 (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87–95 (2001).
[CrossRef]

1994 (2)

1992 (1)

J. C. Wyant, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 1–53 (1992).

1976 (1)

1934 (1)

F. Zernike, “Diffraction theory of the knife-edge test and its improved form: the phase-contrast method,” Physica 1, 689–704 (1934).
[CrossRef]

Aporta, J.

R. Navarro, R. Rivera, and J. Aporta, “Represented the wavefronts in free-form transmission pupils with complex Zernike polynomials,” J. Optom. 4, 41–48 (2012).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999), pp. 905–910.

Collins, M. J.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87–95 (2001).
[CrossRef]

Dai, G.-m.

N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. 31, 2462–2464 (2006).
[CrossRef]

Davis, B.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87–95 (2001).
[CrossRef]

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

Dongmei, M.

S. Ling and M. Dongmei, “A new method for comparing Zernike circular polynomials with Zernike annular polynomials in annular pupils” in International Conference on Computer, Mechatronics, Control and Electronic Engineering (CMCE, 2010).

Iskander, D. R.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87–95 (2001).
[CrossRef]

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

Ling, S.

S. Ling and M. Dongmei, “A new method for comparing Zernike circular polynomials with Zernike annular polynomials in annular pupils” in International Conference on Computer, Mechatronics, Control and Electronic Engineering (CMCE, 2010).

Mahajan, N.

N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

Mahajan, V. N.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (John Wiley, 1992).

Mayers, D. F.

E. Suli and D. F. Mayers, An Introduction to Numerical Analysis (Cambridge, 2003).

Navarro, R.

R. Navarro, R. Rivera, and J. Aporta, “Represented the wavefronts in free-form transmission pupils with complex Zernike polynomials,” J. Optom. 4, 41–48 (2012).

Noll, R. J.

Rivera, R.

R. Navarro, R. Rivera, and J. Aporta, “Represented the wavefronts in free-form transmission pupils with complex Zernike polynomials,” J. Optom. 4, 41–48 (2012).

Suli, E.

E. Suli and D. F. Mayers, An Introduction to Numerical Analysis (Cambridge, 2003).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999), pp. 905–910.

Wyant, J. C.

J. C. Wyant, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 1–53 (1992).

Zernike, F.

F. Zernike, “Diffraction theory of the knife-edge test and its improved form: the phase-contrast method,” Physica 1, 689–704 (1934).
[CrossRef]

Appl. Opt. (2)

Appl. Opt. Opt. Eng. (1)

J. C. Wyant, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 1–53 (1992).

IEEE Trans. Biomed. Eng. (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87–95 (2001).
[CrossRef]

J. Eur. Opt. Soc. (1)

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

J. Opt. Soc. Am. (2)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

J. Optom. (1)

R. Navarro, R. Rivera, and J. Aporta, “Represented the wavefronts in free-form transmission pupils with complex Zernike polynomials,” J. Optom. 4, 41–48 (2012).

Opt. Lett. (1)

Physica (1)

F. Zernike, “Diffraction theory of the knife-edge test and its improved form: the phase-contrast method,” Physica 1, 689–704 (1934).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999), pp. 905–910.

S. Ling and M. Dongmei, “A new method for comparing Zernike circular polynomials with Zernike annular polynomials in annular pupils” in International Conference on Computer, Mechatronics, Control and Electronic Engineering (CMCE, 2010).

D. Malacara, Optical Shop Testing, 2nd ed. (John Wiley, 1992).

E. Suli and D. F. Mayers, An Introduction to Numerical Analysis (Cambridge, 2003).

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Figures (3)

Fig. 1.
Fig. 1.

Wavefront aberration function for a distant point object.

Fig. 2.
Fig. 2.

(a) Unit elliptical aperture of aspect ratio=b. (b) Unit annular ellipse of aspect ratio=b and obscuration ratio=k.

Fig. 3.
Fig. 3.

Polynomials of annular ellipse pupil with k=0.5 and b=0.7 graphically in 2D and 3D.

Tables (6)

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Table 1. Orthonormal Zernike Polynomials for Unit Elliptical Aperture with Aspect Ratio=b(ρ2=x2+y2)a

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Table 2. Orthogonol Zernike Polynomials for Unit Elliptical Aperture Obscured by Elliptical Obscuration (ρ2=x2+y2) with Aspect Ratio=b and Obscuration Ratio=ka

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Table 3. Normalization Constants for the Orthogonal Annular Ellipse Zernike Polynomialsa

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Table 4. Orthonormal Zernike Polynomials for Unit Elliptical Aperture Obscured by Elliptical Obscuration (ρ2=x2+y2), with Aspect Ratio=b and Obscuration Ratio=k=0.5a

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Table 5. Orthonormal Zernike Polynomials for Unit Elliptical Aperture with Elliptical Obscuration with Aspect Ratio=0.7 and Obscuration Ratio=0.5

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Table 6. Orthonormal Annular Ellipse Zernike Polynomials Aspect Ratio=.7 and Obscuration Ratio=0.5 in Polar Coordinates

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

S=1π2|02π01exp(i2πΔW(ρ,θ)ρdρdθ|2.
S=1π2|02π01[1+i2πΔW+12(i2πΔW)2]ρdρdθ|2,
S|1+i2πΔW¯12(2π)2ΔW2¯|21(2π)2[ΔW2¯(ΔW¯)2]=1(2πσ)2,
σ2=ΔW2¯(ΔW¯)2.
W¯(ρ,ϑ)=0102πW(ρ,θ)ρdρdθ0102πρdρdθ,
W¯2(ρ,ϑ)=0102πW2(ρ,θ)ρdρdθ0102πρdρdθ.
W(ρ,θ)=W11ρcosθ+W20ρ+W40ρ2+W31ρ3cosθ+W22ρ2cos2θ,
W(x,y)=Z0+Z1x+Z2y+Z3(2x2+2y21)+Z4(x2y2)+Z5(2xy)+Z6(3x3+3xy22x)+Z7(3x2y+3y32y)+Z8(6x4+12x2y2+6y46x26y2+1)+Z9(x33xy2)+Z10(3x2yy3)+Z11(4x3y4xy3)+Z12(8x3y+8xy36xy)+Z13(4x43x24y4+3y2)+Z14(x46x2y2+y4),
Z2(x,y)=Z2(x,y)11b1x2b1x2Z1(x,y)Z2(x,y)dydx11b1x2b1x2Z12(x,y)dydx.
1=C211b1x2b1x2Z2(x,y)dydx11b1x2b1x2dydx.
Z2(x,y)=Z2(x,y)11b1x2b1x2Z1(x,y)Z2(x,y)dydxkkbk2x2bk2x2Z1(x,y)Z2(x,y)dydx11b1x2b1x2Z12(x,y)dydxkkbk2b2x2bk2b2x2Z12(x,y),
1=C211b1x2b1x2Z2(x,y)dydxkkbk2x2bk2x2Z2(x,y)dydx11b1x2b1x2dydxkkbk2x2bk2x2dydx.
W¯(x,y)=11b1x2b1x2W(x,y)dydxkkbk2x2bk2x2W(x,y)dydx11b1x2b1x2dydxkkbk2x2bk2x2dydx,
W¯2(x,y)=C211b1x2b1x2W2(x,y)dydxkkbk2x2bk2x2W2(x,y)dydx11b1x2b1x2dydxkkbk2x2bk2x2dydx.

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