Abstract

Jacobi circle polynomials, which are orthogonal on the unit circle with orthogonal radial derivatives, have been developed previously. As the classical Zernike mode can be represented as a linear combination of Jacobi modes, Zernike wavefront modes can be reconstructed using Jacobi modes. Comparison of the Jacobi and Zernike modes for the modal approach indicates that a modal approach incorporating the Gram matrix with the Jacobi modes has potential application in high-sampling wavefront gradient sensors. The Gram matrix method using the Jacobi modes can be extended to annular pupils.

© 2018 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Modal wavefront reconstruction from its gradient

Iacopo Mochi and Kenneth A. Goldberg
Appl. Opt. 54(12) 3780-3785 (2015)

Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials

Cosmas Mafusire and Tjaart P. J. Krüger
J. Opt. Soc. Am. A 35(6) 840-849 (2018)

Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts

Virendra N. Mahajan and Maham Aftab
Appl. Opt. 49(33) 6489-6501 (2010)

References

  • View by:
  • |
  • |
  • |

  1. S. Velghe, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005).
    [Crossref]
  2. F. Dai, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt. 51, 5028–5037 (2012).
    [Crossref]
  3. H. Wang, “Improved wavefront reconstruction using difference Zernike polynomials for two double-shearing wavefronts,” Proc. SPIE 8550, 855013 (2012).
    [Crossref]
  4. T. Ling, “Wavefront retrieval for cross-grating lateral shearing interferometer based on differential Zernike polynomial fitting,” Proc. SPIE 8838, 88380J (2013).
    [Crossref]
  5. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. A 69, 972–977 (1979).
    [Crossref]
  6. W. H. Southwell, “Wave-front estimation from wave-front slope measurement,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
    [Crossref]
  7. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. A 71, 989–992 (1981).
    [Crossref]
  8. S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
    [Crossref]
  9. A. Gavrielides, “Vector polynomials orthogonal to the gradient of Zernike polynomials,” Opt. Lett. 7, 526–528 (1982).
    [Crossref]
  10. E. Acosta, S. Bará, M. A. Rama, and S. Ríos, “Determination of phase mode components in terms of local wave-front slopes: an analytical approach,” Opt. Lett. 20, 1083–1085 (1995).
    [Crossref]
  11. C. J. Solomon, G. C. Loos, and S. Ríos, “Variational solution for modal wave-front projection functions of minimum-error norm,” J. Opt. Soc. Am. A 18, 1519–1522 (2001).
    [Crossref]
  12. C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, part I: basis set derived from gradients of Zernike polynomials,” Opt. Express 15, 18014–18024 (2007).
    [Crossref]
  13. C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, part II: completing the basis set,” Opt. Express 16, 6586–6591 (2008).
    [Crossref]
  14. V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. A 34, 1908–1913 (2017).
    [Crossref]
  15. S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
    [Crossref]
  16. S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012).
    [Crossref]
  17. W. Sun, S. Wang, X. He, and B. Xu are preparing a manuscript to be called “Jacobi circle polynomials: basic set on unit circle and extension to concentric scaled pupils, annular pupils, circular sector pupils and annular sector pupils.”
  18. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [Crossref]
  19. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994).
    [Crossref]
  20. M. Born and E. Wolf, “Appendix VII: The circle polynomials of Zernike,” in Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 905–910.
  21. G. Szego, “Preliminaries,” in Orthogonal Polynomials (American Mathematical Society, 1938), p. 3, Chap. I.
  22. G. Szego, “Jacobi polynomials,” in Orthogonal Polynomials (American Mathematical Society, 1938), p. 58, Chap. IV.
  23. R. A. Horn and C. R. Johnson, “Positive definite matrices,” in Matrix Analysis, 1st ed. (Cambridge University, 1990), p. 407, Chap. 7.

2017 (1)

2013 (2)

T. Ling, “Wavefront retrieval for cross-grating lateral shearing interferometer based on differential Zernike polynomial fitting,” Proc. SPIE 8838, 88380J (2013).
[Crossref]

S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

2012 (3)

2011 (1)

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

2008 (1)

2007 (1)

2005 (1)

2001 (1)

1995 (1)

1994 (2)

1982 (1)

1981 (1)

J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. A 71, 989–992 (1981).
[Crossref]

1980 (1)

W. H. Southwell, “Wave-front estimation from wave-front slope measurement,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
[Crossref]

1979 (1)

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. A 69, 972–977 (1979).
[Crossref]

Acosta, E.

Bará, S.

Born, M.

M. Born and E. Wolf, “Appendix VII: The circle polynomials of Zernike,” in Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 905–910.

Burge, J. H.

Cubalchini, R.

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. A 69, 972–977 (1979).
[Crossref]

Dai, F.

Gavrielides, A.

He, X.

W. Sun, S. Wang, X. He, and B. Xu are preparing a manuscript to be called “Jacobi circle polynomials: basic set on unit circle and extension to concentric scaled pupils, annular pupils, circular sector pupils and annular sector pupils.”

Herrmann, J.

J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. A 71, 989–992 (1981).
[Crossref]

Horn, R. A.

R. A. Horn and C. R. Johnson, “Positive definite matrices,” in Matrix Analysis, 1st ed. (Cambridge University, 1990), p. 407, Chap. 7.

Huang, S.

S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Jiang, Z.

S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Johnson, C. R.

R. A. Horn and C. R. Johnson, “Positive definite matrices,” in Matrix Analysis, 1st ed. (Cambridge University, 1990), p. 407, Chap. 7.

Ling, T.

T. Ling, “Wavefront retrieval for cross-grating lateral shearing interferometer based on differential Zernike polynomial fitting,” Proc. SPIE 8838, 88380J (2013).
[Crossref]

Liu, C.

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Loos, G. C.

Mahajan, V. N.

Ning, Y.

S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

Rama, M. A.

Ríos, S.

Solomon, C. J.

Southwell, W. H.

W. H. Southwell, “Wave-front estimation from wave-front slope measurement,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
[Crossref]

Sun, W.

W. Sun, S. Wang, X. He, and B. Xu are preparing a manuscript to be called “Jacobi circle polynomials: basic set on unit circle and extension to concentric scaled pupils, annular pupils, circular sector pupils and annular sector pupils.”

Szego, G.

G. Szego, “Preliminaries,” in Orthogonal Polynomials (American Mathematical Society, 1938), p. 3, Chap. I.

G. Szego, “Jacobi polynomials,” in Orthogonal Polynomials (American Mathematical Society, 1938), p. 58, Chap. IV.

Velghe, S.

Wang, H.

H. Wang, “Improved wavefront reconstruction using difference Zernike polynomials for two double-shearing wavefronts,” Proc. SPIE 8550, 855013 (2012).
[Crossref]

Wang, S.

W. Sun, S. Wang, X. He, and B. Xu are preparing a manuscript to be called “Jacobi circle polynomials: basic set on unit circle and extension to concentric scaled pupils, annular pupils, circular sector pupils and annular sector pupils.”

Wolf, E.

M. Born and E. Wolf, “Appendix VII: The circle polynomials of Zernike,” in Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 905–910.

Xi, F.

S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Xu, B.

W. Sun, S. Wang, X. He, and B. Xu are preparing a manuscript to be called “Jacobi circle polynomials: basic set on unit circle and extension to concentric scaled pupils, annular pupils, circular sector pupils and annular sector pupils.”

Zhao, C.

Appl. Opt. (3)

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

S. Huang, Y. Ning, F. Xi, and Z. Jiang, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

S. Huang, F. Xi, C. Liu, and Z. Jiang, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Proc. SPIE (2)

H. Wang, “Improved wavefront reconstruction using difference Zernike polynomials for two double-shearing wavefronts,” Proc. SPIE 8550, 855013 (2012).
[Crossref]

T. Ling, “Wavefront retrieval for cross-grating lateral shearing interferometer based on differential Zernike polynomial fitting,” Proc. SPIE 8838, 88380J (2013).
[Crossref]

Other (5)

W. Sun, S. Wang, X. He, and B. Xu are preparing a manuscript to be called “Jacobi circle polynomials: basic set on unit circle and extension to concentric scaled pupils, annular pupils, circular sector pupils and annular sector pupils.”

M. Born and E. Wolf, “Appendix VII: The circle polynomials of Zernike,” in Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 905–910.

G. Szego, “Preliminaries,” in Orthogonal Polynomials (American Mathematical Society, 1938), p. 3, Chap. I.

G. Szego, “Jacobi polynomials,” in Orthogonal Polynomials (American Mathematical Society, 1938), p. 58, Chap. IV.

R. A. Horn and C. R. Johnson, “Positive definite matrices,” in Matrix Analysis, 1st ed. (Cambridge University, 1990), p. 407, Chap. 7.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1. Reconstruction error λ obtained using least-squares estimation versus truncation number J of order n for reconstructed Jacobi and Zernike modes. The actual wavefront of the Zernike modes [normalized as Eq. (1)] is Z11+Z31+Z51+Z71+Z91.
Fig. 2.
Fig. 2. Remaining error λr obtained using least-squares estimation versus J of order n for reconstructed Jacobi and Zernike modes, when J of order n is smaller than the actual number of wavefront modes. The actual wavefront of the Zernike modes [normalized as Eq. (1)] is Z11+Z31+Z51+Z71+Z91.
Fig. 3.
Fig. 3. Reconstruction error λ obtained using the Gram matrix versus J of reconstructed Jacobi and Zernike modes on the unit circle. The orders are adjusted as in Table 4 and unified by Jacobi orders. The actual wavefront is chosen to be the first 20 Zernike modes [normalized as Eq. (1)] with coefficients β1=β2==β20=1, corresponding to the first 55 orders of the Jacobi modes. The weight function indices of the Jacobi modes are chosen as p1=p2==p9=2 and q1=q2==q9=2.
Fig. 4.
Fig. 4. Remaining error λr obtained using the Gram matrix versus truncation number J of reconstructed Jacobi and Zernike modes when J is smaller than the actual wavefront. The orders are adjusted as in Table 4 and unified by Jacobi orders. The actual wavefront is chosen as the first 20 Zernike modes [normalized as Eq. (1)] with coefficients β1=β2==β20=1, corresponding to the first 55 orders of the Jacobi modes.
Fig. 5.
Fig. 5. Reconstruction results of the linear combination of Zernike circle polynomials on the unit circle. The actual wavefront is chosen as the first 20 Zernike modes [normalized as Eq. (1)] with coefficients β1=β2==β20=1, corresponding to the first 55 orders of the Jacobi modes. (a) Original wavefront; (b) reconstructed wavefront represented by Jacobi modes, where the zero-order correction of Jacobi coefficients is considered; and (c) residual error.
Fig. 6.
Fig. 6. Reconstruction error λ obtained using the Gram matrix versus truncation number of reconstructed Jacobi and Zernike modes on an annular pupil, for η=0.3 and ε=0.8. The orders are adjusted as in Table 4 and unified by Jacobi orders. The actual wavefront is chosen as the first 20 Zernike circle polynomials [normalized as Eq. (1)] with coefficients β1=β2==β20=1, corresponding to the first 55 orders of the Jacobi modes. The weight function indices of the Jacobi modes are chosen as p1=p2==p9=2 and q1=q2==q9=2.
Fig. 7.
Fig. 7. Remaining error λr obtained using the Gram matrix versus truncation number of reconstructed Jacobi and Zernike modes on an annular pupil, for η=0.3 and ε=0.8. The orders are adjusted as in Table 4 and unified by Jacobi orders. The actual wavefront is chosen as the first 20 Zernike circle polynomials [normalized as Eq. (1)] with coefficients β1=β2==β20=1, corresponding to the first 55 orders of the Jacobi modes.
Fig. 8.
Fig. 8. Reconstruction results of the linear combination of Zernike circle polynomials on an annular pupil, for η=0.3 and ε=0.8. The actual wavefront is chosen as the first 20 Zernike modes [normalized as Eq. (1)] with coefficients β1=β2==β20=1, corresponding to the first 55 orders of the Jacobi modes. (a) Original wavefront; (b) reconstructed wavefront represented by Jacobi modes, where the zero-order correction of Jacobi coefficients is considered; and (c) residual error.

Tables (4)

Tables Icon

Table 1. Coefficients of Zernike Modes Reconstructed by Gram Matrix for Truncation Numbers J of Order n, Where Actual Wavefront of Zernike Modes [Normalized as Eq. (1)] Is Z11+Z31+Z51+Z71+Z91

Tables Icon

Table 2. Coefficients of Jacobi Modes Reconstructed by Gram Matrix for J of Order n, Where Actual Wavefront of Jacobi Modes Is G11(2,2;ρ,θ)+G21(2,2;ρ,θ)+G31(2,2;ρ,θ)+G41(2,2;ρ,θ)+G51(2,2;ρ,θ)+G61(2,2;ρ,θ)+G71(2,2;ρ,θ)+G81(2,2;ρ,θ)+G91(2,2;ρ,θ)a

Tables Icon

Table 3. Coefficients of Jacobi Modes Reconstructed by Gram Matrix for J of Order n, Where Actual Wavefront of Zernike Modes Is Z11+Z31+Z51+Z71+Z91a

Tables Icon

Table 4. Adjusted Order of Jacobi and Zernike Modesa

Equations (30)

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

Znm(ρ,θ)={n+1πRn|m|(ρ)2cos(|m|θ),(m>0),n+1πRn|m|(ρ)2sin(|m|θ),(m<0),n+1πRn0(ρ),(m=0),
Rnm(ρ)=s=0(nm)/2(1)s(ns)!s!(nm2s)!(n+m2s)!ρn2s.
02π01ρZnm(ρ,θ)Znm(ρ,θ)dρdθ=δnnδmm.
Gnm(pm,qm;η,ε;ρηε,θ)={[2πNn(p|m|,q|m|)(εη)pm]12·Gn(p|m|,q|m|,ρηεηεη)2cos(|m|θ),(m>0),[2πNn(p|m|,q|m|)(εη)pm]12·Gn(p|m|,q|m|,ρηεηεη)2sin(|m|θ),(m<0),[2πNn(p0,q0)(εη)pm]12·Gn(p0,q0,ρηεηεη),(m=0),
Gn(p,q,ρ)=F(n,p+n,q,ρ)=1(q)nρ1q(1ρ)qpdndρn[ρq1+n(1ρ)pq+n]=j=0n(1)jn!j!(nj)!(p+n)j(q)jρj,
Nn(p,q)=[Γ(q)]2Γ(p+nq+1)Γ(p+n)Γ(q+n)n!p+2n.
02πηε(ρηεη)qm1(ερηε)pmqmGnm(ρηε,θ)Gnm(ρηε,θ)dρηεdθ=δnnδmm.
02πηε(ρηεη)qm(ερηε)pmqm+1[ρηεGnm(ρηε,θ)][ρηεGnm(ρηε,θ)]dρηεdθ=n(pm+n)δnnδmm.
S˜xk={1AkPkW(x,y)xdxdy,(x,y)Pk,0,(x,y)Pk,
S˜yk={1AkPkW(x,y)ydxdy,(x,y)Pk,0,(x,y)Pk,
W=jαjVj,
S˜xk={1AkjαjPkVjxdxdy,(x,y)Pk,0,(x,y)Pk,={jαjVxjk,(x,y)Pk,0,(x,y)Pk,
S˜yk={1AkjαjPkVjydxdy,(x,y)Pk,0,(x,y)Pk,={jαjVyjk,(x,y)Pk,0,(x,y)Pk,
[S˜x1S˜y1S˜xKS˜yK]=[Vx11Vx21Vxj1Vy11Vy21Vyj1Vx1KVx2KVxjKVy1KVy2KVyjK]·[α1α2αj],
(Wx)meas=kS˜xk,
(Wy)meas=kS˜yk.
(Wρ)meas(ρ,θ)=(Wx)measxρ+(Wy)measyρ,=(Wx)meascosθ+(Wy)meassinθ,
Vjρ=Rj(ρ)exp(imθ).
Wρ=jαjVjρ.
[(Wρ,V1ρ)(Wρ,V2ρ)(Wρ,Vjρ)]=G·[α1α2αj].
G=[(V1ρ,V1ρ)(V1ρ,V2ρ)(V1ρ,Vjρ)(V2ρ,V1ρ)(V2ρ,V2ρ)(V2ρ,Vjρ)(Vjρ,V1ρ)(Vjρ,V2ρ)(Vjρ,Vjρ)].
[((Wρ)meas,V1ρ)((Wρ)meas,V2ρ)((Wρ)meas,VJρ)]=G·[α1α2αJ].
αj=((Wρ)meas,Vjρ)/(Vjρ,Vjρ).
λ=1Mact(αiβi)2+Mact+1Jαi21Mactβi2,
λ=λr+λt=[1J(αiβi)21Mactβi2]+[J+1Mactβi21Mactβi2],
Z11+Z31+Z51+Z71+Z91=2πρ2cosθ+4π(2ρ+3ρ3)2cosθ+6π(3ρ12ρ3+10ρ5)2cosθ+8π(4ρ+30ρ360ρ5+35ρ7)2cosθ+10π(5ρ60ρ3+210ρ5280ρ7+126ρ9)2cosθ=(5.2246ρ72.3730ρ3+292.7397ρ5443.7028ρ7+224.7996ρ9)2cosθ.
Nf(0,pm,qm)α0m=n=1JNf(n,pm,qm)αnm,
Nf(n,pm,qm)=[Nn(p|m|,q|m|)]12.
Nf(0,pm,qm,η,εα0m=n=1JNf(n,pm,qm,η,ε)αnm,
Nf(n,pm,qm,η,ε)=[j=1nn!j!(nj)!(p+n)j(q)j(ηεη)j][Nn(p|m|,q|m|)]12.

Metrics