Abstract

We present a new design of a modal wave-front sensor capable of measuring directly the Zernike components of an aberrated wave front. The sensor shows good linearity for small aberration amplitudes and is particularly suitable for integration in a closed-loop adaptive system. We introduce a sensitivity matrix and show that it is sparse, and we derive conditions specifying which elements are necessarily zero. The sensor may be temporally or spatially multiplexed, the former using a reconfigurable optical element, the latter using a numerically optimized binary optical element. Different optimization schemes are discussed, and their performance is compared.

© 2000 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).
  2. R. K. Tyson, Principles of Adaptive Optics (Academic, London, 1991).
  3. M. J. Booth, M. A. A. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” J. Microsc. 192, 90–98 (1998).
    [CrossRef]
  4. P. Török, P. Varga, G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]
  5. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  6. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  7. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes in Opt. Photon. News, Aug. 1994 [Appl. Opt. 33, 8121–8124 (1994)].
    [CrossRef] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).
  9. A. Gray, G. B. Matthews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).
  10. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. Lab. Notes in Opt. Photon. News, Nov.1994 [Appl. Opt. 33, 8125–8127 (1994)].
    [CrossRef] [PubMed]
  11. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  12. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1978), Chap. 3.
  13. M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
    [CrossRef]

1998 (2)

M. J. Booth, M. A. A. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
[CrossRef]

1995 (1)

1994 (2)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes in Opt. Photon. News, Aug. 1994 [Appl. Opt. 33, 8121–8124 (1994)].
[CrossRef] [PubMed]

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. Lab. Notes in Opt. Photon. News, Nov.1994 [Appl. Opt. 33, 8125–8127 (1994)].
[CrossRef] [PubMed]

1988 (1)

1976 (1)

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Booth, M. J.

M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
[CrossRef]

M. J. Booth, M. A. A. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Gray, A.

A. Gray, G. B. Matthews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1978), Chap. 3.

Mahajan, V. N.

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. Lab. Notes in Opt. Photon. News, Nov.1994 [Appl. Opt. 33, 8125–8127 (1994)].
[CrossRef] [PubMed]

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes in Opt. Photon. News, Aug. 1994 [Appl. Opt. 33, 8121–8124 (1994)].
[CrossRef] [PubMed]

Matthews, G. B.

A. Gray, G. B. Matthews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

Neil, M. A. A.

M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
[CrossRef]

M. J. Booth, M. A. A. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

Nemeth, G.

Noll, R. J.

Roddier, F.

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Török, P.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, London, 1991).

Varga, P.

Wilson, T.

M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
[CrossRef]

M. J. Booth, M. A. A. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Appl. Opt. (1)

Eng. Lab. Notes in Opt. Photon. News (2)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes in Opt. Photon. News, Aug. 1994 [Appl. Opt. 33, 8121–8124 (1994)].
[CrossRef] [PubMed]

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Eng. Lab. Notes in Opt. Photon. News, Nov.1994 [Appl. Opt. 33, 8125–8127 (1994)].
[CrossRef] [PubMed]

J. Microsc. (1)

M. J. Booth, M. A. A. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (6)

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1978), Chap. 3.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

A. Gray, G. B. Matthews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

R. K. Tyson, Principles of Adaptive Optics (Academic, London, 1991).

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Figures (5)

Fig. 1
Fig. 1

Schematic description of the aberration sensor that uses biasing elements and Fourier transform lenses.

Fig. 2
Fig. 2

Output signal from wave-front sensors for Zernike modes 4–10. The bias b=0.7, and the pinhole radius νp=π.

Fig. 3
Fig. 3

Schematic description of the aberration sensor that uses a binary phase optical element to produce the two “biased” spots.

Fig. 4
Fig. 4

(a) Optimized multiplexed wave-front sensing mask (transmission: white = 1, black =-1, and gray = 0), (b) simulated output intensity in focal plane for plane wave-front input (logarithmic scale over 20-dB range).

Fig. 5
Fig. 5

Sensitivity matrices: (a) time multiplexed, (b) spatially multiplexed, optimized for sensitivity with suppression of certain off-diagonal elements and weighting of diagonal elements for better uniformity.

Tables (3)

Tables Icon

Table 1 Sensitivity Matrix, S

Tables Icon

Table 2 Inverse of the Sensitivity Matrix, S-1

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Table 3 First 22 Zernike Circle Polynomials

Equations (47)

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I1(v)=|F{exp[jΨ(r)+jΦ(r)]}|2
I2(v)=|F{exp[jΨ(r)-jΦ(r)]}|2,
I1,2(0)=A exp(jafk±jbfi)dA2.
ΔI=I1(0)-I2(0),
S=ΔIaa=0.
I1,2a=2 ImA exp(jafk±jbfi)dAA fk×exp(jafkjbfi)dA,
S=4 ImA exp(jbfi)dAAfk exp(-jbfi)dA.
S=4bAfidAAfkdA - bAfifkdA×AdA+.
S-4bAAfifkdA=-4bACδik,
exp[jΨ(r,θ)]circ(r)=expjk=0akZk(r,θ)circ(r),
circ(r)=1forr10forr>1,
I1,2(ν,ξ)=18π3 02π01 exp[jaZk(r,θ)±jbZi(r,θ)]×exp[jνr cos(θ-ξ)]rdrdθ2.
ΔWik=02π0νpI1(ν, ξ)νdνdξ-02π0νpI2(ν, ξ)νdνdξ,
Sik=ΔWika a=0=0νp02π I1aa=0-I2aa=0dξνdν.
I1,2(ν, ξ)=18π3 |F[exp(jaZk±jbZi)]|2=18π3 F[exp(jaZk±jbZi)]×F[exp(jaZk±jbZi)]*,
I1,2aa=0=14π3 Im{F[exp(±jbZi)]F[Z˜k exp(jbZ˜i)]},
Z˜i(r,θ)=Zi(r,θ+π).
exp(jz cos ϕ)=p=-jpJp(z)cos(pϕ),
F[exp(±jbZi)]=p=-jpF[Jp(±bRi)cos(pmiθ)].
F[exp(±jbZi)]=2πp=-jp(mi+1)(±)p cos(pmiξ)×01Jp(bRi)Jpmi(νr)rdr,
F[Z˜k exp(jbZ˜i)]
=πq=-jq()pjqmi+mk cos[(qmi+mk)ξ]×01RkJq(bRi)Jqmi+mk(νr)rdr+jqmi-mk cos[(qmi-mk)ξ]×01RkJq(bRi)Jqmi-mk(νr)rdr,
I1aa=0-I2aa=0
=1π Imp=- q=-(p+q) odd(-1)qj(p+q)(mi+1)×jmk cos(pmiξ)cos[(qmi+mk)ξ]×01Jp(bRi)Jpmi(νr)rdr×01RkJq(bRi)Jqmi+mk(νr)rdr+j-mk cos(pmiξ)cos[(qmi-mk)ξ]×01Jp(bRi)Jpmi(νr)rdr×01RkJq(bRi)Jqmi-mk(νr)rdr,
pmi=±(qmi+mk)orpmi=±(qmi - mk).
Sik|mi=mk=0=4 Imp=- q=-(p+q) odd(-1)q(j)(p+q)×0νp01Jp(bRi)J0(νr)rdr×01RkJq(bRi)J0(νr)rdrνdν=4 Im0νp01 exp(jbRi)J0(νr)rdr×01Rk exp(-jbRi)J0(νr)rdrνdν.
Sik|mk/mi=odd=2(-1)mk+mi2mi×0νpp=-×01Jp(bRi)Jpmi(νr)rdr×01Rk[Jp+mkmi(bRi) - Jp-mkmi(bRi)]Jpmi(νr)rdrνdν.
Rnmr2-21-21/2,
Sik|mi=mk=0=2νp2 Im01 exp(jbRi)rdr×01Rk exp(-jbRi)rdr.
Sik|mi=mk=0=2bνp201Ri rdr01Rk rdr-01rdr01RkRi rdr;
Sik=-b2 νp2δik.
Jpmi(νr)1(pmi)! νr2pmi.
Sik|mk/mi=odd(-1)mk+mi2mkνp201J0(bRi)rdr×01Rk[Jmkmi(bRi)-J-mkmi(bRi)]rdr.
Sik|mi/mk=1-bνp201rdr01RkRirdr=-b2 νp2δik.
Sik=-b2 νp2i=kO(b3νp2)+O(bνp4)ik, mkmi isodd0otherwise.
g(x, y)=2π exp[j(ϕ+τ)]+exp[-j(ϕ+τ)]-13 exp[j3(ϕ+τ)]-13 exp[-j3(ϕ+τ)]+.
I(ν,ξ)akak=0=-14π3 Re0102πU(r,θ)×exp[jrν cos(θ-ξ)]dθrdr×0102πjZkU*(r,θ)×exp[-jrν cos(θ-ξ)]dθrdr,
Znm(r,θ)=2(n+1)Rn-m(r)sin(-mθ)m<0n+1Rn0(r)m=02(n+1)Rnm(r)cos(mθ)m>0,
Rnm(r)=s=0(n-|m|)/2(-1)s(n-s)!s![(n+m)/2-s]![(n-m)/2-s]! rn-2scirc(r).
1π 0102πZnm(r,θ)Znm(r,θ)dθrdr=δnnδmm,
01Ri(r)Rk(r)rdr=δik/2mi=mk=0δikmi=mk0.
01Ri(r)rdr=0mi=0.
F{Z(r,θ)}=02π01f(r)cos(mθ)×exp[jνr cos(θ-ξ)]rdrdθ.
F{Z(r,θ)}=01f(r)cos(mξ)02πcos(mϕ)×exp(jνr cos ϕ)dϕ-sin(mξ)×02πsin(mϕ)exp(jνr cos ϕ)dϕrdr.
Jm(νr)=j-m2π 02π cos(mϕ)exp(jνr cos ϕ)dϕ,
F{Z(r,θ)}=2πjm cos(mξ)01f(r)Jm(νr)rdr.
F{Z(r,θ)}=2πjm sin(mξ)01f(r)Jm(νr)rdr.

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