Abstract

Curvature sensors are used in adaptive optics to measure the wave-front aberrations. In practice, their performance is limited by their nonlinear behavior, which we characterize by solving simultaneously the irradiance transport equation and the accompanying wave-front transport equation. We show how the presence of nonlinear geometric terms limits the accuracy of the sensor and how diffraction effects limit the spatial resolution. The effect of photon noise on the sensor is also quantified.

© 2002 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, New York, 1998), pp. 165–168, 377–394.
  2. M. Milman, D. Redding, L. Needels, “Analysis of curvature sensing for large-aperture adaptive optics systems,” J. Opt. Soc. Am. A 13, 1226–1238 (1996).
    [CrossRef]
  3. D. C. Johnston, B. L. Ellerbroek, S. M. Pompea, “Curvature sensing analysis,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 528–538 (1994).
    [CrossRef]
  4. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
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    [CrossRef]
  6. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  7. G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 91–130.
  8. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]
  9. F. Rigaut, B. L. Ellerbroek, M. J. Northcott, “Comparison of curvature-based and Shack–Hartmann-based adaptive optics for the Gemini telescope,” Appl. Opt. 36, 2856–2868 (1997).
    [CrossRef] [PubMed]
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  11. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996), p. 98.
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    [CrossRef]
  13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 63–75.
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  15. F. Roddier, “Error propagation in a closed-loop adaptive optics system: a comparison between Shack–Hartmann and curvature wave-front sensors,” Opt. Commun. 113, 357–359 (1995).
    [CrossRef]

2002 (1)

1997 (1)

1996 (1)

1995 (1)

F. Roddier, “Error propagation in a closed-loop adaptive optics system: a comparison between Shack–Hartmann and curvature wave-front sensors,” Opt. Commun. 113, 357–359 (1995).
[CrossRef]

1990 (1)

1988 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (1)

1976 (1)

1966 (1)

Ellerbroek, B. L.

F. Rigaut, B. L. Ellerbroek, M. J. Northcott, “Comparison of curvature-based and Shack–Hartmann-based adaptive optics for the Gemini telescope,” Appl. Opt. 36, 2856–2868 (1997).
[CrossRef] [PubMed]

D. C. Johnston, B. L. Ellerbroek, S. M. Pompea, “Curvature sensing analysis,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 528–538 (1994).
[CrossRef]

Fried, D. L.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 63–75.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, New York, 1998), pp. 165–168, 377–394.

Johnston, D. C.

D. C. Johnston, B. L. Ellerbroek, S. M. Pompea, “Curvature sensing analysis,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 528–538 (1994).
[CrossRef]

Lane, R. G.

Milman, M.

Needels, L.

Noll, R. J.

Northcott, M. J.

Pompea, S. M.

D. C. Johnston, B. L. Ellerbroek, S. M. Pompea, “Curvature sensing analysis,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 528–538 (1994).
[CrossRef]

Redding, D.

Rigaut, F.

Roddier, F.

F. Roddier, “Error propagation in a closed-loop adaptive optics system: a comparison between Shack–Hartmann and curvature wave-front sensors,” Opt. Commun. 113, 357–359 (1995).
[CrossRef]

F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
[CrossRef] [PubMed]

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[CrossRef] [PubMed]

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996), p. 98.

Rousset, G.

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 91–130.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teague, M. R.

van Dam, M. A.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996), p. 98.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

F. Roddier, “Error propagation in a closed-loop adaptive optics system: a comparison between Shack–Hartmann and curvature wave-front sensors,” Opt. Commun. 113, 357–359 (1995).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Other (5)

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 91–130.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996), p. 98.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 63–75.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, New York, 1998), pp. 165–168, 377–394.

D. C. Johnston, B. L. Ellerbroek, S. M. Pompea, “Curvature sensing analysis,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 528–538 (1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Standard curvature-sensing arrangement comprising two intensity measurements taken on planes at distance l from the focal plane.

Fig. 2
Fig. 2

Effect of the slope on the curvature measurement. The intensity variations measured at P1 are due to the curvature at C2, not at C1, because of the displacement caused by the local tilt in the wave front.

Fig. 3
Fig. 3

Comparison of Fresnel (irregular curve), nonlinear (smooth curve) and linear (straight line) intensity curves for a cubic aberration. Here λ=0.1 μm, z=10 km, and W=100x3 μm.

Fig. 4
Fig. 4

Variance of the error in the estimate of S (parabola) and S (straight line) for ten photons.

Fig. 5
Fig. 5

Variance of the error in the estimate of S (parabola) and S (straight line) for 1000 photons.

Tables (1)

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Table 1 Relative Weight of the H , K , and T Terms in the Curvature-Sensing Signal

Equations (55)

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Iz=-I2W-I  W,
Wz=1-12 |W|2+λ216π2I 2I-λ232π2I2 |I|2,
I1(x)-I2(-x)I1(x)+I2(-x)=f(f-l)lWnfxlδc-P2Wfxl,
Wzdiff=λ216π2I 2I-λ232π2I2 |I|2
dII=-2Wdz-I  WIdz.
Wz=1-12 |W|2.
W(z)=W(0)+zWz(0)+O(z2)W+z-z2 |W|2,
2Wdz=z2W-z24 2|W|2+O(z3)z2W-z22 (Wxx2+Wyy2+2Wxy2+WxWxxx+WxWxyy+WyWxxy+WyWyyy)=z2W-z22 [(2W)2-2WxxWyy+2Wxy2+WxWxxx+WxWxyy+WyWxxy+WyWyyy]=zH-z22 (H2-2K+T).
I(z)=I+z Iz+O(z2)=-z(I2W)=-zI(2W)=-zI[(Wxxx+Wxyy)xˆ+(Wxxy+Wyyy)yˆ].
I  WIdz=-z22 T+O(z3)
ln(I(z)/I0)=-zH+z22 (H2-2K+2T)+O(z3),
ln(I(z)/I0)=-ln(1+zH+z2K-z2T)+O(z3).
I(z)I01+zH+z2(K-T).
S=I1-I2I1+I2,
S=-zH+O(z3)1+z2(K-T).
S=I1-I22I0=-zH+O(z3)(1+zH+z2(K-T))(1-zH+z2(K-T))=-zH+O(z3)1+z2(2K-2T-H2).
ϕ=m=1dmZm,
dm=π-1-11-1-y21-y2ϕZmdxdy.
ϕ=2πλ W.
H=k-1-11-1-y21-y22ϕ(αx)2+2ϕ(αy)2dxdy,
H=192d4α-2k-1,
K=(48d42-24d52-24d62+144d72+144d82-144d92-144d102)α-4k-2,
T=(192d72+192d82+18432d2d8+18432d3d7)α-4k-2.
E[S2]=zE[H2]=z14.1α-7/6r0-5/6k-1.
z2E[(K-T)2]=z21150α-7/3r0-5/3k-2.
dI=-I  Wdz.
I(x, y, 0)=I0-I0U(x-R),
Ix(x, y, 0)=-I0δ(x-R),
Ix(x, y, z)=-I0δ(x-R-zWx(R, y, 0)).
dI=I00zδ(x-R-zWx)Wxdz=I0[1-U(x-R-zWx)]0z=I0[U(x-R)-U(x-R-zWx)].
I(x, y, z)=I0-I0U(x-R-zWx).
I(x, y, z)=-I0δ(x-R-zWx)xˆ+I0zWxyδ(x-R-zWx)yˆ.
(W)z=Wz=-12 |W|2=-12(Wx2+Wy2)xxˆ+(Wx2+Wy2)yyˆ=-(WxWxx+WyWxy)xˆ-(WxWxy+WyWyy)yˆ.
Wx(R+zWx, y, z)=Wx+zWxWxx+zWxz+O(z2)=Wx-zWyWxy.
Wy(R+zWx, y, z)=Wy+zWxWxy+zWyz+O(z2)=Wy-zWyWyy.
dI=-0zI  Wdz=I00z[Wx-zWyWxy-zWxy(Wy-zWyWyy)]×δ(x-R-zWx)dz.
dI=I00z(Wx-2zWyWxy)δ(x-R-zWx)dz.
zδ(x-az)dz=-xa2 U(x-az),
dI=I01-2(x-R) WyWxyWx2×[U(x-R)-U(x-R-zWx)],
I(z)=I01-U(x-R-zWx)-2(x-R) WyWxyWx2×[U(x-R)-U(x-R-zWx)].
I(z)=I01-U(ρ-R-zWρ)-2(ρ-R)R2W0Wρ0Wρ2×[U(ρ-R)-U(ρ-R-zWρ)].
2II=2(I-zI2W)I=-z(Wxxxx+2Wxxyy+Wyyyy),
|I|I2=z2[(Wxxx+Wxyy)2+(Wxxy+Wyyy)2].
Wzdiff=-zλ216π2 (Wxxxx+2Wxxyy+Wyyyy)
W(z)=W(0)-z2λ232π2×(Wxxxx+2Wxxyy+Wyyyy)+O(z3).
ln(I(z)/I0)=0z2z2λ232π2 (Wxxxx+2Wxxyy+Wyyyy)dz=z3λ296π2 (Wxxxxxx+3Wxxxxyy+3Wxxyyyy+Wyyyyyy)+O(z4).
D=λ296π2 (Wxxxxxx+3Wxxxxyy+3Wxxyyyy+Wyyyyyy),
S=-zH+z3D+O(z3)1+z2(K-T).
W(x)=sin2πa x,
z3λ296π22πa6=12 z2πa2.
a=3-1/4πλzλz,
S¯=EI1¯+n1-I2¯-n2I1¯+n1+I2¯+n2I1¯-I2¯I1¯+I2¯,
Var(S)=EI1-I2I1+I2-I1¯-I2¯I1¯+I2¯2=E2(I1I2¯-I1¯I2)(I1+I2)(I1¯+I2¯)2=4En12I2¯2+2n1n2I1¯I2¯+n22I1¯2(I1¯+n1+I2¯+n2)2(I1¯+I2¯)24 I1¯2I2¯+I2¯2I1¯(I1¯+I2¯)2(I1¯+I2¯)2=4 I1¯I2¯(I1¯+I2¯)3=1-S¯2N.
Var(2W)=z-2 Var(S).
Var(S)=EI1-I2I1¯+I2¯-I1¯-I2¯I1¯+I2¯2=1N.

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