Abstract

Hartmann–Shack wave-front sensors produce a distorted grid of spots whose deviation from perfection is linear with the wave-front gradient. Usually, the centroid of each spot is calculated to provide that deviation, but it is also possible to perform the calculation by Fourier demodulation of the spot pattern [Opt. Commun. 215, 285, 2003]. We show that this demodulation can be performed directly on the grid, without reverting to Fourier transforms. Tracking the motion of each centroid individually is limited to well-defined spots with motions smaller than their pitch. In contrast, our method treats the image as a whole, is not limited to nonoverlapping or sharp spots, and allows large spot motions. By replicating the array of spots slightly beyond the edge of the aperture, we reduce the chance for boundary phase dislocations in the reconstruction of the wave front. The method is especially suited to very large arrays.

© 2004 Optical Society of America

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References

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  1. D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1978).
  2. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, New York, 1998).
  3. R. K. Tyson, ed., Adaptive Optics Engineering Handbook (Marcel Decker, New York, 2000).
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    [CrossRef]
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    [CrossRef]
  9. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  10. L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
    [CrossRef]
  11. Y. Carmon, E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288 (2003).
    [CrossRef]
  12. E. N. Ribak, “Separating atmospheric layers in adaptive optics,” Opt. Lett. 28, 613–615 (2003).
    [CrossRef] [PubMed]

2003 (2)

Y. Carmon, E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288 (2003).
[CrossRef]

E. N. Ribak, “Separating atmospheric layers in adaptive optics,” Opt. Lett. 28, 613–615 (2003).
[CrossRef] [PubMed]

2002 (1)

1993 (1)

1991 (1)

1986 (1)

1980 (1)

1977 (2)

Brase, J. M.

Carmon, Y.

Y. Carmon, E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288 (2003).
[CrossRef]

Freischlad, K. R.

Fried, D. L.

Gavel, D. T.

Hudgin, R. H.

Koliopoulos, C. L.

Poyneer, L. A.

Ribak, E. N.

Y. Carmon, E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288 (2003).
[CrossRef]

E. N. Ribak, “Separating atmospheric layers in adaptive optics,” Opt. Lett. 28, 613–615 (2003).
[CrossRef] [PubMed]

Roddier, C.

Roddier, F.

Southwell, W. H.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, New York, 1998).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

Y. Carmon, E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288 (2003).
[CrossRef]

Opt. Lett. (1)

Other (3)

D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1978).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, New York, 1998).

R. K. Tyson, ed., Adaptive Optics Engineering Handbook (Marcel Decker, New York, 2000).

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Figures (3)

Fig. 1
Fig. 1

Extending the Hartmann pattern by one row of spots on each side reduces phase dislocations. The data are shifted by one period to each of the four sides of the original and these four extensions are averaged, then the original is plugged back in. Note also the reduced amplitude near the two highly aberrated regions caused by scintillation.

Fig. 2
Fig. 2

Comparison of results from three Hartmann demodulation techniques using the image from Fig. 1. Top row, x gradients; second row, y gradients; third row, integrated phases; bottom row, reference phase. Left column, convolution with a kernel of size P; center column, smoothing (single pass) as in Eqs. (6) or (B2); right column, Fourier analysis. Positive contours are marked by a central +. The elevation of the two features is 1.2 µm (third row, contour spacing 0.15 µm), to be compared with the much larger 6 µm for the references (bottom row, contour spacing 0.77 µm). The diameter of the image is 3 mm, and the extent of the round aperture is visible in some images. Note how different is the Fourier reference (bottom right), which results from a shift of the lobe to the center by integer frequencies. As a result, a tilt is added to the Fourier reference that also appears in the results and hence subtracts perfectly.

Fig. 3
Fig. 3

Simulation of cuts across the round aperture: Given a wave front (jagged curve), it is converted into a Hartmann pattern with lenslet pitch of 11 pixels. Then Poisson noise at the average levels of 16, 64, 256, and 1024 photons per lenslet was added (four groups starting from the bottom, respectively). The results are always smoother because of the finite size of the lenslets. In addition, the smoothest result is the Fourier method, then the convolution, and, very close to it, the single-pass smoothing.

Equations (51)

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I0s(x, y)=1PxPyjx=-Px/2Px/2jy=-Py/2Py/2I0(x+jxx, y+jyy).
Ir(r)=nx=-ny=-S(r-rnxny)nx=-ny=-S(x-nxPx, y-nyPy),
I(r)=nx=-ny=-S[r-rnxny-Fϕ(rnxny)]=nx=-ny=-S{[x-nxPx-Fϕx(nxPx+nyPy)],  [y-nyPy-Fϕy(nxPx+nyPy)]}.
I(r)=nx=-ny=-S[r-rnxny-Fϕ(rnxny)]lx=-ly=-s(qlxly)exp[-iqlxly·Fϕ(r)]exp(iqlxly·r),
qlxly=(lxqx, lyqy)=(2πlx/Px, 2πly/Py).
I(r)lx=-ly=-s(qlxly)A(r)exp[-iqlxly·Fϕ(r)].
[q0·Fϕ(r)]1,
|A(r)|2/A2(r)q02.
Ie(x, y)=12[I(x-Px/2, y)+I(x+Px/2, y)+I(x, y-Py/2)+I(x, y+Py/2)],
Ie(R)=I0e(R),
Ic(x, y)=Ie(x, y)exp(-2πix/Px),
I1x(x, y)=j=0Px-1Ic(x+j-Px/2, y)/Px,
I1y(x, y)=j=0Py-1I1x(x, y+j-Py/2)/Py,
I2x(x, y)=j=0Px-1I1y(x+j-Px/2, y)/Px,
I2y(x, y)=j=0Py-1I2x(x, y+j-Py/2)/Py,
ϕx(x, y)=arg[I2x(x, y)]Px/2πF.
Iobjx(r)[Irefx(r)]*=[A(r)]2 exp{-iF[ϕobjx(r)-ϕrefx(r)]},
CS(x, y)=ix=-iy=-S(x+ixLx, y+iyLy).
CS(r)=4π2LxLymx=-my=-s(qx, qy)exp(iq·r),
Ir(r)=nx,nyS(r-rnxny)nx,nyCS(r-rnxny),
Ir(x, y)=nx=1Mxny=1MyCS(x-nxPx, y-nyPy).
Ir(r)=4π2PxPymx=-my=-s(qx, qy)exp(iq·r),
I(r)=nx,nyS[r-rnxny-Fϕ(rnxny)]
nx,nyCS[r-rnxny-Fϕ(rnxny)]
nx,nyCS[r-rnxny-Fϕ(r)].
Ir(r)=A(r)4π2PxPymx=-my=-s(qx, qy)×exp(iq·r)exp[-iq·ϕ(r)].
I10S(x, y)=(PxPy)-1-Px/2Px/2dx-Py/2Py/2dy×I10(x+x, y+y)=P-2Pd2rI10(r+r),
I10S(r)=4π2PxPymx=-my=-s(q)V10(q, r),
q=(qx, qy)=2π(mx/Px, my/Py),
V10(q, r)=P-2Pd2rA(r+r)×exp[i(q-q10)·(r+r)-iq·Fϕ(r+r)].
V10(q, r)A¯(r)B(q-q10)×exp[i(q-q10)·r-iq·Fϕ¯(r)],
B(q-q10)=(PxPy)-1-Px/2Px/2dx-Py/2Py/2dy×exp{2πi[x(mx-1)/Px+ymy/Py]}=δ(mx-1)δ(my),
I10S(r)=4π2(PxPy)-1A¯(r)exp[-iq10·Fϕ¯(r)].
Im(r)=I[r+Fϕ¯(r)]/A¯(r).
Bm(q-q10)=P-2Pd2r exp[i(q-q10)·(r+r)]×[exp(-iq·δϑ+δA)],
I10m(r)=4π2P-2dA(r)exp[-iq10·Fdϕ(r)],
dϕ(r)=ϕ(r)-ϕ¯(r)0,dA(r)=A(r)/A¯(r)1
F(r)=12I0(r)+14I0(r-Px/4)+14I0(r+Px/4),
G(r)=12F(r)+14F(r-Px/4)+14F(r+Px/4),
Tf(r)=2F(r)-G(r).
Ic(x, y)=I(x, y)exp(-2πix/Px),
I1y(x, y)=12Ic(x, y)+14Ic(x, y-Py/2)+14Ic(x, y+Py/2),
I2y(x, y)=12I1y(x, y)+14I1y(x, y-Py/4)+14I1y(x, y+Py/4),
I1x(x, y)=12I2y(x, y)+14I2y(x-Px/2, y)+14I2y(x+Px/2, y),
I2x(x, y)=12I1x(x, y)+14I1x(x-Px/4, y)+14I1x(x+Px/4, y),
ϕx(x, y)=arg[I2x(x, y)]Px/2πF,
cos(q)cos(q/2)cos(q/4)=[cos(7q/8)+cos(5q/8)+cos(3q/8)+cos(q/8)]/4,
C1(q)=k=0Ncos(2kq/2N)=2-Nm=12Ncos[(2m-1)q/2N+1],
c1(x)=2-N-1m=-2N+12Nδ[x-(2m-1)/2N+1].
C2(q)=2-Nm=12Ncos[(2m-1)q/2N+1]2=2-2N-2m,k=-2N+12Nexp[i(2m+2k-2)q/2N+1]=2-2N-2p=-2N+12N+1[2N+1-|p|]exp[ipq/2N],
c2(x)=2-2N-2p=-2N+12N+1(2N+1-|p|)δ(x-p/2N).

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