Abstract

Two modified Fried wavefront reconstructors are proposed, both based on an enhanced geometry that combines and balances those of Fried and Hudgin with an additional weight. The optimal weights for both of them are derived with the analytical frequency response functions, which can provide near-unity spatial frequency response over broad bandwidth to the Shack–Hartmann wavefront sensing system. Comparisons between the proposed reconstructors and the classical ones are presented in the frequency domain, and simulations have further confirmed the frequency characteristic and the reconstruction performance of the new reconstructors. It is expected that the optimized reconstructors can help improve the performance of adaptive optics systems for high-power laser wavefront control and other relevant optical systems that require high-accuracy wavefront sensing.

© 2012 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).
  2. R. K. Tyson, Principles of Adaptive Optics (CRC Press, 2011).
  3. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).
  4. B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77, 053103 (2006).
    [CrossRef]
  5. A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
    [CrossRef]
  6. S.-W. Bahk, “Band-limited wavefront reconstruction with unity frequency response from Shack-Hartmann slopes measurements,” Opt. Lett. 33, 1321–1323 (2008).
    [CrossRef]
  7. S.-W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express 19, 18997–19014 (2011).
    [CrossRef]
  8. X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).
  9. J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
    [CrossRef]
  10. K. L. Baker and M. M. Moallen, “Iteratively weighted centroiding for Shack-Hartmann wave-front sensors,” Opt. Express 15, 5147–5159 (2007).
    [CrossRef]
  11. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
    [CrossRef]
  12. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
    [CrossRef]
  13. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
    [CrossRef]
  14. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. A 69, 972–977 (1979).
    [CrossRef]
  15. L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wave-front reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
    [CrossRef]
  16. L. Lessard, M. West, D. MacMynowski, A. Bouchez, and S. Lall, “Experimental validation of single-iteration multigrid wavefront reconstruction at the Palomar Observatory,” Opt. Lett. 33, 2047–2049 (2008).
    [CrossRef]
  17. L. A. Poyneer, D. T. Gavel, and 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]

2011

2008

2007

2006

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77, 053103 (2006).
[CrossRef]

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

2005

A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
[CrossRef]

2003

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

2002

2001

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).

1980

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

1979

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

1977

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
[CrossRef]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
[CrossRef]

Bahk, S.-W.

Baker, K. L.

Bouchez, A.

Brase, J. M.

Brooks, A.

A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
[CrossRef]

Cubalchini, R.

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

Fried, D. L.

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
[CrossRef]

Gao, X.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Gavel, D. T.

Guan, Y.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

He, J.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Hudgin, R. H.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
[CrossRef]

Kelly, T.-L.

A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
[CrossRef]

Lall, S.

Lessard, L.

Lübbecke, M.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77, 053103 (2006).
[CrossRef]

MacMynowski, D.

Mann, K.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77, 053103 (2006).
[CrossRef]

Moallen, M. M.

Munch, J.

A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
[CrossRef]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).

Poyneer, L. A.

Primot, J.

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

Schäfer, B.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77, 053103 (2006).
[CrossRef]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).

Southwell, W. H.

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

Su, Y.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (CRC Press, 2011).

Veitch, P.

A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
[CrossRef]

West, M.

Xie, C.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Ye, Y.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Yuan, X.

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

Gen. Relativ. Gravit.

A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, “An off-axis Hartmann sensor for the measurement of absorption-induced wavefront distortion in advanced gravitational wave interferometers,” Gen. Relativ. Gravit. 37, 1575–1580 (2005).
[CrossRef]

High Power Laser Particle Beams

X. Gao, Y. Su, C. Xie, J. He, X. Yuan, Y. Guan, and Y. Ye, “Centroid position of focal spot of laser beam with aberration,” High Power Laser Particle Beams 18, 717–719 (2006) (in Chinese).

J. Opt. Soc. Am. A

L. A. Poyneer, D. T. Gavel, and 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]

L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wave-front reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
[CrossRef]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
[CrossRef]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
[CrossRef]

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

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

J. Refract. Surg.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).

Opt. Commun.

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Rev. Sci. Instrum.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77, 053103 (2006).
[CrossRef]

Other

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

R. K. Tyson, Principles of Adaptive Optics (CRC Press, 2011).

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Figures (9)

Fig. 1.
Fig. 1.

One-dimensional representation of the principle of the S-H wavefront sensor.

Fig. 2.
Fig. 2.

Reconstruction geometries of classical reconstructors: (a) Hudgin, (b) Fried, and (c) Southwell. Dots are sampling points of the wavefront, dashed arrows represent the first difference of phase, and red solid arrows show the slope measurement in two orthogonal directions, respectively. The red dashed squares stand for the area of the subapertures. sx and sy are the slope measurements as defined in the text.

Fig. 3.
Fig. 3.

Reconstruction geometries of the modified reconstructors: (a) modified Hudgin and (b) weighted Fried I and II. Only the relationship in the x direction is shown for simplicity.

Fig. 4.
Fig. 4.

2-D Amplitude spectrum of system frequency responses derived from analytical functions: (a) Hudgin’s, (b) Fried’s, (c) Southwell’s, (d) modified Hudgin’s, (e) weighted Fried I’s, and (f) weighted Fried II’s. As there are poles in high frequencies of Fried’s and weighted Fried I’s responses, only part of their spectrums are drawn.

Fig. 5.
Fig. 5.

1-D cross-sectional curves of system frequency responses with parameter α=β; for Fried’s and weighted Fried I’s, only parts of their curves are drawn.

Fig. 6.
Fig. 6.

Simulation and analytical system frequency responses (amplitude spectrums). For Fried’s and weighted Fried I’s, only parts of their curves are drawn.

Fig. 7.
Fig. 7.

RMS and P-V of the reconstructions by Fried and weighted Fried I and II.

Fig. 8.
Fig. 8.

Relation between residual wavefront MSE and noise variance, in the case of the reconstructions by Fried and weighted Fried I and II.

Fig. 9.
Fig. 9.

Sensitivity curves of the optimal weight to integration limits in the case of weighted Fried I and weighted Fried II.

Tables (1)

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Table 1. Noise Gains of the Reconstructors

Equations (41)

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{xc,j,k=xIi(x,y)dxdyIi(x,y)dxdyφx,j,kdxdyyc,j,k=yIi(x,y)dxdyIi(x,y)dxdyφy,j,kdxdy,
{φ^x,j,k=12[(φ^j,kφ^j,k+1)+(φ^j+1,kφ^j+1,k+1)]φ^y,j,k=12[(φ^j,kφ^j+1,k)+(φ^j,k+1φ^j+1,k+1)],
{φ^x,j,k=φ^j,kφ^j,k+1φ^y,j,k=φ^j,kφ^j+1,k.
{φ^x,j1,k+φ^x,j,k2=φ^j,kφ^j,k+1φ^y,j,k1+φ^y,j,k2=φ^j,kφ^j+1,k.
LSE[φ^]=[(φ^x,j,ksx,j,k)2+(φ^y,j,ksy,j,k)2],
RI=wI1+wIRFried+11+wIRMHudgin,
φ^=RI(sysx).
{φ^x,j1,k=12[(φ^j1,kφ^j1,k+1)+(φ^j,kφ^j,k+1)]φ^y,j,k1=12[(φ^j,k1φ^j+1,k1)+(φ^j,kφ^j+1,k)].
{φ^x,j1,k+φ^x,j,k2=wII1+wII[(φ^j1,kφ^j1,k+1)2+(φ^j,kφ^j,k+1)+(φ^j+1,kφ^j+1,k+1)2]+1wII1+wII(φ^j,kφ^j,k+1)φ^y,j,k1+φ^y,j,k2=wII1+wII[(φ^j,k1φ^j+1,k1)2+(φ^j,kφ^j+1,k)+(φ^j,k+1φ^j+1,k+1)2]+1wII1+wII(φ^j,kφ^j+1,k),
{sx,j,k=φ^x(x,y)rect(xxo,j,kd)rect(yyo,j,kd)dxdysy,j,k=φ^y(x,y)rect(xxo,j,kd)rect(yyo,j,kd)dxdy,
{FT[sx]=FT[φ]×[ikxsinc(απ)sinc(βπ)]FT[sy]=FT[φ]×[ikysinc(απ)sinc(βπ)],
FTFried[φ^]=i2·sinβcosαFT[sy]+sinαcosβFT[sx]sin2αcos2β+sin2βcos2α.
FTMHudgin[φ^]=i2·sinβcosαFT[sy]+sinαcosβFT[sx]sin2α+sin2β.
FTI[φ^]=11+wI(FTMHudgin[φ^]+wIFTFried[φ^]),
FTII[φ^]=i2(1+wII)[sinβcosα(1+wIIcos2α)FT[sy]sin2α(1+wIIcos2β)2+sinαcosβ(1+wIIcos2β)FT[sx]+sin2β(1+wIIcos2α)2].
GI(α,β)=11+wI(1sin2α+sin2β+wIsin2αcos2β+sin2βcos2α)×(βsinβcosα+αsinαcosβ)×sinc(απ)sinc(βπ)
GII(α,β)=(1+wII)[βsinβcosα(1+wIIcos2α)sin2α(1+wIIcos2β)2+αsinαcosβ(1+wIIcos2β)+sin2β(1+wIIcos2α)2]×sinc(απ)sinc(βπ)
GI(α,α)=11+wI(cosα+wIcosα)sinαα,
GII(α,α)=(1+wII)cosα(1+wIIcos2α)sinαα.
wopt=argminw0.4π0.4π[|G(α,α)|1]2dα,
wI,opt=0.80264/5,
wII,opt=0.66172/3.
GHudgin(α,β)=βsinβexp(iα)+αsinαexp(iβ)sin2α+sin2β×sinc(απ)sinc(βπ),
GFried(α,β)=βsinβcosα+αsinαcosβsin2αcos2β+sin2βcos2α×sinc(απ)sinc(βπ),
GSouthwell(α,β)=βsin2β+αsin2αsin2α+sin2β×sinc(απ)sinc(βπ).
MSE=MSEφ+σn2MSEnp,
MSEnp=Trace(RRT)N2,
F(w)=a1a2[|Gw(α,α)|1]2dα,
F(w)w=a1a2{2[Gw(α,α)1]Gw(α,α)w}dα,
GI(α,α)=11+wI[sin(2α)2αtanαα]+tanαα.
FIwI=2a1a2{11+wI[sin(2α)2αtanαα]+tanαα1}×1(1+wI)2[sin(2α)2αtanαα]dα,
FIwI=1(1+wI)2(C11+wI+C2),
C1=2a1a2[sin(2α)2αtanαα]2dα,
C2=2a1a2[sin(2α)2αtanαα]×(tanαα1)dα.
wI=(C1C2+1).
GII(α,α)=(1+wII)11+w¯cos2αsin(2α)2α.
dFII(wII)dwII=2a1a2[(1+wII)11+w¯cos2αsin(2α)2α1]×11+w¯cos2αsin(2α)2αdα,
dFII(wII)dwII=D1wII+D2D1,
D1=2a1a2[11+w¯cos2αsin(2α)2α]2dα,
D2=2a1a211+w¯cos2αsin(2α)2αdα.
wII=D2D1D1.

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