Abstract

We describe and analyze an interferometer-based virtual modal wavefront sensor (VMWS) that can be configured to measure, for example, Zernike coefficients directly. This sensor is particularly light efficient because the determination of each modal coefficient benefits from all the available photons. Numerical simulations show that the VMWS outperforms state-of-the-art phase unwrapping at low light levels. Including up to Zernike mode 21, aberrations can be determined with a precision of about 0.17 rad (λ/37) using low resolution (65 × 65 pixels) images and only about 400 photons total.

© 2006 Optical Society of America

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    [CrossRef]

2004 (3)

2003 (3)

V. V. Volkov, and Y.M. Zhu, "Deterministic phase unwrapping in the presence of noise," Opt. Lett. 28, 2156-2158 (2003).
[CrossRef] [PubMed]

R. Gens, "Two-dimensional phase unwrapping for radar interferometry: developments and new challenges," Int. J. Remote Sens. 24, 703-710 (2003).
[CrossRef]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

2002 (1)

J. D. Barchers, and T. A. Rhoadarmer, "Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function," Appl Opt. 41, 7499-7509 (2002).
[CrossRef]

2000 (4)

1997 (1)

1996 (1)

1995 (3)

C. A. Primmerman, T. R. Price, R.A. Humphreys, B. G. Zollars, H.T. Barclay, and J. Herrmann, "Atmospheric-Compensation Experiments in Strong-Scintillation Conditions," Appl. Opt. 34, 2081-2088 (1995).
[CrossRef] [PubMed]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, "Fundamental Performance Comparison of a Hartmann and a Shearing Interferometer Wave-Front Sensor," Appl Opt. 34, 4186-4195 (1995).
[CrossRef] [PubMed]

R. Lynch, "The Quantum Phase Problem - a Critical-Review," Phys. Rep. 256,368-436 (1995).
[CrossRef]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, "Two-Photon Laser Scanning Fluorescence Microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

1989 (1)

J. F. Bille, B. Grimm, J. Liang, and K. Mueller, "Imaging of the retina by scanning laser tomography," in New Methods in Microscopy and Low Light ImagingProc. SPIE 1161,417-425 (1989).

1988 (1)

1987 (1)

1985 (1)

1983 (1)

1981 (1)

1976 (1)

1971 (1)

R. V. Shack, and B. C. Platt, "Lenticular Hartmann-screen," Optical Sciences Center Newsletter 5, 15-16 (1971).

1969 (1)

R. Crane, "Interference Phase Measurement," Appl. Opt. 8, 538-542 (1969).

1966 (1)

P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
[CrossRef]

1963 (2)

R. J. Glauber, "Quantum Theory of Optical Coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

R. J. Glauber, "Coherent and Incoherent States of Radiation Field," Phys. Rev. 131, 2766-2788 (1963).
[CrossRef]

1953 (1)

H. W. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229-236 (1953).
[CrossRef]

1934 (1)

F. Zernike, "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica 1, 689-704 (1934).
[CrossRef]

Albert, O.

Babcock, H. W.

H. W. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229-236 (1953).
[CrossRef]

Barchers, J. D.

J. D. Barchers, and T. A. Rhoadarmer, "Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function," Appl Opt. 41, 7499-7509 (2002).
[CrossRef]

Barclay, H.T.

Bille, J. F.

J. F. Bille, B. Grimm, J. Liang, and K. Mueller, "Imaging of the retina by scanning laser tomography," in New Methods in Microscopy and Low Light ImagingProc. SPIE 1161,417-425 (1989).

Booth, M.

Booth, M. J.

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, "Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry," J. Microsc. 213, 11-19 (2004).
[CrossRef]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, "Adaptive aberration correction in a two-photon microscope," J Microsc. 200, 105-108 (2000).
[CrossRef] [PubMed]

M. A. Neil, M. J. Booth, and T. Wilson, "New modal wave-front sensor: a theoretical analysis," J. Opt. Soc. Am. A 17, 1098-1107 (2000).
[CrossRef]

Brohinsky, W. R.

Carré, P.

P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Chen, C. W.

Crane, R.

R. Crane, "Interference Phase Measurement," Appl. Opt. 8, 538-542 (1969).

Dai, G.

Denk, W.

M. Feierabend, M. Ruckel, and W. Denk, "Coherence-gated wave-front sensing in strongly scattering samples," Opt. Lett. 29, 2255-2257 (2004).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, "Two-Photon Laser Scanning Fluorescence Microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Elbaum, M.

Ellerbroek, B. L.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, "Fundamental Performance Comparison of a Hartmann and a Shearing Interferometer Wave-Front Sensor," Appl Opt. 34, 4186-4195 (1995).
[CrossRef] [PubMed]

Feierabend, M.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Gens, R.

R. Gens, "Two-dimensional phase unwrapping for radar interferometry: developments and new challenges," Int. J. Remote Sens. 24, 703-710 (2003).
[CrossRef]

Ghiglia, D. C.

Glauber, R. J.

R. J. Glauber, "Quantum Theory of Optical Coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

R. J. Glauber, "Coherent and Incoherent States of Radiation Field," Phys. Rev. 131, 2766-2788 (1963).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Grimm, B.

J. F. Bille, B. Grimm, J. Liang, and K. Mueller, "Imaging of the retina by scanning laser tomography," in New Methods in Microscopy and Low Light ImagingProc. SPIE 1161,417-425 (1989).

Hee, M.R.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Herrmann, J.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Humphreys, R.A.

Juskaitis, R.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, "Adaptive aberration correction in a two-photon microscope," J Microsc. 200, 105-108 (2000).
[CrossRef] [PubMed]

Kawata, S.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, "Adaptive aberration correction in a two-photon microscope," J Microsc. 200, 105-108 (2000).
[CrossRef] [PubMed]

Kinnstaetter, K.

Liang, J.

J. F. Bille, B. Grimm, J. Liang, and K. Mueller, "Imaging of the retina by scanning laser tomography," in New Methods in Microscopy and Low Light ImagingProc. SPIE 1161,417-425 (1989).

Liang, J. Z.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Lohmann, A. W.

Lynch, R.

R. Lynch, "The Quantum Phase Problem - a Critical-Review," Phys. Rep. 256,368-436 (1995).
[CrossRef]

Mastin, G. A.

Miller, D. T.

Mourou, G.

Mueller, K.

J. F. Bille, B. Grimm, J. Liang, and K. Mueller, "Imaging of the retina by scanning laser tomography," in New Methods in Microscopy and Low Light ImagingProc. SPIE 1161,417-425 (1989).

Neil, M. A.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

M. A. Neil, M. J. Booth, and T. Wilson, "New modal wave-front sensor: a theoretical analysis," J. Opt. Soc. Am. A 17, 1098-1107 (2000).
[CrossRef]

Neil, M. A. A.

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, "Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry," J. Microsc. 213, 11-19 (2004).
[CrossRef]

M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, "Adaptive aberration correction in a two-photon microscope," J Microsc. 200, 105-108 (2000).
[CrossRef] [PubMed]

Noll,

Norris, T. B.

Nowakowski, J.

Ota, T.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

Oughstun, K. E.

Pennington, T. L.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, "Fundamental Performance Comparison of a Hartmann and a Shearing Interferometer Wave-Front Sensor," Appl Opt. 34, 4186-4195 (1995).
[CrossRef] [PubMed]

Platt, B. C.

R. V. Shack, and B. C. Platt, "Lenticular Hartmann-screen," Optical Sciences Center Newsletter 5, 15-16 (1971).

Price, T. R.

Primmerman, C. A.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Rhoadarmer, T. A.

J. D. Barchers, and T. A. Rhoadarmer, "Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function," Appl Opt. 41, 7499-7509 (2002).
[CrossRef]

Roggemann, M. C.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, "Fundamental Performance Comparison of a Hartmann and a Shearing Interferometer Wave-Front Sensor," Appl Opt. 34, 4186-4195 (1995).
[CrossRef] [PubMed]

Romero, L. A.

Ruckel, M.

Schuman, J.S.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Schwertner, M.

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, "Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry," J. Microsc. 213, 11-19 (2004).
[CrossRef]

M. Schwertner, Booth, M. J.  and T. Wilson, "Characterizing specimen induced aberrations for high NA adaptive optical microscopy," Opt. Express 12, 6540 - 6552 (2004).
[CrossRef] [PubMed]

Schwider, J.

Shack, R. V.

R. V. Shack, and B. C. Platt, "Lenticular Hartmann-screen," Optical Sciences Center Newsletter 5, 15-16 (1971).

Sherman, L.

Stetson, K. A.

Stinson, W.G.

D. Huang, E. A. Swanson, C. P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Streibl, N.

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, "Two-Photon Laser Scanning Fluorescence Microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Sugiura, T.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, "Enhancement of laser trapping force by spherical aberration correction using a deformable mirror," Jpn. J. Appl. Phys. 42, L701-L703 (2003).
[CrossRef]

Swanson, E. A.

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Figures (13)

Fig. 1.
Fig. 1.

Block diagram of a virtual modal wavefront sensor: Sample beam S and reference beam R are combined to interfere on camera C where interferograms with different phase shifts (introduced by the phase stepper PS) are recorded. After that all processing occurs in a computer.

Fig. 2.
Fig. 2.

Detector response for defocus: Estimated defocus (Eq. (4) using method B) vs. actual defocus contained in the wavefront for scaling factors of b = 0.7 rad (Solid line) and 0.9 rad (Dashed line). The negative abscissa is truncated, because of the point symmetry (f(-x)=-f(x)) of the response function. Numbered circles: Iterations for an actual defocus of 3.20 rad, which gives an initial estimate of 0.34 rad, which is then subtracted from the wavefront. This gives after one iteration a remaining defocus of 2.86 rad (point 2) and so on. After 19 iterations the residual wavefront is flat and no further defocus is sensed. Note the sign reversal in the detected aberration. Calculations were performed on a grid of 33 × 33 points.

Fig. 3.
Fig. 3.

Detector responses for Zernike modes 5,7,9,16. Note that the range and scale are different from Fig. 2.

Fig. 4.
Fig. 4.

Convergence behavior of the VMWS (using method B, grid size 65 × 65 points). For each wave shape the aberration strength was gradually increased from 0.2 to 6 rad: Panel (a): Final wavefront measurement error for 100 different initial distortion wave shapes (all containing modes z1 - z21 ). Panel (b): Failure probability of convergence vs. aberration strength.

Fig. 5.
Fig. 5.

Convergence behavior of the VMWS. For several examples the deviation between estimated and actual wavefront is shown as the iteration progresses. In four cases, the wavefronts (circles, stars, points, crosses) contained the modes z1 - z21 , all with a total aberration of 3.5 rad but different coefficient compositions; z2 - z21 were sensed. One wavefront (squares) had an aberration of 2 rad. Another wavefront (diamonds) contained only defocus (z4 , 3.2 rad) and only that mode was sensed. For the traces starting at 3.5 rad, only every other data point is shown.

Fig. 6.
Fig. 6.

Range of convergence for the different normalization methods (A, B, C) and different grid spacings. Circles: Wavefronts containing modes z1 - z21 . Squares: Wavefronts without tilt and defocus. The symbols are spread out slightly in horizontal direction to show error bars more clearly. For method C the convergence range on the 17 × 17 points grid was smaller than 0.2 rad (smallest aberration tested). Note that the error bars indicate the standard deviation of distribution of the convergence ranges, which vary strongly with wave shape.

Fig. 7.
Fig. 7.

Number of iterations required for convergence to better than 2 mrad as a function of the aberration strength. Data for methods A and B are shown. Computation was on a 33 × 33 grid. The data points are averages over 100 different wave-shapes, all containing the aberration modes z1 - z21 .

Fig. 8.
Fig. 8.

Dependence of convergence ranges on contained and sensed aberration modes for different grid spacings; evaluation method B was used. The symbols are slightly offset horizontally to show error bars more clearly (grids were 17 × 17, 33 × 33, 65 × 65, 129 × 129, 257 × 257 points). Error bars show the spread (rms) of the convergence ranges for different wavefronts. Simulations were done with 100 different wavefronts for each data point.

Fig. 9.
Fig. 9.

Precision of the wavefront measurement in the bright-illumination limit. Plotted is the difference between measured and original wavefront as a function of the number of grid points per direction. (a): Aberrations up to mode z10 present and measurements up to mode z10 (circles) and mode z21 (points). The symbols are spread out slightly horizontally to show error bars more clearly. (b): Aberrations up to mode z21 present and measured. (c): Aberrations up to mode z28 present and measured.

Fig. 10.
Fig. 10.

Fraction of correctly converging calculations (out of 2500 for each data point), vs. the average photon number in the sample arm. Grid sizes were 65 × 65 and 33 × 33 points.

Fig. 11.
Fig. 11.

Simulated interference patterns (a) for bright-illumination and (b) for low light levels (100 photons from the sample arm). Note the actual number of photons impinging on the detector is much higher due to light from the reference arm; but only sample-arm photons carry the wavefront information. Scaling in (b) is such that averaging of many noisy interferograms would produce an image identical to (a). The numbers next to the gray level calibration bar indicates the number of detected photons per pixel in the noisy case. The reference arm contained on average 200 times more photons than the sample arm.

Fig. 12.
Fig. 12.

rms error for modes z2 - z21 . The number of photons refers to the average total photon number from the sample.

Fig. 13.
Fig. 13.

Comparison between the VMWS and phase unwrapping (PU). Error of the wavefront measurement vs. the number of photons from the sample (total number in all four interferograms). The solid line (Δφ = 3.4/n0.51) is a fit to the VMWS data (2 rad initial rms deviation, different data set from Fig. 12), with the first three points excluded as outliers. Initial aberrations were 1 rad and 2 rad. Phase unwrapping was performed on grids of 64 × 64 and 128 × 128 pixels. The VMWS showed the same accuracy for grids of 33 × 33 and 65 × 65 points (not shown).

Equations (8)

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φ ( x , y ) = arctan { [ I 4 ( x , y ) I 2 ( x , y ) ] [ I 1 ( x , y ) I 3 ( x , y ) ] }
h i ± = E φ E i ± 2 = A exp [ ( x , y ) ] × exp [ j b z i ( x , y ) ] dxdy 2 ,
Method A : g i = h i + h i ,
Method B : g i = ( h i + h i ) ( h i + + h i ) ,
Method C : g i = ( h i + h i ) ( h i + + h i + γ h 0 ) ,
d i = p i × g i .
I ( k ) ( x , y ) = I r ( x , y ) + I s ( x , y ) + 2 I r ( x , y ) I s ( x , y ) cos ( Φ ( x , y ) + ( k 1 ) π 2 ) ,
Δφ = 1 2 [ 1 n s ( 1 + n s n r + 4 n c n r ) ] 1 2 ,

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