## Light-efficient, quantum-limited interferometric wavefront estimation by virtual mode sensing

Optics Express, Vol. 14, Issue 9, pp. 3700-3714 (2006)

http://dx.doi.org/10.1364/OE.14.003700

Acrobat PDF (245 KB)

### 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

## 1. Introduction

1. H. W. Babcock, “The possibility of compensating astronomical seeing,” Publications of the Astronomical Society of the Pacific **65**, 229–236 (1953). [CrossRef]

4. J. Z. Liang, D. R. Williams, and D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A **14**, 2884–2892 (1997). [CrossRef]

5. 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]

6. K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,” J. Opt. Soc. Am. **71**, 862–872 (1981). [CrossRef]

7. 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]

8. 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]

9. D. T. Miller, J. Qu, R. S. Jonnal, and K. Thorn, “Coherence Gating and Adaptive Optics in the Eye,” in *Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII*, ValeryV. Tuchin, Joseph A. Izatt, and James G. Fujimoto, eds., Proc. SPIE **4956**, 65–72 (2003). [CrossRef]

10. O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. **25**, 52–54 (2000). [CrossRef]

11. 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]

12. 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]

13. J. Nowakowski and M. Elbaum, “Fundamental Limits in Estimating Light Pattern Position,” J. Opt. Soc. Am. **73**, 1744–1758 (1983). [CrossRef]

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

16. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia **2**, 13–23 (1966). [CrossRef]

20. 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]

22. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-Automata Method for Phase Unwrapping,” J. Opt. Soc. Am. A **4**, 267–280 (1987). [CrossRef]

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

^{0}norm in polynomial time [24

24. C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A **17**, 401–414 (2000). [CrossRef]

26. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica **1**, 689–704 (1934). [CrossRef]

27. R. J. Noll, “Zernike Polynomials and Atmospheric-Turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976). [CrossRef]

28. M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wavefront sensor,” in *Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

*z*

_{5},

*z*

_{6}) and coma (

*z*

_{7},

*z*

_{8}). Modal decomposition into other complete function sets, such as disk harmonic functions [29

29. N. M. Milton and M. Lloyd-Hart, “Disk harmonic functions for adaptive optics simulations,” in *Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal
Recovery and Synthesis; Topical Meetings on CD-ROM* (The Optical Society of America, Washington, DC, 2005),AW3.

30. 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]

11. 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]

32. G. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A **13**, 1218–1225 (1996). [CrossRef]

28. M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wavefront sensor,” in *Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

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

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

## 2. Methodology

28. M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wavefront sensor,” in *Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

30. 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]

*i*

^{th}Zernike mode) is added and subtracted, respectively, by the phase plates. The difference in the signals from two pinhole detectors, onto which the wavefronts are focused, is a measure of the amount of the tested aberration mode contained in the beam. While for small aberrations the modes are sensed independently, larger aberrations cause modal crosstalk and require an iterative procedure whereby the estimated aberrations are removed using a wavefront shaping element in the input path and a new modal measurement is made, which increases the amount of light required. Furthermore, to measure multiple modes, the available light has to be split or measurements of different modes have to be performed sequentially.

*I*

_{1}(

*x*,

*y*) to

*I*

_{4}(

*x*,

*y*)) are recorded with reference-path shifts of 0, λ/4, λ/2 and 3λ/4. The spatial location (

*x*,

*y*) is confined to a circular aperture, and is expressed in units of the aperture radius. While three phase steps are minimally needed we chose a four-step algorithm because it is less sensitive to second-order detector nonlinearities [35

35. K. Kinnstaetter, A. W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of Phase-Shifting Interferometry,” Appl. Opt. **27**, 5082–5089 (1988). [CrossRef] [PubMed]

36. K. A. Stetson and W. R. Brohinsky, “Electrooptic Holography and its Application to Hologram Interferometry,” Appl. Opt. **24**, 3631–3637 (1985). [CrossRef] [PubMed]

*x*,

*y*) is calculated [37] using:

*h*

_{i+}and

*h*

_{i-}(which correspond to the intensities measured at the pinhole detector in a regular MWS) are calculated

*E*

_{φ}= exp[

*j*φ(

*x*,

*y*)],

*E*

_{i±}=exp[±

*jbz*

_{i}(

*x*,

*y*)],

*z*

_{i}is the

*i*

^{th}function in the estimator set,

*b*is a scaling factor (typically around 0.7), chosen to minimize crosstalk [28

*Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

*A*the aperture, and

*j*= √-1. The next step is to calculate for each mode a “deviation signal”

*g*

_{i}. Three slightly different normalization methods have been introduced [28

*Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

30. 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]

*h*

_{0}= 〈

*E*

_{φ}|

*E*

_{φ}〉 and γ is a constant (typically around 1) adjusted for best linearity [28

*Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

*Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

*Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

*g*

_{i}

*s*are, for small aberrations, proportional to the amount of mode

*i*contained in the wavefront φ(

*x*,

*y*) and we can make first estimates of the Zernike coefficients

*d*

_{i}by using

*p*

_{i}

*s*can be calculated analytically [30

**17**, 1098–1107 (2000). [CrossRef]

*Advanced Wavefront Control: Methods, Devices, and Applications*,
John D. Gonglewski, Mikhail A. Vorontsov, and Mark T. Gruneisen, eds., Proc. SPIE **5162**, 79–90 (2003). [CrossRef]

^{(0)}(

*x*,

*y*) = ∑

_{i}

*z*

_{i}(

*x*,

*y*). The difference between estimate and actual wavefront, Δ

^{(0)}= φ -

^{(0)}, is now taken as the new input to the estimation algorithm, yielding coefficients

^{(n+1)}(

*x*,

*y*) = ∑

_{i}

*z*

_{i}(

*x*,

*y*); Δ

^{(n+1)}= Δ

^{n}-

^{(n+1)}is the new input for the next step. The final estimate is ∑

_{n}∑

_{i}

*z*

_{i}(

*x*,

*y*).

^{(n)}|

^{(n)}〉)

^{5 0}went below 0.01 rad (which reliably leads to very small final errors). Here, as in the following all deviations and errors are given as root mean square (RMS). Convergence to “false” fixpoints does occur but is easily detectable (see below). The modal representation is determined directly (without unwrapping) form the measured phase.

## 3. Performance of the VMWS:

### 3.1 Methods

*D*

_{i}a wavefront Φ(

*x*,

*y*) was calculated: Φ(

*x*,

*y*) = ∑

_{i}

*D*

_{i}×

*z*

_{i}(

*x*,

*y*), whereby

*z*

_{i}(

*x*,

*y*), is the

*i*

^{th}Zernike polynomial with the normalized lateral position (

*x*,

*y*) limited to a circular aperture (

*x*

^{2}+

*y*

^{2}≤ 1). Then the phase-shifted interferograms (

*I*

_{1}(

*x*,

*y*) to

*I*

_{4}(

*x*,

*y*)) were calculated:

*I*

_{r}(

*x*,

*y*) and

*I*

_{s}(

*x*,

*y*) are the intensities in the reference and sample arms of the interferometer, respectively;

*k*∈{1,2,3,4} is the phase-shift index. In the shot-noise regime a random integer (number of photons) was generated for each pixel using a Poisson distribution with a mean equal to the light flux. In the bright-illumination limit (high photon numbers) shot noise was neglected. In both cases uniform illumination (independent of x,y) was assumed.

*z*

_{1}to

*z*

_{21}) using uniform amplitude distributions. For each set (corresponding to a particular wave shape) the overall amplitude was increased until convergence, as tested in simulations, failed. We considered the convergence as failing if iteration yielded no further change while the sensing error still exceeded 0.3 rad (≈λ/20) or if there were still changes after more than 800 iterations, indicating oscillations. Piston,

*z*

_{1}, was not sensed but included in the set of initial aberrations. In these simulations convergence can be assessed by comparing the estimated coefficients with those given; in a real application this is not possible. In that case the correct convergence can be tested by repeating the last iteration step using a different value of the scaling factor

*b*, which shifts the false zeros in the response function (Fig. 2). For bright illumination this test was found to be equivalent to the residual-deviation criterion. Mean convergence ranges were determined by averaging the maximum aberration strengths for which the algorithm still converged.

*b*, i.e. without knowledge of the initial wavefront. The accuracies of the estimated wavefronts were calculated only for the correctly converging cases. Calculations were repeated on grids of 33 × 33 and 65 × 65 points. The reference arm was always 200 times brighter than the sample arm. The data were fitted with Matlab.

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

### 3.2 Detector response and convergence range

*z*

_{4}), using normalization method B. As expected, the detector response is almost linear for small aberrations but becomes highly nonlinear, including sign reversals, for large aberrations (Fig. 2).

*b*. This can be used to detect erroneous convergence. See Fig. 3 for the response curves for different modes.

*z*

_{4}, defocus) with only that mode present as an aberration. In this case sensor crosstalk is, of course, nonexistent but in any realistic situation, where multiple aberration modes will always be present, crosstalk needs to be considered. The interaction of different modal measurements also makes it difficult, if not impossible, to determine the region of convergence analytically and makes it meaningless to estimate the convergence behavior for each component independently.

### 3.3 Region of convergence for different grid spacings

*z*

_{21}present). This range depends on the exact aberration shape, with some shapes having considerably smaller convergence ranges.

*b*(Eq. (2)), whose value is reduced in consecutive iteration steps because higher

*b*-values shift the “false” zeros to higher values, but at the price of decreasing the linearity and hence slowing convergence for small aberrations (Fig. 2).

### 3.4 Aberration order and convergence range

*z*

_{10}and sensed to

*z*

_{21}.

### 3.5 Modal decomposition in the presence of higher-order wavefront noise

*z*

_{21}) than are contained in the estimator set might affect the region of convergence of the iterative algorithm. Therefore, we repeated the estimation of convergence ranges with wavefronts that contained the aberration modes

*z*

_{1}to

*z*

_{28}while only the modes

*z*

_{2}to

*z*

_{21}were measured with the VMWS. Deviations were again calculated only for modes

*z*

_{2}to

*z*

_{21}. We found substantially reduced convergence ranges (e.g. to 1.0 ± 0.2 rad (mean ± SE) from 3.5 ± 0.8 for method B, grid 33 × 33). This shows that it is essential to sense aberrations to a sufficiently high order.

### 3.6 Precision of the wavefront estimation

*z*

_{10},

*z*

_{21}, and

*z*

_{28}(Fig. 9). While the estimator set usually contained modes to the same order as aberrations were present we additionally tested (using calculation method B) some wavefronts that containing aberrations only to

*z*

_{10}but were measured to

*z*

_{21}. Similar to the behavior of the convergences ranges, we found that with only lower modes (up to

*z*

_{10}) present the final error is larger when including extra modes (up to

*z*

_{21}) in the estimator. In all cases average errors below 0.3 mrad were reached on a grid of 65 × 65 points, which is the grid size for which the error typically was minimal (Fig. 9). These errors are not a fundamental limit. They rather reflect numeric inaccuracies and, of course, depend on the termination criterion for the iteration (data not shown). Precision is usually improved, if the termination threshold is lowered, which comes at the expense of more iteration steps and has a higher chance of oscillations. Note, that the final error can be much smaller than the termination threshold because only the deviation from linearity in the last iteration step is seen as the final error. As for the convergence range (Fig. 6), performance is slightly reduced for very fine grids.

### 3.7 The influence of noise

*n*

_{r}and

*n*

_{s}are the total photon numbers in the reference and sample arm for four interferograms (

*I*

_{1}(

*x*,

*y*) to

*I*

_{4}(

*x*,

*y*)), respectively, nc is the camera dark noise, which was modeled as a Poisson process. The error approaches an asymptotic lower limit (Δφ

_{∞}= 1/√2

*n*

_{s}) for large numbers of reference arm photons, which is independent not only of the reference arm intensity but also of the camera dark noise (with

*n*

_{r}≥ 200

*n*

_{s}Δφ ≤ 1.005 Δφ

_{∞}). This limit is slightly higher than the naïve quantum limit for coherent states [38

38. R. J. Glauber, “Quantum Theory of Optical Coherence,” Phys. Rev. **130**, 2529–2539 (1963). [CrossRef]

39. R. J. Glauber, “Coherent and Incoherent States of Radiation Field,” Phys. Rev. **131**, 2766–2788 (1963). [CrossRef]

_{∞}≥ 1/(2√

*n*

_{s})[40

40. R. Lynch, “The Quantum Phase Problem - a Critical-Review,” Phys. Rep. **256**, 368–436 (1995). [CrossRef]

*z*

_{1}-

*z*

_{21}) could be reconstructed (correct convergence, see “Methods”) in most (97%) cases for an aberration strength of 1 rad, and for half (50%) of the cases for an aberration of 2 rad. The error of the wavefront estimate (at the endpoint of the iteration) for a termination threshold of 10 mrad was on the average 0.17 rad (lambda/37), or 0.038 rad per mode, independent of the initial aberration strength. We did not find a substantial improvement when using the finer of the two grids tested (33 × 33 and 65 × 65). Figure 11 shows a simulation of a noiseless and a noisy interference pattern with 100 photons in the sample arm.

*const*/√

*n*

_{s}for each mode with constants of proportionality that were on average 0.72 ± 0.01 (± standard error of the mean). The error expected, using Eq. (6), is ≈ 0.71/√

*n*

_{s}. The errors are roughly equal for each of the modes for a given photon number (Fig. 12). The total error thus scales as Δφ ≈ (0.72 ± 0.01)√

*n*

_{m}/

*n*

_{s}where

*n*

_{m}is the number of modes and

*n*

_{s}is the number of photons from the sample arm (Fig. 13). This value is independent of the grid spacing and the initial aberration strength (data not shown).

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

## 4. Summary and discussion

**28**, 2156–2158 (2003). [CrossRef] [PubMed]

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

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

## Acknowledgments

## References and links

1. | H. W. Babcock, “The possibility of compensating astronomical seeing,” Publications of the Astronomical Society of the Pacific |

2. | J. W. Hardy, |

3. | J. F. Bille, B. Grimm, J. Liang, and K. Mueller, “Imaging of the retina by scanning laser tomography,” in |

4. | J. Z. Liang, D. R. Williams, and D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A |

5. | 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. |

6. | K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,” J. Opt. Soc. Am. |

7. | 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. |

8. | 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 |

9. | D. T. Miller, J. Qu, R. S. Jonnal, and K. Thorn, “Coherence Gating and Adaptive Optics in the Eye,” in |

10. | O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. |

11. | 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. |

12. | 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. |

13. | J. Nowakowski and M. Elbaum, “Fundamental Limits in Estimating Light Pattern Position,” J. Opt. Soc. Am. |

14. | M. Minsky, |

15. | W. Denk, J. H. Strickler, and W. W. Webb, “Two-Photon Laser Scanning Fluorescence Microscopy,” Science |

16. | P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia |

17. | R. Crane, “Interference Phase Measurement,” Appl. Opt. |

18. | 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. |

19. | M. Feierabend, M. Ruckel, and W. Denk, “Coherence-gated wave-front sensing in strongly scattering samples,” Opt. Lett. |

20. | J. D. Barchers and T. A. Rhoadarmer, “Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function,” Appl Opt. |

21. | D. C. Ghiglia, |

22. | D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-Automata Method for Phase Unwrapping,” J. Opt. Soc. Am. A |

23. | R. Gens, “Two-dimensional phase unwrapping for radar interferometry: developments and new challenges,” Int. J. Remote Sens. |

24. | C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A |

25. | R. K. Tyson, |

26. | F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica |

27. | R. J. Noll, “Zernike Polynomials and Atmospheric-Turbulence,” J. Opt. Soc. Am. |

28. | M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wavefront sensor,” in |

29. | N. M. Milton and M. Lloyd-Hart, “Disk harmonic functions for adaptive optics simulations,” in |

30. | M. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A |

31. | R. V. Shack and B. C. Platt, “Lenticular Hartmann-screen,” Optical Sciences Center Newsletter |

32. | G. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A |

33. | M. Schwertner, M. J. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express |

34. | V. V. Volkov and Y.M. Zhu, “Deterministic phase unwrapping in the presence of noise,” Opt. Lett. |

35. | K. Kinnstaetter, A. W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of Phase-Shifting Interferometry,” Appl. Opt. |

36. | K. A. Stetson and W. R. Brohinsky, “Electrooptic Holography and its Application to Hologram Interferometry,” Appl. Opt. |

37. | D. Malacara, |

38. | R. J. Glauber, “Quantum Theory of Optical Coherence,” Phys. Rev. |

39. | R. J. Glauber, “Coherent and Incoherent States of Radiation Field,” Phys. Rev. |

40. | R. Lynch, “The Quantum Phase Problem - a Critical-Review,” Phys. Rep. |

**OCIS Codes**

(000.2170) General : Equipment and techniques

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(040.3780) Detectors : Low light level

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Atmospheric and ocean optics

**History**

Original Manuscript: February 17, 2006

Revised Manuscript: April 24, 2006

Manuscript Accepted: April 24, 2006

Published: May 1, 2006

**Virtual Issues**

Vol. 1, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Marcel A. Lauterbach, Markus Ruckel, and Winfried Denk, "Light-efficient, quantum-limited interferometric wavefront estimation by virtual mode sensing," Opt. Express **14**, 3700-3714 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-9-3700

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### References

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