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

doi:10.1364/OE.14.003700

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

Marcel A. Lauterbach, Markus Ruckel, and Winfried Denk »View Author Affiliations

Optics Express, Vol. 14, Issue 9, pp. 3700-3714 (2006)
http://dx.doi.org/10.1364/OE.14.003700

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▸ Article Outline
– Abstract
1. Introduction
2. Methodology
3. Performance of the VMWS:
4. Summary and discussion
– References and links

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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.

1. Introduction

Fast and light-efficient wavefront detectors are essential for adaptive optics (AO) [1

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

], which is used in applications that are affected by wavefront distortions due to refractive index inhomogeneities in the beam path, such as astronomy [2

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (A. Hasegawa, et al., Oxford, 1998).

], ophtalmoscopy [3

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 Imaging Proc. SPIE 1161, 417–425 (1989).

, 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]

], free space communication [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]

], laser resonators [6

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

], optical trapping [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]

], optical coherence tomography [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]

] and confocal and multi-photon microscopy [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]

]. A reference wave, which allows the direct determination of the local phase by interferometry, is not always available, notably in astronomy. Non-interferometric wavefront measurement systems need to divide the available photons, spatially or temporally, to measure the aberration’s different degrees of freedom. This imposes a tradeoff between measurement accuracy of different wavefront parameters [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]

]. Particularly in scanned-laser imaging microscopy [14

M. Minsky, Microscopy Apparatus , U.S. Patent 3013467, USA (1961).

, 15

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

] a reference wave can usually be obtained and thus direct measurements of the local phase are possible, employing, for example, a phase-shifting interferometer [16 – 19

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

] (which has been analyzed in context of point diffraction interferometers by [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]

]). Even when the local phase is known the wavefront has to be reconstructed to determine the correction parameters for the wave-shaping element, such as a deformable mirror. Reconstruction can be by direct phase unwrapping [21

D. C. Ghiglia, Two-Dimensional Phase Unwrapping (New York, Chichester, Weinheim, Brisbane, Singapore, Totonto, 1998).

], which is, however, complicated by path inconsistencies due to noise, dislocations [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]

], and discontinuities [23

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

] and may not be solvable with the desired L⁰ norm in polynomial time [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]

]. Alternatively, the wavefront can be quantified in terms of a set of modes [25

R. K. Tyson, Principles Of Adaptive Optics (Boston, 1997).

]. Using as such a set Zernike polynomials [26

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

] (for which we use the Noll numbering scheme [27

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

]) has the advantage that they form an orthogonal set and that lower-order functions closely correspond to conventional aberrations [25

R. K. Tyson, Principles Of Adaptive Optics (Boston, 1997).

, 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]

], such as astigmatism (z₅ , z₆ ) and coma (z₇ , z₈ ). Modal decomposition into other complete function sets, such as disk harmonic functions [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.

], could be used as well.

A modal wavefront sensor (MWS) where the wavefront is passed through specially designed phase plates and the resulting intensity distribution is detected by light detectors with only a few elements was proposed [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]

] and implemented [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]

]. This approach directly yields a modal representation of the wavefront and is, due to the typically smaller number of detectors, faster than zonal sensors, such as the Hartmann-Shack sensor [31

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

], which are limited by camera readout speed. Multiple modes can be measured simultaneously if the input beam is split into the appropriate number of sub-beams but this, of course, reduces the sensitivity. Simple MWSs suffer from non-linear crosstalk, which is of a different character than fitting crosstalk encountered when fitting modes to zonal detector measurements [32

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

], but can be reduced by iteration [28

]. Recently an interferometric modal sensor was used to detect distortions in biological specimens [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]

Instead of passing the wavefront through phase plates and measuring the ensuing intensity one can also measure the phase locally and perform the propagation through the phase plate numerically, creating a “virtual” MWS (VMWS). Such a sensor has better sensitivity due to interferometric detection and because it does not require a division of the available light between different modes. In this paper, we describe a Zernike VMWS in detail and analyze its performance. We investigate measurement range and speed (number of required iterations) for different sampling-grids. Special attention is paid to the performance under low light conditions, where we compare the VMWS to a state-of-the-art phase unwrapping algorithm [34

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

]. We expect the VMWS to be especially appropriate when only little light is available from the sample, but a reference wavefront can be obtained, such as in laser microscopy.

2. Methodology

Conceptually, a modal sensor using real phase plates [28

, 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]

] consists of two beam paths where a deliberate aberration containing only one mode (e. g. the 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.

The virtual modal wavefront sensor (Fig. 1), which is analyzed here theoretically and numerically, physically measures interferograms. The propagation through phase-plates as well as the iterative removal of aberrations are done by computation using the originally measured data. Thus no additional light is needed for iteration or the measurement of more modes.

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.

To acquire the interferograms the wavefront to be measured and the plane reference wave are superimposed. Four interference patterns (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

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]

]. The wrapped phase φ (x, y) is calculated [37

D. Malacara, Optical Shop testing (J. Wiley, New York, 1992).

] using:

φ (x, y) = \arctan {\frac{[I_{4} (x, y) - I_{2} (x, y)]}{[I_{1} (x, y) - I_{3} (x, y)]}}

(1)

and expanded [37

D. Malacara, Optical Shop testing (J. Wiley, New York, 1992).

] from [-π/2, π/2] to [-π, π]. Next, the “intensity coefficients” h _i+ and h _i- (which correspond to the intensities measured at the pinhole detector in a regular MWS) are calculated

h_{i \pm} = {∣ 〈 E_{φ} ∣ E_{i \pm} 〉 ∣}^{2} = {∣ \int_{A} \exp [jφ (x, y)] \times \exp [\mp j b z_{i} (x, y)] dxdy ∣}^{2},

(2)

where E _φ = exp[jφ(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

],〈∙|∙〉 denotes the scalar product, 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

, 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]

Method A : g_{i} = h_{i +} - h_{i -},

(3A)

Method B : g_{i} = \frac{(h_{i +} - h_{i -})}{(h_{i +} + h_{i -})},

(3B)

Method C : g_{i} = \frac{(h_{i +} - h_{i -})}{(h_{i +} + h_{i -} + γ h_{0})},

(3C)

where h ₀ = 〈E_φ |E_φ 〉 and γ is a constant (typically around 1) adjusted for best linearity [28

]. Method A is computationally more efficient, but also more sensitive to crosstalk (see below) than B and C [28

]. Since it was not apparent from a previous theoretical analysis [28

] which one would perform best, in particular with an iterative algorithm, we investigated all three methods. In each case the g_is 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

d_{i} = p_{i} \times g_{i} .

(4)

The p_is can be calculated analytically [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]

], but we found that proportionality coefficients obtained from simulations using wavefronts that contain known small single-mode aberrations are actually more precise, possibly because analytical calculation does not take into account the effect of spatial discretization.

While for small aberrations the aberration coefficients can be determined with good accuracy in a single step, for larger aberrations the first estimates increasingly deviate from the true values because of the nonlinearity of the deviation signal (Fig. 2). Correct estimation even for much larger deviations is, however, possible with the help of iteration, first suggested by Booth [28

] in the context of deformable mirrors as biasing elements.

The iterative algorithm used in this paper works as follows. First a set of coefficients

d_{i}^{(0)}

is estimated using the (wrapped) phaseφ (Eq. (4)). Next, an initial wavefront estimate is calculated,

\hat{φ}

⁽⁰⁾(x, y) = ∑_i

d_{i}^{(0)}

× z_i (x, y). The difference between estimate and actual wavefront, Δ⁽⁰⁾ = φ -

\hat{φ}

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

d_{i}^{(1)}

, and so forth. At each step of the iteration the estimate for the residual is given by

\hat{φ}

⁽ⁿ⁺¹⁾(x, y) = ∑_i

d_{i}^{(n + 1)}

× z_i (x, y); Δ⁽ⁿ⁺¹⁾ = Δⁿ -

\hat{φ}

⁽ⁿ⁺¹⁾ is the new input for the next step. The final estimate is ∑_n∑_i

d_{i}^{(n)}

× z_i (x, y).

We usually terminated the iteration when (〈

\hat{φ}

⁽ⁿ⁾|

\hat{φ}

⁽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:

To test various aspects of VMWS performance we used numerically simulated inputs.

3.1 Methods

All computations were carried out using Matlab (The MathWorks, Inc, Natick, MA, USA). The interferometer was represented as follows: For a given set of Zernike coefficients 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_{(k)} (x, y) = I_{r} (x, y) + I_{s} (x, y) + 2 ∙ \sqrt{I_{r} (x, y) ∙ I_{s} (x, y)} ∙ \cos (Φ (x, y) + (k - 1) ∙ \frac{π}{2}),

(5)

where 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.

The four interferograms were then used as the input for the VMWS. All calculations were performed on rectangular grids with a constant and fixed spacing cropped to a circular aperture.

To map regions of convergence (see below) for different grid spacings we used 100 sets of random Zernike coefficients (z₁ to z₂₁ ) 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, z₁ , 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.

To analyze the precision of wavefront estimation in the bright illumination limit, we used the same data that had been used for the determination of the convergence ranges. We selected cases with correct convergence and then calculated the errors of the wavefront estimation. Because the error does not depend on the initial aberration strength (data not shown) we averaged the errors for all wavefront shapes and aberration strengths.

To test the effect of shot noise on the performance of the VMWS, 50 different wavefronts were generated for each of the different sample-arm intensities tested. Calculations were performed for initial aberration strengths of 1 and 2 rad. For each wavefront 50 different sets of interferograms with independent random photon distributions were calculated to account for the effect that convergence might be affected not only by the wavefront but also by the actual photon distribution. For these 2500 sets (50 wavefronts × 50 photon distributions) of interferograms, the wavefronts were estimated using the VMWS (calculation method B). The percentage of correct estimations was recorded. Here, the correctness of the convergence was tested by changing the scaling factor 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.

For the phase unwrapping calulations [34

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

] we used 15 wavefronts and 15 photon distributions on grids of 32 × 32, 64 × 64, and 128 × 128 points. Wavefronts were excluded, if they were not unwrapped correctly in the bright illumination limit.

3.2 Detector response and convergence range

We first considered only defocus (z₄ ), 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).

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

In particular, we find points (zeros) where, even with a strong aberration present, the detected defocus is zero. These are “false fixpoints” towards which the algorithm can erroneously converge if, in addition, the slope is between 0 and 2. The positions of false fixpoints depend, however, strongly on the parameter b. This can be used to detect erroneous convergence. See Fig. 3 for the response curves for different modes.

So far we have considered a sensor for a single mode (z₄ , 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.

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.

3.3 Region of convergence for different grid spacings

As discussed above (see “Detector response and convergence range”) the iterative algorithm can converge to incorrect results. We will refer to the region in Zernike coefficient space, where the algorithm converges to the correct results, as the “region of convergence” and to its average extent in coefficient space as “range of convergence”. Lacking an analytical way to estimate the ranges of convergence we used numerical simulations. For actual computation the interferograms need, furthermore, to be spatially sampled (which roughly, but not exactly, corresponds to the pixelation in a physical detector). Because a finer grid spacing should increase accuracy but also computation time we determined how the grid spacing affects convergence regions and accuracy. We first tested all three calculation methods (A, B, C) in the bright illumination (noiseless) limit.

For weak aberrations we found final wavefront errors (between the reconstructed and the introduced aberration) below 1 mrad (Fig. 4 (a)). Correct convergence is ensured for small (below 2 rad) aberrations but at large aberrations errors began to rise steeply, indicating convergence to false fixpoints (Fig. 4 (b)). The probability of converge failure rises roughly linearly for aberrations above 2 rad. In this range, points that will converge are interspersed with points that will not. The convergence region is thus not clearly delineated in the space of Zernike coefficients and there is no strictly defined “radius” of convergence. Nevertheless, a mean range of convergence can be determined and was found to be ≈4.9 rad (for a grid of 65 × 65 points, method B, and aberrations up to z₂₁ present). This range depends on the exact aberration shape, with some shapes having considerably smaller convergence ranges.

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.

For initial wavefront aberrations above 2.8 rad we found a few cases of oscillations and hence no convergence. The error as a function of iteration number for some typical cases is shown in Fig. 5. For small aberrations the convergence is very fast (Fig. 5, squares). While for coarser grids the convergence ranges for methods A and B are similar (Fig. 6), method B is substantially better for finer grids (starting at 65 × 65 points) and converges much faster in general (Fig. 7). For method C the mean convergence range is considerably smaller and convergence failed, in particular, on a coarse grid (17 × 17 points) even for aberrations as small as 0.2 rad.

In addition, we investigated whether the range of convergence depends on the presence of tilt and defocus in the wavefront in order to evaluate if a substantial increase in the region of convergence could be achieved if tilt and defocus, often the dominant aberrations and easily measurable by alternative methods [25

R. K. Tyson, Principles Of Adaptive Optics (Boston, 1997).

], are removed beforehand. We found (Fig. 6) that tilt and defocus do not affect the radius of convergence more strongly than other aberrations but their preemptive removal allows stronger higher order aberrations before the convergence range is exceeded. It might also be possible to extend the convergence range, e.g. by using a collection of different scaling factors 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).

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.

As the grid becomes very fine the convergence range starts to decrease again. This effect is reduced if the outer sampling points in the aperture are omitted (data not shown) indicating that it might be related to the fact that Zernike polynomials and their derivatives are often large near the aperture edge.

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 .

3.4 Aberration order and convergence range

We next explored how the maximal aberration order that is present in the wavefront affects convergence (Fig. 8). We tested what happens when the sensing is performed to the same order as aberrations are present. In addition, we explored, what happens if aberrations are present to z₁₀ and sensed to z₂₁ .

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.

We found (using calculation method B) that the convergence range generally increases with finer grids (but see above) and decreases for a higher maximum aberration order. For a given maximum aberration order, the convergence range was always largest if as many aberration modes were sensed as were present in the wavefront.

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

Crosstalk from orders higher (in our case beyond z₂₁ ) 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 z₁ to z₂₈ while only the modes z₂ to z₂₁ were measured with the VMWS. Deviations were again calculated only for modes z₂ to z₂₁ . 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

We first investigated how well wavefronts can be measured in the bright illumination limit. This was done on different grids for wavefronts containing aberrations up to modes z₁₀ , z₂₁ , and z₂₈ (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 z₁₀ but were measured to z₂₁ . Similar to the behavior of the convergences ranges, we found that with only lower modes (up to z₁₀ ) present the final error is larger when including extra modes (up to z₂₁ ) 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.

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.

3.7 The influence of noise

All detection of light is ultimately limited by quantum noise (photon shot noise). We, therefore, explored the performance of the VMWS as the number of available photons decreases and the (relative) noise thus increases.

To estimate the light level needed in the reference arm, for the reference arm intensity not to influence the accuracy significantly, we calculated the error of the estimated (wrapped) phase, using Eqs (1) and (5), as a function of the reference light level. Using error propagation we find for the errorΔφ of the phase estimate

Δφ = \frac{1}{\sqrt{2}} {[\frac{1}{n_{s}} (1 + \frac{n_{s}}{n_{r}} + \frac{4 n_{c}}{n_{r}})]}^{\frac{1}{2}},

(6)

where 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/√2n_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 ≥ 200n_s Δφ ≤ 1.005 Δφ_∞). This limit is slightly higher than the naïve quantum limit for coherent states [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]

] of (Δφ_∞ ≥ 1/(2√n_s )[40

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

We found (Fig. 10) that for as few as 400 sample-arm photons (100 per interferogram, the reference arm contained about 20 000 photons per interferogram, see above) the wavefronts (containing modes z₁ - z₂₁ ) 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.

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. 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.

Considering only those cases for which convergence to the correct wavefront is reached, we found for the final estimate an error Δφ ≈ 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).

Fig. 12. rms error for modes z₂ - z₂₁ . The number of photons refers to the average total photon number from the sample.

We also compared the performance of the VMWS with a modern phase unwrapping method that is considered noise resistant [34

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

]. We found that the estimation error for phase unwrapping follows that for the VMWS at high photon numbers (> 20 000) but exceeds the VMWS value dramatically for smaller photon numbers (Fig. 13). With increasing aberration strength the grid resolution needs to be raised (33 × 33 pixels is too coarse even with aberrations as small as 1 rad, data not shown) but the higher relative photon noise per pixel results in a higher reconstruction error. The VMWS, in contrast, can operate with low resolution images.

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/n^0.51) 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).

4. Summary and discussion

We have shown, using numerical simulations, that a virtual modal wavefront sensor (VMWS), based on interferometric detection and using iterative elimination of modal crosstalk and detection nonlinearity, can accurately detect wavefront distortions up to almost 5 rad. Under low light conditions the quantum limit for phase detection is reached. Convergence and phase error are optimized when the estimator set matches the set of distortion modes present in the wavefront. VMWS performance depends somewhat on discretization. A method to check correct convergence is available and has been demonstrated. The convergence properties of the VMWS compare favorably to those of phase unwrapping algorithms [34

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

] at low light levels and at low discretization resolution.

The numerical calculations needed for the operation of a VMWS can be implemented in a highly parallel fashion, which is important for closed-loop adaptive optics systems, which will, in addition, benefit from the high convergence rate for small aberrations. Because the convergence range of the VMWS is limited it may be necessary to roughly estimate the dominant low-order aberrations first, using, for example, a virtual Hartman-Shack sensor [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]

] or a quadrant photo diode for tilt [25

R. K. Tyson, Principles Of Adaptive Optics (Boston, 1997).

]. A VMWS can be constructed without specialized optical components, such as phase plates or lens arrays and, if operated with a low coherence source, such as a ultra short-pulse laser, inherently performs coherence gating [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]

In conclusion, we have shown that a new type of wavefront sensor, the virtual modal sensor, performs well, especially under low light conditions. It combines the advantages of interferometric detection with a direct modal representation of the measured wavefront.

Acknowledgments

We thank I. Janke and M. Feierabend for helpful discussions and suggestions on the manuscript. ML was supported by the Studienstiftung des deutschen Volkes.

References and links

1.	H. W. Babcock, “The possibility of compensating astronomical seeing,” Publications of the Astronomical Society of the Pacific 65, 229–236 (1953). [CrossRef]
2.	J. W. Hardy, Adaptive Optics for Astronomical Telescopes (A. Hasegawa, et al., Oxford, 1998).
3.	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 Imaging Proc. SPIE 1161, 417–425 (1989).
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]
14.	M. Minsky, Microscopy Apparatus , U.S. Patent 3013467, USA (1961).
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]
17.	R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538–542 (1969).
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. 213, 11–19 (2004). [CrossRef]
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]
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]
21.	D. C. Ghiglia, Two-Dimensional Phase Unwrapping (New York, Chichester, Weinheim, Brisbane, Singapore, Totonto, 1998).
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]
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]
25.	R. K. Tyson, Principles Of Adaptive Optics (Boston, 1997).
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]
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]
31.	R. V. Shack and B. C. Platt, “Lenticular Hartmann-screen,” Optical Sciences Center Newsletter 5, 15–16 (1971).
32.	G. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218–1225 (1996). [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]
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]
37.	D. Malacara, Optical Shop testing (J. Wiley, New York, 1992).
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]
40.	R. Lynch, “The Quantum Phase Problem - a Critical-Review,” Phys. Rep. 256, 368–436 (1995). [CrossRef]

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 Oceanic 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

H. W. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229-236 (1953). [CrossRef]
J. W. Hardy, Adaptive Optics for Astronomical Telescopes (A. Hasegawa, et al., Oxford, 1998).
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).
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]
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]
K. E. Oughstun, "Intracavity adaptive optic compensation of phase aberrations. I: Analysis," J. Opt. Soc. Am. 71, 862 - 872 (1981). [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]
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]
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, James G. Fujimoto, eds., Proc. SPIE 4956, 65-72 (2003). [CrossRef]
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]
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]
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]
J. Nowakowski, and M. Elbaum, "Fundamental Limits in Estimating Light Pattern Position," J. Opt. Soc. Am. 73, 1744-1758 (1983). [CrossRef]
M. Minsky, Microscopy Apparatus, U.S. Patent 3013467, USA (1961).
W. Denk, J. H. Strickler, and W. W. Webb, "Two-Photon Laser Scanning Fluorescence Microscopy," Science 248, 73-76 (1990). [CrossRef] [PubMed]
P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966). [CrossRef]
R. Crane, "Interference Phase Measurement," Appl. Opt. 8, 538-542 (1969).
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. Feierabend, M. Ruckel, and W. Denk, "Coherence-gated wave-front sensing in strongly scattering samples," Opt. Lett. 29, 2255-2257 (2004). [CrossRef] [PubMed]
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]
D. C. Ghiglia, Two-Dimensional Phase Unwrapping (New York, Chichester, Weinheim, Brisbane, Singapore, Totonto, 1998).
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]
R. Gens, "Two-dimensional phase unwrapping for radar interferometry: developments and new challenges," Int. J. Remote Sens. 24, 703-710 (2003). [CrossRef]
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]
R. K. Tyson, Principles Of Adaptive Optics (Boston, 1997).
F. Zernike, "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica 1, 689-704 (1934). [CrossRef]
Noll, R. J. , "Zernike Polynomials and Atmospheric-Turbulence," J. Opt. Soc. Am. 66, 207-211 (1976). [CrossRef]
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, Mark T. Gruneisen, eds., Proc. SPIE 5162, 79 - 90 (2003). [CrossRef]
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.
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]
R. V. Shack, and B. C. Platt, "Lenticular Hartmann-screen," Optical Sciences Center Newsletter 5, 15-16 (1971).
G. Dai, "Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions," J. Opt. Soc. Am. A 13, 1218-1225 (1996). [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]
V. V. Volkov, and Y.M. Zhu, "Deterministic phase unwrapping in the presence of noise," Opt. Lett. 28, 2156-2158 (2003). [CrossRef] [PubMed]
K. Kinnstaetter, A. W. Lohmann, J. Schwider, and N. Streibl, "Accuracy of Phase-Shifting Interferometry," Appl. Opt. 27, 5082-5089 (1988). [CrossRef] [PubMed]
K. A. Stetson, and W. R. Brohinsky, "Electrooptic Holography and its Application to Hologram Interferometry," Appl. Opt. 24, 3631-3637 (1985). [CrossRef] [PubMed]
D. Malacara, Optical Shop testing (J. Wiley, New York, 1992).
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]
R. Lynch, "The Quantum Phase Problem - a Critical-Review," Phys. Rep. 256,368-436 (1995). [CrossRef]

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