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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 1, Iss. 6 — Jun. 13, 2006
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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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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

2. Methodology

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

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

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

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

] using:

φ(x,y)=arctan{[I4(x,y)I2(x,y)][I1(x,y)I3(x,y)]}
(1)

and expanded [37

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

hi±=EφEi±2=Aexp[(x,y)]×exp[jbzi(x,y)]dxdy2,
(2)

MethodA:gi=hi+hi,
(3A)
MethodB:gi=(hi+hi)(hi++hi),
(3B)
MethodC:gi=(hi+hi)(hi++hi+γh0),
(3C)

di=pi×gi.
(4)

The pis can be calculated analytically [30

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

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]

] in the context of deformable mirrors as biasing elements.

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 Di a wavefront Φ(x, y) was calculated: Φ(x, y) = ∑i Di × zi (x, y), whereby zi (x, y), is the ith 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)=Ir(x,y)+Is(x,y)+2Ir(x,y)Is(x,y)cos(Φ(x,y)+(k1)π2),
(5)

where Ir (x,y) and Is (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

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.

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

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.

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

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=12[1ns(1+nsnr+4ncnr)]12,
(6)

where nr and ns 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/√2ns ) 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 nr ≥ 200ns 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

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

] of (Δφ ≥ 1/(2√ns )[40

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/√ns 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/√ns . 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)√nm /ns where nm is the number of modes and ns 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

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/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).

4. Summary and discussion

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

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

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

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

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

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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 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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