## 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

Fast and light-efficient wavefront detectors are essential for adaptive optics (AO) [1], which is used in applications that are affected by wavefront distortions due to refractive index inhomogeneities in the beam path, such as astronomy [2], ophtalmoscopy [3, 4], free space communication [5], laser resonators [6], optical trapping [7], optical coherence tomography [8, 9] and confocal and multi-photon microscopy [10, 11]. 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, 13]. Particularly in scanned-laser imaging microscopy [14, 15] 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] (which has been analyzed in context of point diffraction interferometers by [20]). 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], which is, however, complicated by path inconsistencies due to noise, dislocations [22], and discontinuities [23] and may not be solvable with the desired L^{0} norm in
polynomial time [24]. Alternatively, the wavefront can be quantified in terms of a set of modes [25]. Using as such a set Zernike polynomials [26] (for which we use the Noll numbering scheme [27]) has the advantage that they form an orthogonal set and that lower-order functions closely correspond to conventional aberrations [25, 28], such as astigmatism (*z*_{5}
, *z*_{6}
) and coma (*z*_{7}
, *z*_{8}
). Modal decomposition into other complete function sets, such as disk harmonic functions [29], 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] and implemented [11]. 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], 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], but can be reduced by iteration [28]. Recently an interferometric modal sensor was used to detect distortions in biological specimens [33].

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]. 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] 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.

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, 36]. The wrapped phase φ (*x*, *y*) is calculated [37] using:

and expanded [37] 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

where *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],〈∙|∙〉 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]:

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*_{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

The *p*_{i}*s* can be calculated analytically [30], 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 ${d}_{i}^{\left(0\right)}$ is estimated using the (wrapped) phaseφ (Eq. (4)). Next, an initial wavefront estimate is calculated, $\widehat{\phi}$
^{(0)}(*x*, *y*) = ∑_{i}
${d}_{i}^{\left(0\right)}$ × *z*_{i}
(*x*, *y*). The difference between estimate and actual wavefront, Δ^{(0)} = φ - $\widehat{\phi}$
^{(0)}, is now taken as the new input to the estimation algorithm, yielding coefficients ${d}_{i}^{\mathit{\left(}\mathit{1}\mathit{\right)}}$
, and so forth. At each step of the iteration the estimate for the residual is given by $\widehat{\phi}$
^{(n+1)}(*x*, *y*) = ∑_{i}
${d}_{i}^{(n+1)}$ × *z*_{i}
(*x*, *y*); Δ^{(n+1)} = Δ^{n} - $\widehat{\phi}$
^{(n+1)} is the new input for the next step. The final estimate is ∑_{n}∑_{i}
${d}_{i}^{\left(n\right)}$ × *z*_{i}
(*x*, *y*).

We usually terminated the iteration when (〈$\widehat{\phi}$
^{(n)}|$\widehat{\phi}$
^{(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:

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 (*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.

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] 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 (*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).

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 (*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

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

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], 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).

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.

#### 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 *z*_{10}
and sensed to *z*_{21}
.

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 *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

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 *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

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

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/√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, 39] of (Δφ_{∞} ≥ 1/(2√*n*_{s}
)[40].

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 *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.

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).

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

## 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] 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] or a quadrant photo diode for tilt [25]. 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].

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.

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