Abstract

Coherence-gated wavefront sensing (CGWS) allows the determination of wavefront aberrations in strongly scattering tissue and their correction by adaptive optics. This allows, e.g., the restoration of the diffraction limit in light microscopy. Here, we develop a model, based on ray tracing of ballistic light scattered from a set of discrete scatterers, to characterize CGWS performance as it depends on coherence length, scatterer density, coherence-gate position, and polarization. The model is evaluated by using Monte Carlo simulation and verified against experimental measurements. We show, in particular, that all aberrations needed for adaptive wavefront restoration are correctly sensed if circularly polarized light is used.

© 2007 Optical Society of America

1. INTRODUCTION

Light microscopy in biological tissue can be hampered by a degradation of the focus due to refractive index inhomogeneities within the specimen [[1], [2], [3], [4], [5]]. Using adaptive wavefront correction the wavefront of the illumination light can be shaped in such a way that the specimen-introduced distortions are precompensated and a diffraction-limited focus is restored. Two fundamentally different approaches to adaptive wavefront correction can be distinguished [[6]]: (1) iterative optimization of the fluorescence signal using trial wavefront perturbations on the incident light, and (2) direct wavefront sensing and subsequent wavefront correction using the phase-conjugation approach. The first approach can be slow and requires the excitation of fluorescence light, which causes photobleaching and photodamage [[1], [2], [3], [7], [8]].

In contrast, coherence-gated wavefront sensing (CGWS) implements direct wavefront sensing and can measure the specimen-introduced distortions even in strongly scattering samples [[5], [9]]. CGWS is based on a low-coherence interferometer, which can select that portion of the backscattered sample light that originates from a region close to the focus. In combination with phase-shifting interferometry [[10]] and a real [[11]] or virtual Shack–Hartmann sensor (vSHS [[9], [12]]), the aberrations are reconstructed and can then be used for wavefront correction. The CGWS approach allows fast wavefront correction since the appropriate information about the distortions in the optical path can be obtained with a single set of measurements [[5]]. Here we investigate CGWS theoretically and experimentally and characterize CGWS performance, in particular with respect to coherence length, density of scatterers, coherence-gate (CG) position, and the polarization of illumination and reference light. Monte Carlo simulations (MCSs) of the model are compared with experimental results on test samples.

2. MODEL OF CGWS

In the following a model is developed to derive the characteristic properties of CGWS by simulating all experimentally implemented steps [[5], [9]]. While our model resembles models [[13], [14], [15]] of optical coherence tomography (OCT [[16]]), we pay particular attention to the phase of the backscattered light instead of its intensity. Based on ray tracing, the model determines the optical travel distances of light scattered by a random distribution of discrete scatterers within the geometrical focal (double) cone. Only singly scattered light is taken into account.

The electric field of the backscattered sample light ES (complex-valued vector) interferes with that of the reference light ER on a spatially resolving detector, whereupon the location on the detector is parameterized by u and v. The intensity of the interferogram is

I(u,v)=IR(u,v)+IS(u,v)+2Re{ER*(u,v,t)kES(k)[u,v,t+τ(k)(u,v)]¯},
where IR(u,v)=ER(u,v,t)2¯ and IS(u,v)=ES(u,v,t)2¯. Averaging over time, t, is denoted by the overbar and complex conjugation by the asterik. The electric field in the sample arm is ES=kES(k)[u,v,t+τ(k)(u,v)], where τ(k)(u,v) is the time delay between light scattered by the scatterer k and light that has traveled along the reference arm. The sum is over all scatterers within the focal cone.

The averaging over time can be carried out by using the (complex) degree of self-coherence γ(τ), which is also called the normalized self-coherence function of the scalar electric field E [[17]],

γ(τ)=E*(t)E(t+τ)¯E*(t)E(t)¯=γ(τ)exp(iωτ),
whereby ω is the center frequency of the light source, which we assume (as is reasonable for narrowband light) has a symmetrical spectrum. For wavelength-independent scattering the spectra of scattered and reference light are identical, and Eq. (1) becomes
I=IS+IR+2Re[ERk(pR*pS(k))ES(k)γ(τ(k))exp(iωτ(k))].
The amplitudes of the electric fields of reference and sample light scattered by the scatterer k are ER and ES(k), respectively, and the polarizations are pR and pS(k), respectively. For clarity, the dependence on u and v has been omitted, and in the following the dot product pR*pS(k) is replaced by WP(k).

The first two terms in Eq. (3) are the total intensities of the light backscattered from the sample and of the reference light, respectively, both of which need to be subtracted to isolate the interference (third) term. This can be done, for example, by using phase-shifting interferometry [[10]]. The self-coherence function γ(τ(k)) weights the interference term and thus can select backscattered light that arrives within a certain time window [defined by γ(τ(k))] at the detector.

Usually, the detector is located in a plane that is optically conjugate to the back focal plane (BFP) of the objective [[5], [9]]. If one assumes a flat reference wavefront, a simple analytical expression for τ(k) can be derived. The path difference between light scattered at P(x,y,z) [light path A–P–B, Fig. 1a ] and the reference light is

τ[r(k),w(u,v)]=nc[sgn(z)r(k)+r(k)w(u,v)]+ll0c,
where r(k) connects the focus (F) to the scatterer k at P(x,y,z), w is the unit vector in the direction of the scattered light, c is the velocity of light in vacuum, and n is the refractive index of the sample. If the CG position is at the focus, the length of the reference arm is l0; i.e., the time traveled by the reference light matches the time traveled by light that has been backscattered at the focus. For a different reference-arm length l the CG position within the specimen is shifted from the focus by (ll0)(2n). The origin of the coordinate system is at the focus.

Because of the finite width of γ(τ(k)) for low-coherence light, mainly light scattered within a certain region, which we call the coherence volume (CV), contributes to the interference term in Eq. (3). Since γ(τ(k)) never becomes strictly zero, the definition of the CV is somewhat arbitrary. We use the region where γ[τ(r,w)]>0.5. It is important to realize that, since τ[r,w(u,v)] depends on the detection position (u,v), the CV depends on the location in the BFP [Figs. 1b, 1c] and is therefore different for different sublenses.

The electric field of the coherence-gated sample light ECGWS(u,v) can be directly obtained from the interference term in Eq. (3),

ECGWS(u,v)kWP(k)(u,v)ES(k)(u,v)γ[τ(k)(u,v)]exp[iωτ(k)(u,v)].
The amplitude of the reference light ER [Eq. (3)], as long as it is uniform, does not affect the wavefront reconstruction and is thus taken to be 1 (a known nonuniform amplitude could also be accounted for). Note that ECGWS(u,v) is a scalar quantity, since only the projection of pS(k) on pR can be measured interferometrically. Because the local phase of ECGWS(u,v) is only determined modulo 2π (for example in the interval [0;2π[), it needs to be unwrapped to reconstruct the actual wavefront. All available phase-unwrapping methods are encumbered by path inconsistencies due to noise, singularities [[18]], and discontinuities [[19]] and may not be solvable in polynomial time [[20]]. However, singularities are unavoidable when using CGWS in a scattering sample, where backscattered light from randomly distributed scatterers superimposes coherently. Therefore, we need to use a method that can reliably reconstruct the wavefront in the presence of such singularities. The virtual and also the real Shack–Hartmann sensor fulfills this requirement if averaging over different ensembles of scatterers is performed [[5], [9], [12]]. The vSHS numerically propagates ECGWS(u,v) through a virtual lenslet array onto a spatially resolved virtual detector (see below), closely mimicking a real Shack–Hartman sensor. The wavefront is then reconstructed as in a real Shack–Hartman sensor, for example by least-squares fitting to the focus displacements of all lenslets [[21]]. For a single lenslet the electric field on the virtual detector (parameterized by r and q) is
ECGWS(r,q)=AECGWS(u,v)exp[i2πλf(ur+vq)]dudv=kES(k)(r,q),
where λ is the wavelength; f and A are the focal length and the area of the lenslet, respectively. The Fourier transformation of the coherence-gated electric field, scattered by the scatterer k, is ES(k)(r,q). The intensity of the diffraction patterns on the virtual detector is then
I(r,q)=ECGWS(r,q)2=kES(k)(r,q)2+mnm[ES(m)(r,q)]*ES(n)(r,q).
The term kES(k)(r,q)2 alone is the superposition of the intensities of the light backscattered by the individual scatterers (incoherent superposition), and together with mnm[ES(m)(r,q)]*ES(n)(r,q) corresponds to the coherent superposition of their electric fields. Using the Wiener–Khintchine relation [[22], [23]], the ensemble-averaged (⟨⟩) diffraction pattern in the focus of a single lenslet can be written as
I(r,q)=Adsdtexp[i2πλf(sr+tq)]AdudvECGWS(u,v)ECGWS*(u+s,v+t).
The term ECGWS(u,v)ECGWS*(u+s,v+t) is the mutual intensity function μ(x1,x2) (spatial coherence [[17]]) between point x1=(u,v) and x2=(u+s,v+t) in the BFP, which can be used to determine (see below) whether incoherent or coherent superposition of the backscattered light dominates I(r,q). The mutual intensity can be written as
ECGWS(x1)ECGWS*(x2)=kES(k)(x1)ES(k)*(x2)+mnmES(m)(x1)ES(n)*(x2)=kES(k)(x1)ES(k)*(x2)+mnmES(m)(x1)ES(n)*(x2)=NES(x1)ES*(x2)+N(N1)ES(x1)ES*(x2)=N(αβ)+N2β,
where N is the number of scatterers inside the focal cone; α=ES(x1)ES*(x2) and β=ES(x1)ES*(x2) depend on the self-coherence function γ(τ), WP, CG position, x1, and x2 [Eq. (5)]. Note that the terms Nα and N(N1)β are due to the incoherent and coherent superposition of the scattered light, respectively. We will show below that the coherent superposition of the coherence-gated backscattered light can nearly always be neglected for randomly distributed scatterers.

3. METHODS

Our Monte Carlo simulation (MCS) is based on sets of discrete scatterers randomly sampled from a uniform distribution, closely follows the model derived in the previous section, and is similar to an approach [[24]] commonly used to analyze OCT.

First one ensemble of scatterers was established. Then the sum in Eq. (5) was calculated separately for each pixel on the detector. The time delay, τ depends essentially only on the location, r, of the scatterer and on the scattering direction, w, [Eq. (4), Fig. 1a]. Phase (time) shifts due to scattering (depending on the scattering angle) are included in WP(k), but can be neglected, in particular for Rayleigh scatterers (see Section 4). By keeping track of the propagation of the polarization, which is changed by a generic objective [[25]] and by Mie scattering [[26]], the weight function WP(k) can be calculated for each scattering event (for a detailed description see Appendix A). Since it is computationally expensive to keep track of the polarization for all light rays, a simplified version of the MCS was also implemented with WPol(k)=1; i.e., the polarization of light is neglected and the Mie scattering phase functions are replaced by an isotropic scattering function. The simplified version of the MCS was used except when the polarization dependence was to be investigated.

For the calculation of ECGWS further simplifications were made. First, in the limit of geometrical optics the amplitude of the backscattered light depends only on z (as 1z) for scatterers inside the illumination cone, assuming laterally uniform illumination. This is justified because MCSs showed the same results for both the uniform and the, experimentally used [[5]], Gaussian profiles. Second, the self-coherence function was assumed to be Gaussian, which is computationally simpler but very similar to a squared hyperbolic secant, which best describes the self-coherence function of passively mode-locked lasers, such as the Ti:sapphire oscillator used in our experiments [[27]].

To save computational time, the detector used for MCS calculations had 105  pixels across the aperture diameter even though the detector used in the actual experiments sampled the aperture using 375  pixels across the diameter. As long as the speckles are larger than 2  pixels in diameter, this restriction does not lead to significant deviations [[12]]. Furthermore, scatterers were distributed uniformly but only up to 4 coherence lengths from the center of the CG position in both axial directions; Gaussian and squared hyperbolic secant functions have very small weight outside of this volume.

After the coherence-gated electric field was calculated, the wavefront was reconstructed by using the vSHS. The circular aperture was covered by 37 lenslets, each containing 15×15  pixels. For precise detection of the focus displacement for each lenslet the complex electric field was zero padded to a field of 65×65  pixels before Fourier propagation. In addition to the centroid estimation, which was used unless noted otherwise, the peak position of the diffraction patterns was in some cases determined by least-squares fitting a Gaussian distribution to the central part of each diffraction pattern. Then the wavefront was reconstructed by least-squares fitting a linear combination of 21 Zernike modes (⩽ fifth order [[28]]) to the peak- or centroid-position displacements in all lenslets. For ensemble averaging at least three different ensembles of scatterers were used.

For the experimental measurements 441 lenslets, covering a circular aperture with a diameter of 375  pixels, were used. Before Fourier transformation of the coherence-gated electric field the 15×15  pixels region of each lenslet was zero padded to 65×65  pixels [[5], [12]]. Centroid estimation was used, except for the measurements with linearly polarized illumination light, where peak fitting was used [[9]]. The use of peak fitting had an effect only on rotationally symmetric aberrations, such as defocus (data not shown). The wavefront, averaged over 20 different ensembles of scatterers, was fitted by a linear combination of 28 Zernike modes (⩽ sixth order). Details can be found in [[5]].

4. RESULTS

4A. Properties of the Lenslets’ Diffraction Patterns

We first investigated how CGWS-measured wavefronts and their errors due to speckle depend on CG position, density of scatterers, and coherence length. The speckle error increases with decreasing speckle size in the BFP [[12]], because then the peaks in the virtual detection plane of the vSHS become broader. The speckle size scales inversely [[29]] with the number of phase singularities in CGWS-measured electric fields, which we determined by detecting whether the phase integrals along closed paths were zero or not [[30]]. We found (by MCS) that the number of singularities did not change with the density of scatterers but changed with CG position and coherence length [Figs. 2a, 2c, 2e ]. This is because the number of singularities depends linearly on the solid angle under which the CV is seen from the BFP (speckle size scales inversely with the solid angle [[31]]). Thus, with a smaller coherence length or with proximity of the CG position to the focus, the CV becomes smaller and the speckle error for a given number of ensemble averages is reduced.

First, no distortions were included in the optical path for these MCSs, and only the defocus component of the wavefront (Zernike coefficient c4) was investigated. When the wavefront was determined by centroid estimation, c4 remained constant as the coherence length or the density of scatterers was varied, but changed linearly with CG position [Figs. 2b, 2d, 2f]. Interestingly, c4 was always smaller when determined by peak fitting than by centroid estimation with the difference increasing with increasing coherence length [Fig. 2d]; c4 also changed linearly with CG position [with a smaller slope than centroid estimation, Fig. 2b] but remained constant as the density of scatterers was varied [Fig. 2f]. This means that centroid and fitted peak position diverge as the CG moves farther from the focus and the coherence length becomes longer [Figs. 2b, 2d]. This is likely due to the fact that the diffraction patterns of the vSHS (which are the projections of the CVs) are, in particular for peripheral lenslets, rather asymmetrical, and as a consequence that peak position and centroid are rather different [Figs. 3a, 3b ].

For a single scatterer located at the center of the CG on the optical axis, the change of c4 with CG position should be 102nmμm. For MCS the slopes found by centroid estimation and by peak fit were 119±2 and 78±3nmμm, respectively (Fig. 2b).

The centroid-estimation slope (MCS) is in good agreement with the experimental values of 117±1 and 101±1nmμm for a scattering phantom containing 110nm scattering beads (for preparation see [[5]]) and a chemically fixed organotypic rat hippocampus slice [[32]], respectively [Fig. 2g].

4B. CGWS-Measured Wavefront Aberrations

Next we investigated whether the distortions present in the optical path are accurately reflected in the wavefront measured by CGWS. This is crucial for fast and complete wavefront correction. A systematic bias might occur, for example, because the light encounters the distortions in the sample arm twice, on the way to the focus and on the way back (double pass), while for the preemptive wavefront correction in two-photon microscopy only the information about the distortions on the way to the focus is needed. The behavior of the wavefront sensor can depend strongly on the sample properties, as becomes very apparent when comparing a mirror and a single scatterer at the focus location. For a mirror double passing doubles point-symmetric aberrations, such as astigmatism or defocus, but eliminates point antisymmetric aberrations, such as coma, which therefore cannot be detected at all. For a single scatterer, from which a spherical wavelet emanates, the aberrations of the illumination light are lost completely, and only inhomogeneities encountered during backpropagation are seen (single-pass aberrations). Therefore, with a single scatterer single-pass aberrations are correctly detected and, as a consequence of optical reciprocity [[33]], are equal to the aberrations encountered by the incoming light on the way to the focus.

For a collection of randomly distributed scatterers we used MCSs with distortions introduced by phase plates to investigate whether the single-pass aberrations are measured correctly by CGWS. To account for the change in travel time caused by the distortions, we replaced τ(r,w) in Eq. (5) with

τaberr.(r,w)=τ(r,w)+τillum.(r)+τscatt.(r,w),
where τ(r,w) is the time calculated by using Eq. (4) while τillum.(r) and τscatt.(r,w) are the time changes due to the distortions for incoming and backpropagating light, respectively. Direction changes of the light rays due to refraction were neglected, since only small distortions were investigated. The shape of the CV will be distorted because of its dependence on the travel times, albeit only little for typical distortion strengths [Fig. 4a ].

As an example, we inserted at the BFP of the objective a phase plate carrying astigmatism, c6=0.3μm (c5=0μm), and coma, c8=0.5μm (c7=0μm), which was passed by the incoming and the scattered light. For all coherence lengths tested the wavefronts determined by MCS (the density of scatterers was 10μm3) were those expected for the single-pass aberrations [Fig. 4b]. As in the single-scatterer case, aberrations encountered by the incoming light are lost completely, and therefore the wavefront detected corresponds to the incoherent superposition of the coherence-gated light and is the amplitude-weighted average of the wavefronts independently emanating from all the scatterers within the CV. This result does not depend on the type and amount of aberration and is consistent with measurements described below and in [[5], [9]]. However, the assumption of randomly distributed scatterers is essential, since the backscattered light from, for example, a dense layer of scatterers in a single plane (acting as a mirror) would, of course, be sensitive to distortions in the incoming light path.

Another assumption needed for the incoherent superposition to hold [Eq. (9)] is that the number of scatterers within the focal cone is lower than ρlim=(αβ)β, beyond which the coherent term would dominate the incoherent term, even if the scatterers are randomly distributed. To roughly estimate α and β, we assumed an objective with a low numerical aperture (NA). Then, because x,yz, only the axial coordinates of the scatterers need to be taken into account. Furthermore, we neglect polarization effects (WPol=1) and assume that the detection points are x1=x2=0, which, from Eq. (5), yields

α=ES(0)ES*(0)=1V(z0)V(z0)dxdydzES2(z,0)γ[τ(z,0)]2,
β=ES(0)ES*(0)=1V(z0)V(z0)dxdydzES(z,0)γ[τ(z,0)]exp[iωτ(z,0)]2,
where V(z0) is the volume of the focal cone up to an axial distance of z0 in both directions from the focus, ES(z,0)=1z is the amplitude of the backscattered light (see Section 3), γ[τ(z,0)] is a Gaussian self-coherence function with τ(z,0)2n(z+dCG)c, and dCG is the CG position. For a NA of 0.1, z0=100μm, a CG position shift of 0μm, and a coherence length of 5μm, the numerically calculated density of scatterers was 1073μm3, which exceeds by more than 60 orders of magnitude any possible density of atoms. The maximum density of scatterers expected for biological tissue, where the size of the dominant backscatterers is λ4λ2 [[34], [35]], is about 100μm3. For larger coherence lengths, cl, ρlim scales roughly as exp(κcl2), where κ is a constant. Thus the second (coherent superposition) term of Eq. (9) can be safely neglected. The extremely large value for ρlim makes it unlikely that coherent superposition will play a significant role even if the assumptions used for this estimate are not strictly valid.

It was also verified by MCS for an experimentally realistic coherence length of 50μm [[5]] that for scatterer densities tested up to 6×105μm3 coherent effects do not affect wavefront measurements.

All distortions (Zernike modes higher than c5) are therefore correctly measured by CGWS, but can focus displacements due to spatially varying tip–tilt and defocus, which are not accompanied by a change in focus shape but cause image distortions, also be measured? For episcopic illumination focus displacements due to tip and tilt cannot be sensed [[36]], since a conjugate displacement is introduced on the way back to the objective. An axial displacement of the focus due to a defocus introduced into the optical path is, however, detectable, since CGWS is sensitive to the travel time of the scattered light.

4C. Aberrations Close to Focus

A bias toward a flat wavefront may, however, be introduced if the distortion layer is located close to the focus rather than far above. Then the wavefront as measured by CGWS no longer accurately reflects the actual specimen-introduced distortions. The reason is that in this case, due to the lateral extent of the CV, light scattered at different locations is affected by refractive index inhomogeneities differently, which then leads to spatial averaging of distortions [Fig. 4a]. We explored (using MCS) this effect by inserting phase plates at various distances above the focus. We chose for the phase plates the same nominal distortion (c6=0.5μm and c8=0.3μm) across the diameter of the focal cone at the point of insertion. As a result the same aberrations are caused in a spherical wave emanating directly from the focus, but note that phase plates closer to the focus contain aberrations with higher spatial frequencies.

This is in contrast to real samples, where distortions with a certain refractive index variation would typically cause less aberration closer to the focus.

MCSs show that nearly correct distortion values are obtained as long as the phase plate is sufficiently far from the focus with substantial deviations visible only at distances below 50μm for a coherence length of 50μm and for a CG position at 5μm [Fig. 4c]. This deviation toward a flat wavefront is due solely to spatial averaging of the distortions and not to scatterers lying above the phase plate, because those were not included in the MCS. For a CG position centered at the focus no significant deviations are observed, presumably because of the smaller lateral extent of the CV [Fig. 4c].

4D. Aberrations Caused by a Tilted Glass Plate

Next we tested, using experimental measurements and MCSs, whether CGWS correctly detects the distortions due to a tilted glass plate (BK7, thickness 145μm, 10° tilt angle) inserted between the scattering sample and the objective [Fig. 5a ]. To model this situation realistically, the time delays for illumination and scattered light due to retardation and refraction by the glass plate need to be taken into account, which can be done by ray tracing using straightforward geometrical considerations. Since the effect of the glass plate on a transmitted ray is invariant under lateral and axial translations, a displacement of the CG position will not change the measured aberrations in this particular case. Note, however, that for an objective that meets the sine condition [[33]] all rotationally symmetric aberrations, such as, e.g., c4 (defocus) or c11 (first-order spherical aberration) vary with CG position even when no actual aberrations are present, albeit with steeply declining coefficients as the order increases.

The aberrations caused by the tilted glass plate and calculated by ray tracing for a single scatterer on the optical axis were mainly astigmatism, c6=90nm (c5 was zero because the tilt was along one of the principal axes, in this case the y axis), coma, c8=220nm (c7=0nm), and spherical aberration, c11=170nm. For a random collection of scatterers MCS also showed astigmatism, c6, and coma, c8, which remained almost constant when the CG position was varied [Fig. 5b], with average values c6=100±20nm and c8=220±10nm. The defocus, c4, and the spherical aberration, c11, changed linearly with slopes of 108±5 and 6.0±0.5nmμm, respectively, with the CG position at the focus c11=167±5nm. All other Zernike coefficients tested (up to c11) were below the speckle-noise level of about 10nm.

Using a scattering sample with 110nm beads [[5], [9]] we measured the aberrations caused by a tilted glass plate experimentally [Fig. 5c] and found c6=90±20nm and c8=230±10nm (averaged over all measured CG positions), in agreement with the values found by MCS. For the measured value of c11=120±10nm (CG position centered at the focus) there is a substantial discrepancy with the value obtained by MCS (c11=167±5nm), possibly because the refractive index of the scattering sample is slightly different from that of water. The slope of c4 versus CG position was 120±5nmμm, and that for c11 was 4±1nmμm, both in good agreement with slopes found with MCSs (see above). Experimentally, c6 changed slightly with CG position, possibly due to the illumination-light polarization’s being not strictly circularly (see below).

These aberrations obtained experimentally, by MCS, and calculated for a single scatterer are in agreement, showing that single-pass aberrations caused by the tilted glass plate are correctly measured.

4E. Polarization Effects

The phase and amplitude of the coherence-gated electric field in the BFP are affected not only by the self-coherence function γ[τ(k)(u,v)] and the amplitude distribution across the focal cone ES(k)(u,v) but also by the polarization dependence of scattering as described [Eq. (5)] by WP(k)(u,v), which can be calculated exactly for spherical scatterers by using Mie scattering theory [[37]], which, in addition to providing polarization-resolved amplitudes also provides the phase delay for any scattering angle. Here, the effects of the incident- and reference-light polarizations on CGWS-measured wavefronts is investigated (preliminary results have been published in [[38]]).

First, measurements with linearly polarized light, with parallel directions for the incident sample and reference light, were performed by using the optical setup described in [[9]]. Two scattering samples (with beads of 100nm and 1μm diameter) with distinct polarization-dependent scattering properties were examined. As the CG position was changed, not only did the defocus c4, change (not shown), which was expected, but also the astigmatism c6 [Figs. 6a, 6b ], which was not expected. The slope for c6 (c50μm for linear polarization along the x axis) depended on the bead size with values of 370±3 and 8±3nmμm for larger and smaller beads, respectively. None of the other Zernike modes varied with CG position. The measured astigmatism cannot be due to actual distortions, because it depends on the bead size.

The origin of spurious astigmatism is somewhat complicated. It is not caused by the polarization dependence of the phase shift during scattering, which MCSs show to be too small to account for the observed astigmatism. This is obvious for scatterers that are small compared with the wavelength (Rayleigh scatterers), where the phase delays are almost independent of scattering angle within the acceptance cone of the objective (NA of 0.9). What is responsible instead for the spurious astigmatism is an angle-, polarization-, BFP-position-, and scatterer-position-dependent weight, which can be written as WP(r(k),u,v,pR,pI), where r(k)=(x,y,z) is the position of the scatterer k, (u,v) is the detection position, and pR and pI are the polarizations of the reference and incident sample light, respectively [see Eq. (5)]. Note that the time delay τ(k)(u,v)=τ(r(k),u,v) does not change when both the scatterer, (x,y), and the detection point, (u,v), rotate around the optical axis, but WP(r(k),u,v,pR,pI) will generally change.

For linearly polarized light rotational symmetry is obviously broken, but WP still shows two mirror symmetries, parallel and perpendicular to the direction of the linear polarization. If the linear polarization is along either the x or the y axis

WP(x,y,z,u,v,pR,pI)=WP(x,y,z,u,v,pR,pI),
WP(x,y,z,u,v,pR,pI)=WP(x,y,z,u,v,pR,pI),
where the u axis is along the x axis. Any polarization-dependent spurious effects should also show these two axial symmetries, as do Zernike modes 6, 12, and 14.

Symmetry arguments do not, however, allow quantitative predictions. For those we used MCSs. To model the effects that a generic objective lens has on polarization, refraction by a prism was used [[25]]. The steps performed to keep track of the polarization state are detailed in Appendix A. Because keeping track of the polarization is computationally expensive, a coherence length of 2μm and a density of scatterers of 1μm3 were chosen to keep the number of scatterers small. As shown above for the polarization-insensitive case, wavefronts calculated by the MCSs are, except for the size of the speckle error, rather insensitive to the coherence length and density of scatterers. We confirmed this for the polarization-sensitive case, where we performed MCSs with a coherence length of 8μm for some CG positions and found that the results were unchanged (data not shown).

For linearly polarized illumination and reference light (polarizations parallel), MCSs showed a linear change of c6 with CG position [Figs. 6c, 6d]. The slopes of 24±4 and 50±5nmμm for 100nm and 1μm beads, respectively, were, however, significantly larger than the experimental values of 8±3 and 37±3nmμm. In addition we found, by MCS but not in our experiments, for 1μm beads a spurious c12 with a slope of 6±1nmμm.

Possible explanations for these discrepancies are, first, that only singly scattered light is taken into account for the MCSs, but experimentally there was no way to exclude coherence-gated multiply scattered light. Multiple scattering should reduce the observed spurious aberrations, since then polarization effects are averaged over a range of scattering angles. Another explanation could be the depolarization of the incident sample light caused by polarization-dependent transmission losses [[39]] at individual lens components within the objective [[40], [41]], which was not taken into account in MCSs.

Important for the application of CGWS to wavefront correction is that all spurious aberrations vanish when the CG is centered at the focus. But polarization-mediated spurious aberrations can be avoided completely (for all CG positions) if circularly polarized illumina tion and reference light is used, since then WP(x,y,z,u,v,pR,pI) is rotationally symmetric.

For the experimental measurements we used the optical setup described in [[5]], where the incident sample light was circularly polarized, but the reference light was linearly polarized. However, since the backscattered sample light passes a quarter-wave plate, this is equivalent to circularly polarized reference light. We found only a very small amount of spurious astigmatism, c6, with slopes of 2±2 and 6±3nmμm for 100nm or 1μm beads, respectively [Figs. 7a, 7b ]. For the 1μm beads CG position-dependent c12 was significant, with a slope of 3±1nmμm. No significant CG position-dependent c6 and c12 were seen by MCS [Figs. 7c, 7d]. The slopes of c4 (defocus), obtained by MCS for circularly polarized light, were 99±4 and 104±4nmμm for 100nm and 1μm beads, respectively, very similar to the slope obtained by polarization-insensitive MCSs where point scatterers had been assumed (119±2nmμm; see above).

5. DISCUSSION

Experimentally measured wavefronts using CGWS were compared with wavefronts obtained by Monte-Carlo simulations (MCS) and, in some cases, to direct calculations. All experimentally observed properties of CGWS-measured wavefronts, such as the change of the Zernike defocus with the variation of the CG position and that the measured wavefront is not affected by aberrations in the incident beam, are consistent with MCS results. Only for the dependence of the CGWS-measured wavefronts on the incident-light polarization did we find a discrepancy between experiment and MCS. However, in the model a number of additional simplifications were made, which need to be discussed. First, attenuation of the illumination intensity with depth due to scattering and absorption was ignored. Attenuation results in more weight being given to scatterers near the surface of the specimen and thus shifts the centroid of the CV toward the surface, in particular if the attenuation length is comparable to the axial extent of the CV. Such a shift will significantly affect rotationally symmetric Zernike modes.

Second, our model neglects multiply scattered light, which can have a total travel time that allows it to pass the CG while having been scattered by scatterers outside the single-scattering CV. In strongly scattering specimens multiple scattering can be neglected only for low probing depths. To estimate at which focus depth multiply scattered light might begin to dominate singly scattered light, results from OCT can be used. For OCT the maximum probing depth is, even with confocal detection [[13], [15]], which strongly suppresses multiply scattered light, limited by imaging contrast to several (5–8) mean free path lengths, depending on the scattering properties of the tissue and on the optical configuration [[14], [24], [42]]. For CGWS, however, a pinhole, which acts as a spatial filter, in the sample arm could lead to an underestimation of actual distortions. A subresolution pinhole size, for example, would not only suppress the signal size by orders of magnitude but would also completely filter out all aberrations. However, a properly sized pinhole should suppress multiply scattered light without much affecting the singly scattered light, reducing both bias and noise. Even if aberrations are underestimated, proper correction might still be possible either by correcting, for a known degree of underestimation or by using a larger number of measure correction iterations [[5]].

Since the coherence-gated multiply scattered light that is high-angle scattered more than once is predominantly scattered above the CV (of the singly scattered light) and therefore originates from a laterally more extended volume, the CGWS is affected in two ways. First, the speckle size of the coherence-gated backscattered light is reduced, leading to a larger speckle error (see above), which then requires averaging over more ensembles of scatterers to achieve a certain wavefront error. Second, lateral averaging of distortions deeper inside the sample (closer to the focus) is more severe (see above) leading to an underestimation of the actual distortions.

A third assumption we made is that of discrete, randomly located scattering particles. This assumption relies on studies that show that the scattering properties of biological specimens can be mimicked by distributions of discrete particles of different sizes [[35], [43], [44]]. Most of the extinction by scattering is caused by sizes 2λ4λ [[35], [43], [44]], but high-angle scattering (backscattering) is due mainly to sizes λ4λ2 [[34], [35]]. Cellular organelles, in particular mitochondria (0.30.7μm in diameter), lysosomes (0.20.5μm), and structures within the nucleus thus contribute most to backscattering [[34], [45]]. For particles in the range of λ4λ2 backscattering is largely independent of size and thus behaves as pointlike.

The assumption that the distribution of scatterers is spatially homogenous is likely to be violated to a varying degree in real tissue.

6. CONCLUSION

We have demonstrated both experimentally and by Monte Carlo simulation that wavefronts measured by CGWS represent single-pass specimen-introduced distortions correctly. Since for a two-photon microscope [[46]] only the distortions for the incident light need to be compensated, the necessary wavefront aberrations are directly measured by CGWS and can thus be used for wavefront correction in a single step. This allows fast feed-forward wavefront correction.

APPENDIX A: PROPAGATION OF POLARIZED LIGHT IN THE SAMPLE ARM

The interference of reference and sample light in terms of polarization is described by WP(k)=pR*pS(k) [Eq. (5)], where pR and pS(k) are the polarizations of reference light and of light backscattered at the scatterer k, respectively. The effects by the objective on polarization can be locally approximated by that of a prism [[25], [47]], and the polarization dependence of scattering is contained in Mie’s theory [[37]].

First, the light rays were traced through the sample arm, and at each change of direction a new local basis (eϑ,eφ,er) was defined (Fig. 8 ) relative to a fixed basis (ex,ey,ez) with ex×ey=ez. For each local basis eϑ(i)×eφ(i)=er(i), whereby er(i) points in the current direction of propagation and eϑ(i) lies in the plane that contains both the current and the previous direction of propagation. An exception is the first basis, which essentially corresponds to the fixed basis, but er(1) is in the direction of propagation for the incident sample light (see Fig. 8).

Basis 1. For the incoming light rays

er(1)=ez;eφ(1)=ey;eϑ(1)=ex.

Basis 2. Rotation of basis 1 about er(1) such that eϑ(1) lies in the plane of refraction (containing both er(1) and er(3)) caused by the objective

er(2)=er(1);eφ(2)=eφ(3);eϑ(2)=eφ(2)×er(2).

Basis 3. After passing the objective but before scattering at P(x,y,z),

er(3)=sgn(z)x2+y2+z2(xex+yey+zez);
eφ(3)=er(1)×er(3)er(1)×er(3);eϑ(3)=eφ(3)×er(3).

Basis 4. Rotation of basis 3 about er(3) such that eϑ(3) lies in the scattering plane (containing er(3) and er(5))

er(4)=er(3);eφ(4)=er(4)×er(5)er(4)×er(5);eϑ(4)=eφ(4)×er(4).

Basis 5. After scattering in the direction w,

er(5)=wxex+wyey+wzez;eφ(5)=eφ(4);eϑ(5)=eφ(5)×er(5).

Basis 6. Rotation of basis 5 about er(5) such as eϑ(5) lies in the plane of refraction (containing er(5) and er(7)) caused by the objective

er(6)=er(5);eφ(6)=eφ(7);eϑ(6)=eφ(6)×er(6).

Basis 7. After passing the objective on the way back (fback is the back focal length),

er(7)=(xzwzwx)ex+(yzwzwy)ey+fbackez,
eφ(7)=er(5)×er(7)er(5)×er(7);eϑ(7)=eφ(7)×er(7).
In the next step, the polarization, described by using the Jones formalism [[48]], is traced for each change of the local basis. The polarization of the incoming light, expressed in the fixed basis, is pini=ainiex+biniey, which is the same as the polarization of the reference light pR.

Fixed BasisBasis 1. Transformation into the local basis

(a1b1)=(1001)(ainibini)
with pini=ainiex+biniey and pk(1)=a1eϑ(1)+b1eφ(1).

Basis 1Basis 2. Rotation into the plane of refraction

(a2b2)=(cos(β)sin(β)sin(β)cos(β))(a1b1)
with cos(β)=eϑ(2)eϑ(1) and sin(β)=eϑ(2)eφ(1).

Basis 2Basis 3. Change due to the objective [[25]]

(a3b3)=(1001)(a2b2).

Basis 3Basis 4. Rotation into the scattering plane

(a4b4)=(cos(φ)sin(φ)sin(φ)cos(φ))(a3b3)
with cos(φ)=eϑ(4)eϑ(3) and sin(φ)=eϑ(4)eφ(3).

Basis 4Basis 5. Mie scattering functions S1(ϑ) and S2(ϑ) [[26]]

(a5b5)=(S2(ϑ)00S1(ϑ))(a4b4)
with cos(ϑ)=er(4)er(5).

Basis 5Basis 6. Rotation into the plane of refraction

(a6b6)=(cos(ε)sin(ε)sin(ε)cos(ε))(a5b5)
with cos(ε)=eϑ(6)eϑ(5) and sin(ε)=eϑ(6)eφ(5).

Basis 6Basis 7. Change due to objective [[25]]

(a7b7)=(1001)(a6b6).

Basis 7Fixed Basis.

(afbf)=(eϑ(7)exeφ(7)exeϑ(7)eyeφ(7)ey)(a7b7)
with pS(k)=afex+bfey. Since af and bf are related to aini and bini, the polarization weight WP(k)=pR*pS(k) can be calculated for each scattered light ray.

ACKNOWLEDGMENTS

We thank Manfred Hauswirth, Michael Müller, and Jürgen Tritthardt for technical support, and Jonas Binding, Marcus Feierabend, and Marcel Lauterbach for helpful discussions and comments on the manuscript. The work was supported by the Max-Planck Society.

 

Fig. 1 (a) Calculation of τ(k) for sample light scattered by a scatterer at P(x,y,z) based on ray tracing. The incident ray passes the BFP at point A, is refracted by the objective lens toward the focus F, scatters at P in the direction w, and crosses the BFP at B on its return path (red online). For the CG centered at the focus the reference light takes a path that equals in length C-F-C (blue online) and A-F-B. Typical CVs, shown in cross section, for detection points (b) in the center and (c) at the edge of the BFP of the objective, respectively. The dotted curves correspond to γ(τ)=0.5 and delineate the CV. The coherence length was 50μm (FWHM, normalized Gaussian self-coherence function), corresponding to a CG length of 18.8μm within the specimen, and the CG position was +5μm.

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Fig. 2 (a), (c), (e) Number of phase singularities and (b), (d), (f) size of c4 as they change with (a), (b) CG position, (c), (d), coherence length, (e), (f), and density of scatterers. Plotted are c4 values as determined by estimating the centroid and those determined by fitting the peak by a Gaussian distribution. When not varied, the CG position, the density of scatterers, and the coherence length were +5μm, 100μm3, and 30μm (FWHM), respectively. Linear fits to the data in (b) yield slopes of 119±0.002nmμm (centroid) and of 78±0.003nmμm (peak fit). Error bars are not shown in (b) because they are too small. (g) Experimentally measured c4 as a function of the CG position for a scattering sample with 110nm scattering beads (crosses, black) and for a chemically fixed organotypic rat hippocampus slice (dots, red). Also shown are linear fits to the data giving slopes of 101±0.001 and 117±0.001nmμm (very close to the theoretical expected value of 119±0.002nmμm). Note that the vertical displacement between the data series is not meaningful, since only relative CG positions were measured. Except for (b) and (g), the data points were connected by straight lines.

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Fig. 3 Asymmetric diffraction patterns for off-axis vSHS lenslets. (a) Illustration of how the CV is projected through a sublens. Peak fit (P) and centroid estimation (C) of the diffraction pattern differ. (b) Diffraction pattern numerically calculated for a edge lenslet and averaged over nine different scatterer ensembles for a CG position, density of scatterers, and coherence length (FWHM) of +5μm, 100μm3, and 24μm, respectively. Note that, for better visibility, CV and focal cone are not drawn to scale.

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Fig. 4 MCSs showing the influence of distortions on the CGWS-measurement process. (a) The CV is deformed by distortions, and due to the extended CV distortions are averaged laterally, depending on the distance of the distortion layer to the focus. (b) The aberrations (Zernike coefficients c5 to c8) as a function of the coherence length with the phase plate located in the BFP. The dashed lines show the expected aberrations due to a single pass through a phase plate with c6=0.5μm and c8=0.3μm, respectively. The density of scatterers was 10μm3, and the CG was at +5μm. (c) Phase plates with c6=0.5μm and c8=0.3μm located at varying distances above the focus and simulated for CG positions at 0 and 5μm. The coherence length was 50μm, and the density of scatterers was 1μm3. Only Zernike coefficients c6 and c8 are shown. The dashed lines indicate the expected aberrations for c6 and c8.

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Fig. 5 (a) Distortions due to a tilted glass plate (10°, 145μm thick) inserted between the objective and the 100nm bead sample. (b) Aberrations (Zernike coefficients c4 to c11) determined by MCS for a coherence length of 6μm and a density of scatterers of 1μm3 for CG positions in steps of 2μm. (c) Experimentally measured aberrations for the tilted glass plate with the same parameters for CG positions in steps 1μm. The data points were connected by straight lines.

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Fig. 6 Detected wavefronts using linearly polarized light. Measured wavefronts using scattering samples with (a) 100nm and (b) 1μm sized beads. The CG position was varied in steps of 2 and 3μm, respectively. The absolute CG position was not determined, only relative positions are depicted. For comparison, wavefronts determined by MCSs using (c) 100nm and (d) 1μm sized beads are shown. A scatterer density of 1μm3 and a coherence length of 2μm were used for the calculation. The CG position was varied from 8 to 8μm in steps of 4μm. Only spurious (varying with CG position) Zernike modes are shown: c6 and c12. The slopes, obtained by linear fitting, are (a) 8±3, (b) 37±3, (c) 24±4, and (d) 50±5nmμm for c6 and (a) 1±1, (b) 1±1, (c) 2±1, and (d) 6±1nmμm for c12.

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Fig. 7 Detected wavefronts using circularly polarized light. Experimentally measured wavefronts using scattering samples with (a) 100nm (b) 1μm sized beads. The CG position was varied in steps of 1μm. For comparison, wavefronts determined by MCSs using (c) 100nm and (d) 1μm sized beads. A scatterer density of 1μm3 and a coherence length of 2μm were used. The CG position was varied from 8 to 8μm in steps of 4μm. Only spurious (varying with CG position) Zernike modes are shown: c6 and c12. Linear fitting gave slopes of (a) 2±2, (b), 6±3, (c) 1±3, and (d) 2±3nmμm for c6, and (a) 1±1, (b) 3±1, (c) 0±3, and (d) 1±2nmμm for c12.

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Fig. 8 Propagation of the polarization through sample arm. The local bases are depicted as small arrows (red online). The fixed basis is shown at the focus. The scattering angle is ϑ.

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References

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  1. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Adaptive aberration correction in a confocal microscope,' Proc. Natl. Acad. Sci. U.S.A. 99, 5788-5792 (2002).
    [CrossRef] [PubMed]
  2. 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]
  3. P. N. Marsh, D. Burns, and J. M. Girkin, 'Practical implementation of adaptive optics in multiphoton microscopy,' Opt. Express 11, 1123-1130 (2003).
    [CrossRef] [PubMed]
  4. 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]
  5. M. Rueckel, J. A. Mack-Bucher, and W. Denk, 'Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,' Proc. Natl. Acad. Sci. U.S.A. 103, 17137-17142 (2006).
    [CrossRef] [PubMed]
  6. J. W. Hardy, 'Active optics: a new technology for the control of light (review article),' Proc. IEEE 66, 651-697 (1978).
    [CrossRef]
  7. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, 'Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,' J. Microsc. 206, 65-71 (2002).
    [CrossRef] [PubMed]
  8. A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, 'Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,' Microsc. Res. Tech. 67, 36-44 (2005).
    [CrossRef] [PubMed]
  9. M. Feierabend, M. Ruckel, and W. Denk, 'Coherence-gated wave-front sensing in strongly scattering samples,' Opt. Lett. 29, 2255-2257 (2004).
    [CrossRef] [PubMed]
  10. D. Malacara, Optical Shop Testing (Wiley, 1992).
  11. R. V. Shack and B. C. Platt, 'Abstract. Production and use of a lenticular Hartmann screen,' J. Opt. Soc. Am. 61, 656-660 (1971).
  12. M. Rueckel and W. Denk, 'Coherence-gated wavefront sensing using a virtual Shack-Hartmann sensor,' Proc. SPIE 6306, 63060H (2006).
    [CrossRef]
  13. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, 'Optical coherence tomography--principles and applications,' Rep. Prog. Phys. 66, 239-303 (2003).
    [CrossRef]
  14. J. M. Schmitt, 'Optical coherence tomography (OCT): A review,' IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999).
    [CrossRef]
  15. B. Karamata, M. Laubscher, M. Leutenegger, S. Bourquin, T. Lasser, and P. Lambelet, 'Multiple scattering in optical coherence tomography. I. Investigation and modeling,' J. Opt. Soc. Am. A 22, 1369-1379 (2005).
    [CrossRef]
  16. 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).
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  32. J. J. Zhu, J. A. Esteban, Y. Hayashi, and R. Malinow, 'Postnatal synaptic potentiation: Delivery of GluR4-containing AMPA receptors by spontaneous activity,' Nat. Neurosci. 3, 1098-1106 (2000).
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    [CrossRef]
  37. G. Mie, 'Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,' Ann. Phys. 25, 377-445 (1908).
    [CrossRef]
  38. M. Rueckel and W. Denk, 'Polarization effects in coherence-gated wave-front sensing,' in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings (CD-ROM), Technical Digest (Optical Society of America, 2005), paper AThC4.
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  40. S. Inoue and W. L. Hyde, 'Studies on depolarization of light at microscope lens surfaces. 2. The simultaneous realization of high resolution and high sensitivity with the polarizing microscope,' J. Biophys. Biochem. Cytol. 3, 831-838 (1957).
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  41. K. Bahlmann and S. W. Hell, 'Depolarization by high aperture focusing,' Appl. Phys. Lett. 77, 612-614 (2000).
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2006

M. Rueckel, J. A. Mack-Bucher, and W. Denk, 'Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,' Proc. Natl. Acad. Sci. U.S.A. 103, 17137-17142 (2006).
[CrossRef] [PubMed]

M. Rueckel and W. Denk, 'Coherence-gated wavefront sensing using a virtual Shack-Hartmann sensor,' Proc. SPIE 6306, 63060H (2006).
[CrossRef]

2005

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, 'Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,' Microsc. Res. Tech. 67, 36-44 (2005).
[CrossRef] [PubMed]

M. J. Booth, M. Schwertner, and T. Wilson, 'Specimen-induced aberrations and adaptive optics for microscopy,' Proc. SPIE 5894, 589403 (2005).
[CrossRef]

B. Karamata, M. Laubscher, M. Leutenegger, S. Bourquin, T. Lasser, and P. Lambelet, 'Multiple scattering in optical coherence tomography. I. Investigation and modeling,' J. Opt. Soc. Am. A 22, 1369-1379 (2005).
[CrossRef]

J. D. Wilson and T. H. Foster, 'Mie theory interpretations of light scattering from intact cells,' Opt. Lett. 30, 2442-2444 (2005).
[CrossRef] [PubMed]

2004

2003

P. N. Marsh, D. Burns, and J. M. Girkin, 'Practical implementation of adaptive optics in multiphoton microscopy,' Opt. Express 11, 1123-1130 (2003).
[CrossRef] [PubMed]

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

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, 'Optical coherence tomography--principles and applications,' Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

2002

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Adaptive aberration correction in a confocal microscope,' Proc. Natl. Acad. Sci. U.S.A. 99, 5788-5792 (2002).
[CrossRef] [PubMed]

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, 'Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,' J. Microsc. 206, 65-71 (2002).
[CrossRef] [PubMed]

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, 'Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,' J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, 'Three-dimensional position detection of optically trapped dielectric particles,' J. Appl. Phys. 91, 5474-5488 (2002).
[CrossRef]

2000

K. Bahlmann and S. W. Hell, 'Depolarization by high aperture focusing,' Appl. Phys. Lett. 77, 612-614 (2000).
[CrossRef]

J. J. Zhu, J. A. Esteban, Y. Hayashi, and R. Malinow, 'Postnatal synaptic potentiation: Delivery of GluR4-containing AMPA receptors by spontaneous activity,' Nat. Neurosci. 3, 1098-1106 (2000).
[CrossRef] [PubMed]

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]

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]

1999

1998

1997

1994

N. Shvartsman and I. Freund, 'Vortices in random wave-fields--nearest-neighbor anticorrelations,' Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef] [PubMed]

1991

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]

1990

W. Denk, J. H. Strickler, and W. W. Webb, 'Two-photon laser scanning fluorescence microscopy,' Science 248, 73-76 (1990).
[CrossRef] [PubMed]

1987

1986

1979

R. Cubalchini, 'Modal wavefront estimation from phase derivative measurements,' J. Opt. Soc. Am. 69, 973-977 (1979).
[CrossRef]

1978

J. W. Hardy, 'Active optics: a new technology for the control of light (review article),' Proc. IEEE 66, 651-697 (1978).
[CrossRef]

1976

1971

R. V. Shack and B. C. Platt, 'Abstract. Production and use of a lenticular Hartmann screen,' J. Opt. Soc. Am. 61, 656-660 (1971).

1957

S. Inoue and W. L. Hyde, 'Studies on depolarization of light at microscope lens surfaces. 2. The simultaneous realization of high resolution and high sensitivity with the polarizing microscope,' J. Biophys. Biochem. Cytol. 3, 831-838 (1957).
[CrossRef] [PubMed]

1941

1934

A. Khintchine, 'Correlation theories of stationary stochastic processes,' Math. Ann. 109, 604-615 (1934).
[CrossRef]

1930

N. Wiener, 'Generalized harmonic analysis,' Acta Math. 55, 117-258 (1930).
[CrossRef]

1908

G. Mie, 'Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,' Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Acta Math.

N. Wiener, 'Generalized harmonic analysis,' Acta Math. 55, 117-258 (1930).
[CrossRef]

Ann. Phys.

G. Mie, 'Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,' Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

K. Bahlmann and S. W. Hell, 'Depolarization by high aperture focusing,' Appl. Phys. Lett. 77, 612-614 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. M. Schmitt, 'Optical coherence tomography (OCT): A review,' IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999).
[CrossRef]

Int. J. Remote Sens.

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

J. Appl. Phys.

A. Rohrbach and E. H. K. Stelzer, 'Three-dimensional position detection of optically trapped dielectric particles,' J. Appl. Phys. 91, 5474-5488 (2002).
[CrossRef]

J. Biomed. Opt.

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, 'Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,' J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef] [PubMed]

J. Biophys. Biochem. Cytol.

S. Inoue and W. L. Hyde, 'Studies on depolarization of light at microscope lens surfaces. 2. The simultaneous realization of high resolution and high sensitivity with the polarizing microscope,' J. Biophys. Biochem. Cytol. 3, 831-838 (1957).
[CrossRef] [PubMed]

J. Microsc.

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]

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, 'Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,' J. Microsc. 206, 65-71 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

R. V. Shack and B. C. Platt, 'Abstract. Production and use of a lenticular Hartmann screen,' J. Opt. Soc. Am. 61, 656-660 (1971).

R. Cubalchini, 'Modal wavefront estimation from phase derivative measurements,' J. Opt. Soc. Am. 69, 973-977 (1979).
[CrossRef]

R. C. Jones, 'A new calculus for the treatment of optical systems I. Description and discussion of the calculus,' J. Opt. Soc. Am. 31, 488-493 (1941).
[CrossRef]

R. J. Noll, 'Zernike polynomials and atmospheric-turbulence,' J. Opt. Soc. Am. 66, 207-211 (1976).
[CrossRef]

J. Opt. Soc. Am. A

Math. Ann.

A. Khintchine, 'Correlation theories of stationary stochastic processes,' Math. Ann. 109, 604-615 (1934).
[CrossRef]

Microsc. Res. Tech.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, 'Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,' Microsc. Res. Tech. 67, 36-44 (2005).
[CrossRef] [PubMed]

Nat. Neurosci.

J. J. Zhu, J. A. Esteban, Y. Hayashi, and R. Malinow, 'Postnatal synaptic potentiation: Delivery of GluR4-containing AMPA receptors by spontaneous activity,' Nat. Neurosci. 3, 1098-1106 (2000).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

N. Shvartsman and I. Freund, 'Vortices in random wave-fields--nearest-neighbor anticorrelations,' Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef] [PubMed]

Proc. IEEE

J. W. Hardy, 'Active optics: a new technology for the control of light (review article),' Proc. IEEE 66, 651-697 (1978).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Adaptive aberration correction in a confocal microscope,' Proc. Natl. Acad. Sci. U.S.A. 99, 5788-5792 (2002).
[CrossRef] [PubMed]

M. Rueckel, J. A. Mack-Bucher, and W. Denk, 'Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,' Proc. Natl. Acad. Sci. U.S.A. 103, 17137-17142 (2006).
[CrossRef] [PubMed]

Proc. SPIE

M. Rueckel and W. Denk, 'Coherence-gated wavefront sensing using a virtual Shack-Hartmann sensor,' Proc. SPIE 6306, 63060H (2006).
[CrossRef]

M. J. Booth, M. Schwertner, and T. Wilson, 'Specimen-induced aberrations and adaptive optics for microscopy,' Proc. SPIE 5894, 589403 (2005).
[CrossRef]

Rep. Prog. Phys.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, 'Optical coherence tomography--principles and applications,' Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Science

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]

W. Denk, J. H. Strickler, and W. W. Webb, 'Two-photon laser scanning fluorescence microscopy,' Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Other

J. W. Goodman, Statistical Optics (Wiley, 2000).

C. F. Bohren and D. R. Huffmann, Absorption and Scattering by Small Particles (Wiley, 1983).

A. E. Siegman, Lasers (University Science, 1986).

D. Malacara, Optical Shop Testing (Wiley, 1992).

M. Rueckel and W. Denk, 'Polarization effects in coherence-gated wave-front sensing,' in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings (CD-ROM), Technical Digest (Optical Society of America, 2005), paper AThC4.

E. Hecht, Optics (Addison-Wesley, 1990).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

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Figures (8)

Fig. 1
Fig. 1

(a) Calculation of τ ( k ) for sample light scattered by a scatterer at P ( x , y , z ) based on ray tracing. The incident ray passes the BFP at point A, is refracted by the objective lens toward the focus F, scatters at P in the direction w, and crosses the BFP at B on its return path (red online). For the CG centered at the focus the reference light takes a path that equals in length C-F-C (blue online) and A-F-B. Typical CVs, shown in cross section, for detection points (b) in the center and (c) at the edge of the BFP of the objective, respectively. The dotted curves correspond to γ ( τ ) = 0.5 and delineate the CV. The coherence length was 50 μ m (FWHM, normalized Gaussian self-coherence function), corresponding to a CG length of 18.8 μ m within the specimen, and the CG position was + 5 μ m .

Fig. 2
Fig. 2

(a), (c), (e) Number of phase singularities and (b), (d), (f) size of c 4 as they change with (a), (b) CG position, (c), (d), coherence length, (e), (f), and density of scatterers. Plotted are c 4 values as determined by estimating the centroid and those determined by fitting the peak by a Gaussian distribution. When not varied, the CG position, the density of scatterers, and the coherence length were + 5 μ m , 100 μ m 3 , and 30 μ m (FWHM), respectively. Linear fits to the data in (b) yield slopes of 119 ± 0.002 nm μ m (centroid) and of 78 ± 0.003 nm μ m (peak fit). Error bars are not shown in (b) because they are too small. (g) Experimentally measured c 4 as a function of the CG position for a scattering sample with 110 nm scattering beads (crosses, black) and for a chemically fixed organotypic rat hippocampus slice (dots, red). Also shown are linear fits to the data giving slopes of 101 ± 0.001 and 117 ± 0.001 nm μ m (very close to the theoretical expected value of 119 ± 0.002 nm μ m ). Note that the vertical displacement between the data series is not meaningful, since only relative CG positions were measured. Except for (b) and (g), the data points were connected by straight lines.

Fig. 3
Fig. 3

Asymmetric diffraction patterns for off-axis vSHS lenslets. (a) Illustration of how the CV is projected through a sublens. Peak fit (P) and centroid estimation (C) of the diffraction pattern differ. (b) Diffraction pattern numerically calculated for a edge lenslet and averaged over nine different scatterer ensembles for a CG position, density of scatterers, and coherence length (FWHM) of + 5 μ m , 100 μ m 3 , and 24 μ m , respectively. Note that, for better visibility, CV and focal cone are not drawn to scale.

Fig. 4
Fig. 4

MCSs showing the influence of distortions on the CGWS-measurement process. (a) The CV is deformed by distortions, and due to the extended CV distortions are averaged laterally, depending on the distance of the distortion layer to the focus. (b) The aberrations (Zernike coefficients c 5 to c 8 ) as a function of the coherence length with the phase plate located in the BFP. The dashed lines show the expected aberrations due to a single pass through a phase plate with c 6 = 0.5 μ m and c 8 = 0.3 μ m , respectively. The density of scatterers was 10 μ m 3 , and the CG was at + 5 μ m . (c) Phase plates with c 6 = 0.5 μ m and c 8 = 0.3 μ m located at varying distances above the focus and simulated for CG positions at 0 and 5 μ m . The coherence length was 50 μ m , and the density of scatterers was 1 μ m 3 . Only Zernike coefficients c 6 and c 8 are shown. The dashed lines indicate the expected aberrations for c 6 and c 8 .

Fig. 5
Fig. 5

(a) Distortions due to a tilted glass plate (10°, 145 μ m thick) inserted between the objective and the 100 nm bead sample. (b) Aberrations (Zernike coefficients c 4 to c 11 ) determined by MCS for a coherence length of 6 μ m and a density of scatterers of 1 μ m 3 for CG positions in steps of 2 μ m . (c) Experimentally measured aberrations for the tilted glass plate with the same parameters for CG positions in steps 1 μ m . The data points were connected by straight lines.

Fig. 6
Fig. 6

Detected wavefronts using linearly polarized light. Measured wavefronts using scattering samples with (a) 100 nm and (b) 1 μ m sized beads. The CG position was varied in steps of 2 and 3 μ m , respectively. The absolute CG position was not determined, only relative positions are depicted. For comparison, wavefronts determined by MCSs using (c) 100 nm and (d) 1 μ m sized beads are shown. A scatterer density of 1 μ m 3 and a coherence length of 2 μ m were used for the calculation. The CG position was varied from 8 to 8 μ m in steps of 4 μ m . Only spurious (varying with CG position) Zernike modes are shown: c 6 and c 12 . The slopes, obtained by linear fitting, are (a) 8 ± 3 , (b) 37 ± 3 , (c) 24 ± 4 , and (d) 50 ± 5 nm μ m for c 6 and (a) 1 ± 1 , (b) 1 ± 1 , (c) 2 ± 1 , and (d) 6 ± 1 nm μ m for c 12 .

Fig. 7
Fig. 7

Detected wavefronts using circularly polarized light. Experimentally measured wavefronts using scattering samples with (a) 100 nm (b) 1 μ m sized beads. The CG position was varied in steps of 1 μ m . For comparison, wavefronts determined by MCSs using (c) 100 nm and (d) 1 μ m sized beads. A scatterer density of 1 μ m 3 and a coherence length of 2 μ m were used. The CG position was varied from 8 to 8 μ m in steps of 4 μ m . Only spurious (varying with CG position) Zernike modes are shown: c 6 and c 12 . Linear fitting gave slopes of (a) 2 ± 2 , (b), 6 ± 3 , (c) 1 ± 3 , and (d) 2 ± 3 nm μ m for c 6 , and (a) 1 ± 1 , (b) 3 ± 1 , (c) 0 ± 3 , and (d) 1 ± 2 nm μ m for c 12 .

Fig. 8
Fig. 8

Propagation of the polarization through sample arm. The local bases are depicted as small arrows (red online). The fixed basis is shown at the focus. The scattering angle is ϑ.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

I ( u , v ) = I R ( u , v ) + I S ( u , v ) + 2 Re { E R * ( u , v , t ) k E S ( k ) [ u , v , t + τ ( k ) ( u , v ) ] ¯ } ,
γ ( τ ) = E * ( t ) E ( t + τ ) ¯ E * ( t ) E ( t ) ¯ = γ ( τ ) exp ( i ω τ ) ,
I = I S + I R + 2 Re [ E R k ( p R * p S ( k ) ) E S ( k ) γ ( τ ( k ) ) exp ( i ω τ ( k ) ) ] .
τ [ r ( k ) , w ( u , v ) ] = n c [ sgn ( z ) r ( k ) + r ( k ) w ( u , v ) ] + l l 0 c ,
E CGWS ( u , v ) k W P ( k ) ( u , v ) E S ( k ) ( u , v ) γ [ τ ( k ) ( u , v ) ] exp [ i ω τ ( k ) ( u , v ) ] .
E CGWS ( r , q ) = A E CGWS ( u , v ) exp [ i 2 π λ f ( u r + v q ) ] d u d v = k E S ( k ) ( r , q ) ,
I ( r , q ) = E CGWS ( r , q ) 2 = k E S ( k ) ( r , q ) 2 + m n m [ E S ( m ) ( r , q ) ] * E S ( n ) ( r , q ) .
I ( r , q ) = A d s d t exp [ i 2 π λ f ( s r + t q ) ] A d u d v E CGWS ( u , v ) E CGWS * ( u + s , v + t ) .
E CGWS ( x 1 ) E CGWS * ( x 2 ) = k E S ( k ) ( x 1 ) E S ( k ) * ( x 2 ) + m n m E S ( m ) ( x 1 ) E S ( n ) * ( x 2 ) = k E S ( k ) ( x 1 ) E S ( k ) * ( x 2 ) + m n m E S ( m ) ( x 1 ) E S ( n ) * ( x 2 ) = N E S ( x 1 ) E S * ( x 2 ) + N ( N 1 ) E S ( x 1 ) E S * ( x 2 ) = N ( α β ) + N 2 β ,
τ aberr. ( r , w ) = τ ( r , w ) + τ illum. ( r ) + τ scatt. ( r , w ) ,
α = E S ( 0 ) E S * ( 0 ) = 1 V ( z 0 ) V ( z 0 ) d x d y d z E S 2 ( z , 0 ) γ [ τ ( z , 0 ) ] 2 ,
β = E S ( 0 ) E S * ( 0 ) = 1 V ( z 0 ) V ( z 0 ) d x d y d z E S ( z , 0 ) γ [ τ ( z , 0 ) ] exp [ i ω τ ( z , 0 ) ] 2 ,
W P ( x , y , z , u , v , p R , p I ) = W P ( x , y , z , u , v , p R , p I ) ,
W P ( x , y , z , u , v , p R , p I ) = W P ( x , y , z , u , v , p R , p I ) ,
e r ( 1 ) = e z ; e φ ( 1 ) = e y ; e ϑ ( 1 ) = e x .
e r ( 2 ) = e r ( 1 ) ; e φ ( 2 ) = e φ ( 3 ) ; e ϑ ( 2 ) = e φ ( 2 ) × e r ( 2 ) .
e r ( 3 ) = sgn ( z ) x 2 + y 2 + z 2 ( x e x + y e y + z e z ) ;
e φ ( 3 ) = e r ( 1 ) × e r ( 3 ) e r ( 1 ) × e r ( 3 ) ; e ϑ ( 3 ) = e φ ( 3 ) × e r ( 3 ) .
e r ( 4 ) = e r ( 3 ) ; e φ ( 4 ) = e r ( 4 ) × e r ( 5 ) e r ( 4 ) × e r ( 5 ) ; e ϑ ( 4 ) = e φ ( 4 ) × e r ( 4 ) .
e r ( 5 ) = w x e x + w y e y + w z e z ; e φ ( 5 ) = e φ ( 4 ) ; e ϑ ( 5 ) = e φ ( 5 ) × e r ( 5 ) .
e r ( 6 ) = e r ( 5 ) ; e φ ( 6 ) = e φ ( 7 ) ; e ϑ ( 6 ) = e φ ( 6 ) × e r ( 6 ) .
e r ( 7 ) = ( x z w z w x ) e x + ( y z w z w y ) e y + f back e z ,
e φ ( 7 ) = e r ( 5 ) × e r ( 7 ) e r ( 5 ) × e r ( 7 ) ; e ϑ ( 7 ) = e φ ( 7 ) × e r ( 7 ) .
( a 1 b 1 ) = ( 1 0 0 1 ) ( a ini b ini )
( a 2 b 2 ) = ( cos ( β ) sin ( β ) sin ( β ) cos ( β ) ) ( a 1 b 1 )
( a 3 b 3 ) = ( 1 0 0 1 ) ( a 2 b 2 ) .
( a 4 b 4 ) = ( cos ( φ ) sin ( φ ) sin ( φ ) cos ( φ ) ) ( a 3 b 3 )
( a 5 b 5 ) = ( S 2 ( ϑ ) 0 0 S 1 ( ϑ ) ) ( a 4 b 4 )
( a 6 b 6 ) = ( cos ( ε ) sin ( ε ) sin ( ε ) cos ( ε ) ) ( a 5 b 5 )
( a 7 b 7 ) = ( 1 0 0 1 ) ( a 6 b 6 ) .
( a f b f ) = ( e ϑ ( 7 ) e x e φ ( 7 ) e x e ϑ ( 7 ) e y e φ ( 7 ) e y ) ( a 7 b 7 )

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