direct wavefront sensing in adaptive optical microscopy ... · direct wavefront sensing in adaptive...

Direct wavefront sensing in adaptive opticalmicroscopy using backscattered light

Saad A. Rahman1 and Martin J. Booth1,2,*1Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK

2Centre for Neural Circuits and Behavior, University of Oxford, Mansfield Road, Oxford, OX1 3SR, UK

*Corresponding author: [email protected]

Received 3 May 2013; revised 3 July 2013; accepted 5 July 2013;posted 10 July 2013 (Doc. ID 189946); published 30 July 2013

Adaptive optics has been used to compensate the detrimental effects of aberrations in a range ofhigh-resolution microscopes. We investigate how backscattered laser illumination can be used as thesource for direct wavefront sensing using a pinhole-filtered Shack–Hartmann wavefront sensor. It isfound that the sensor produces linear response to input aberrations for a given specimen. The gradientof this response is dependent upon experimental configuration and specimen structure. Cross sensitivitybetween modes is also observed. The double pass nature of the microscope system leads in general tolower sensitivity to odd-symmetry aberration modes. The results show that there is potential for use ofthis type of wavefront sensing in microscopes. © 2013 Optical Society of AmericaOCIS codes: (010.7350) Wave-front sensing; (180.1790) Confocal microscopy; (180.5810) Scanning

microscopy; (220.1080) Active or adaptive optics.http://dx.doi.org/10.1364/AO.52.005523

1. Introduction

Optical microscopes are widely used in the biologicalsciences to image cellular structures and function.Laser scanning methods, such as confocal ormultiphoton microscopes, are particularly useful asthey provide 3D images of thick specimens. Theeffectiveness of such microscopes is compromisedby specimen-induced aberrations, so adaptive opticshas been employed to compensate for these aberra-tions through the introduction of a conjugate aberra-tion using an adaptive element, such as a deformablemirror (DM) [1].

Aberrations can be measured either using directwavefront sensing or indirect sensorless optimiza-tion methods. Sensorless optimization uses imageinformation to determine the specimen-inducedaberration from a sequence of measurements. Thishas successfully been demonstrated in several

adaptive microscope applications [2,3]. Recent workhas shown that direct wavefront sensing can also beemployed. For this to be effective, a method is neededto ensure that only light from the focal region con-tributes to the aberration measurement. Severalimplementations have used fluorescence emissionfor aberration measurement. In two-photon excita-tion fluorescence microscopes, fluorescence is onlygenerated in the focal spot, which can be used as thesource for a wavefront sensor [4,5]. Another approachinvolves the placement of small fluorescent beadsin the specimen that can act as point sources forwavefronts that are detected by a Shack–Hartmannwavefront sensor (SHWFS) [6,7].

For weakly fluorescent or sensitive specimens, itwould be desirable to use backscattered illuminationlight, whose intensity is often orders of magnitudegreater than the resulting fluorescence. However,aberrations encountered in backscattered light candepend upon the specimen structure and the result-ing measurements can be ambiguous. For example,reflection by a planar mirror in the focus of the

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microscope would result in a doubling of even-symmetry aberrations, such as astigmatism orspherical aberration, whereas odd-symmetry aberra-tions, such as coma, are cancelled out [8,9]. Scatter-ing from isolated point objects would, however, resultin the correct detection of aberrations. Intermediate-sized objects with dimensions similar to the wave-length would exhibit more complex scatteringcharacteristics.

One method has used backscattered illuminationlight and coherence gating to ensure that thewavefront sensor only responded to light scatteredfrom the focal region [10]. This approach is onlypossible with low-coherence illumination and re-quires a complicated interferometric arrangement.Alternatively, a pinhole can be used to exclude out-of-focus light, in a similar way to the pinhole usedin a confocal microscope. The pinhole has the addi-tional advantage of smoothing the light field thatreaches the wavefront sensor pupil. The size of thepinhole must be carefully chosen to provide a com-promise between spatial selectivity and the degree ofsmoothing. Such an arrangement has been used forfluorescence-based wavefront sensing in a confocalmicroscope, but not using backscattered illuminationlight [11,12].

In this paper, we expand upon previous prelimi-nary work [13] and present experimental data illus-trating the effectiveness of direct sensing using apinhole-filtered SHWFS for mirror and scatteringspecimens. The effects of different pinhole sizes onthe measured aberrations are determined, as arethe effects of different scattering structures. We alsoinvestigate the differing sensitivity of the SHWFS toeven- and odd-symmetry aberration modes and mea-surements. This effect, arising from the dual passnature of the optical system, has been investigatedpreviously for ophthalmic applications, but this isthe first detailed investigation of these phenomenain high numerical aperture microscopy [8,9].

2. Experimental System

Figure 1 shows a simplified schematic of the wave-front sensing system used for the experimental mea-surements. Light of wavelength 633 nm from ahelium neon laser was expanded by a combinationof lenses to form a beam that illuminated the mem-brane DM (Flexible Optical, Netherlands). The de-fault position of the DM was set at approximatelyhalf of its maximum deflection. The illuminatingbeam was set to be divergent, hence compensatingfor the DM curvature arising from the default shapeof the mirror. The membrane mirror, which allowscorrection of low-order optical aberrations, has a15 mm diameter, Al-coated circular mirror mem-brane, and its 37 electrodes are arranged in a hex-agonal structure within a 12 mm diameter circle.Experimentally, it was found that optimum repro-duction of Zernike modes was achieved when usingaround 10 mm diameter active aperture on the DM.Light reflected by the DM was passed through a 4f

system and focused by a water immersion objectivelens (Zeiss, 63×, 1.2 NA) to probe a specimen thatwas placed on a piezo scanning stage, which allowedraster scanning of the specimen. Light backscatteredfrom the specimen was collected by the same objec-tive lens, then was incident on the DM once more inthe detection path. The light reflected by the DMwascoupled off by a beam splitter and passed through a4f lens system. First, this 4f system served to de-magnify the beam so that it can be picked up com-pletely by the CCD camera on the SHWFS (Thorlabs,WFS150, with custom software). Second, it wasused to collimate the convergent beam reflected bythe DM. Third, the lens system was used to focus thelight through a pinhole, which performed the essen-tial tasks of filtering out light from out-of-focusregions and smoothing of the light distributionincident on the SHWFS.

A range of pinholes with different diameters wasused. The pinhole excluded out-of-focus light in asimilar manner to the pinhole used in a confocalmicroscope. The effects of the pinhole size on theconfocal microscope have been well studied [14,15].Optimal axial sectioning is obtained for a vanish-ingly small pinhole, although pinhole diameters upto around 70% of the first zero of the Airy patternprovide essentially equivalent sectioning with theadditional benefit of increased signal. For larger pin-hole sizes, the sectioning depth increases in an ap-proximately linear manner. For the pinhole filteredSHWFS, an increase in pinhole size similarly in-creases the axial depth from which scattered lightcontributes to the sensor measurement.

In an adaptive microscope, the specimen’s refrac-tive index structure would induce similar aberra-tions in both the illumination and detection paths.In order to simulate specimen-induced aberrations,the DM was used to introduce individual aberrationmodes simultaneously into both paths. This configu-ration models very closely the effect of specimens,with the restriction that only aberrations with the

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Fig. 1. Schematic of the microscope wavefront sensing system.Several optical elements have been omitted for clarity. Arrowsshow the light propagation direction in the illumination anddetection paths.

5524 APPLIED OPTICS / Vol. 52, No. 22 / 1 August 2013

capabilities of the DM are included. The controlsignals required to generate Zernike polynomialmodes were obtained through calibration of the sys-tem using the SHWFS. Linearized deflection of themirror surface was achieved by using actuator drivevoltages that were proportional to the square root ofthe control signal [16]. The amplitude of the Zernikemodes was swept from −0.6 to 0.6 units, in steps of0.2 units, where one unit corresponds to a root meansquare phase of 1 rad. Higher amplitudes caused theDM electrode voltage to saturate for some modes. Asthe backscattered light incident on the SHWFS hadcomplex structure like a speckle pattern, scanningand averaging was necessary in order to obtain reli-able measurements. For each applied amplitude ofthe Zernike modes, averaging was achieved, first,via long exposure of the specimen while it was beingscanned. Second, several of the resulting pupil im-ages were recorded by the SHWFS camera. TheZernike coefficients were estimated from the aver-aged pupil image by calculating the centroid of eachHartmann spot and using the least-squares methodof Southwell [17]. The exposure time and number ofimages averaged were selected so as to achieve a uni-form pupil intensity at the SHWFS, which in turnproduced steady coefficient measurements. In orderto illustrate the variability between measurements,several sets of the coefficients were recorded in thisway. The multiple readings were used to calculatethe mean and standard deviation of the data so asto provide an indication of the repeatability of wave-front sensing for a given scenario.

The effects of scattering in the specimen and spa-tial filtering by the detector pinhole create darkregions in the pupil that correspond to missing spotsin the Hartmann spot pattern. The nonuniform in-tensity also gives rise to brighter regions in thepupil that may cause saturation of the CCD sensor,which can lead to errors in the calculation of the spotcentroid. These missing or corrupted centroid mea-surements inevitably introduce errors into the wave-front measurements. The control software wasprogrammed to use the reference (i.e., undeflected)spot position where a spot was missing or saturated,yielding a zero local phase gradient.

3. Measurements for Different Specimen Structures

The system described in the previous section wasused to investigate wavefront sensing using back-scattered light in different specimens. In this section,we present results illustrating the effectiveness ofaberration measurements using a range of speci-mens, ranging from a specular mirror-like specimenand scattering structures consisting of beads andnatural specimens.

It can be easily shown that a specular specimen,such as a mirror, yields twice the induced even-symmetry aberrations and no odd-symmetry aberra-tions in a dual pass setup, such as the confocalmicroscope. To demonstrate this phenomenon, a mir-ror was placed on the stage of the dual pass wavefront

sensing system of Fig. 1. An aperture of 10 mmdiameter was illuminated on the DM. The reflectedbeam was imaged to fill exactly the back aperture ofthe objective lens. The light reflected back by themir-ror specimen and off the DM was demagnified to4.1 mm diameter to form the input to the SHWFS.As a result, a total of 540 lenslets were illuminatedand used by the SHWFS for aberration estimation.Figure 2 shows the measurements made by theSHWFSusing themirror specimenwith the introduc-tion of even symmetry aberration modes (astigma-tism and spherical) by the DM. A range of pinholeswith different diameters was used to provide axialselectivity and smoothing for the SHWFS. The diam-eter of the aberration-free Airy disc (referred to as theAiry unit) at the position of the SHWFS pinhole was30.9 μm.For the astigmatismmode, it canbe seen thatthe measurements made with the 200, 300, and600 μm pinholes were almost identical to the mea-surementsmadewithout a pinhole in place. Themea-sured aberration amplitude was approximately twicethat of the induced aberration, as predicted by theory.The asymmetry in these latter plots is due to residualaberrations. For the smaller pinhole sizes of 100and 50 μm, there was deviation from the ideal meas-urement. For the spherical aberration mode, themeasurements were more sensitive to effects of thepinhole, with the gradient of the response decreasingnoticeably as the pinhole diameterwas reduced below600 μm. The deviation of these gradients from theideal is due to the spatial filtering effect of the pinhole.The spatial filtering caused smoothing of the pupilphase and, for regions with large phase gradient, areduction in intensity. These dark pupil regions ledtomissingSHWFSspotswhich affected themeasuredaberration. A comparison of this effect for the Z5 andZ11 modes is shown in Figs. 3(a) and 3(b). The greatersensitivity ofZ11 to this phenomenon canbe explainedby consideration of the phase gradients of the differ-ent modes [Fig. 3(c)]. In a ray optics approximation,the gradient of thewavefront is proportional to thede-flection of the corresponding ray in the focal planefrom the optic axis. The larger gradients of the Z11mode for a given amplitude mean that more light isfiltered by the pinhole and more spots are lost fromthe SHWFS pattern.

Equivalent measurements were also taken for low-order, odd-symmetry modes, such as coma. In thiscase, the experimental measurements showed negli-gible detection of coma with the mirror specimen.This is entirely in agreement with theoretical mod-elling, which shows that the inversion of symmetryon reflection in the focus causes cancellation of oddaberration modes. A summary of the sensor responsegradients to each Zernike mode for the 100 and300 μm pinholes and with no pinhole is shown inFig. 4. This figure (and subsequent figures) showstwo quantities for each aberration mode. The firstquantity is the sensitivity, denoted jai;ij and shownas a black bar, which is the gradient of the sensor re-sponse to the mode i induced by the DM. The second


quantity is the modal crosstalk, denoted jfai≠kgj andshown as a gray bar, which is the root mean squaresum of the other detected aberration coefficientswhen mode i was induced by the DM. An ideal meas-urement would therefore correspond to jai;ij � 1 andjfai≠kgj � 0. The ratio jai;ij∕jfai≠kgj quantifies thepurity of the coefficient measurement.

To test the behavior of the dual pass wavefrontsensing system in a scattering sample, a model speci-men was created using polystyrene beads of 200 nmdiameter. A suitably diluted suspension of distilledwater and beads was mixed with powdered gelatinand then allowed to set between a glass slide anda glass coverslip, resulting in the beads being fixed

in a three-dimensional distribution. The resultingspecimen exhibited significant scattering from thedense arrangement of beads, as illustrated in theconfocal microscope image (Fig. 5). As a large reflec-tion was created by the interface between the glasscoverslip and the (water-based) gelatin, it was impor-tant to be able to focus several tens of micrometersinto the specimen to ensure that the light enteringthe SHWFS was scattered from the beads. TheSHWFS pinhole ensured that only light scatteredfrom near the focal region could contribute signifi-cantly to the aberration measurement.

Figure 6 shows the response of the SHWFS toeven-symmetry aberration modes (Z5 astigmatism

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Fig. 2. SHWFSmeasurements of the DM-induced even aberration modes, (a, b) Z5 (astigmatism) and (c, d) Z11 (spherical), for the mirrorspecimen. (a, c): Measured aberration amplitude as a function of induced aberration amplitude. (b, d): Maximum measured range of allZernike modes when�0.6 rad (rms) of Z5 or Z11 was induced, showing crosstalk betweenmodes. The inset in (b) shows the definition of theZernike modes.


and Z11 spherical) for the scattering bead specimen.Note that measurements for Z6 (astigmatism) werealso taken and were found to be similar to thoseobtained for the homologous mode Z5. In this case,pinholes of 200, 300, and 600 μmdiameter were used.It was observed that the responses to eachmode wereapproximately linear, although the gradient was con-siderably lower than one in each case. The gradientof the response was also dependent upon the pinholesize and differed between the astigmatism andspherical modes.

Figure 6 shows the response of the SHWFS to theodd aberration modes Z7 (coma) and Z9 (trefoil) forthe scattering bead specimen. Similar results werealso obtained for the modes Z8 and Z10, respectively.Again, an approximately linear relationship betweenthe induced and measured aberration was alsoobserved. In these cases, however, the gradient wasmuch smaller than that seen for the even modes.Dependence of the gradient on the pinhole size wasalso observed. A summary of the sensor response toeach Zernike mode for the scattering bead specimenis shown in Fig. 7. The large degree of crosstalk could

be caused by a combination of multiple scatteringand the nonpoint-like nature of the 200 nm beads,which would both affect the SHWFS measurements.

A sample of artificial collagen was immersed inwater and placed between a cover glass and micro-scope slide. This was chosen as the matrix of collagenfibrils scatter light in a different manner to beadsand so provide a further test of wavefront sensingusing light scattered from different specimens.Figure 8 shows a confocal reflection microscope im-age of the collagen sample, indicating strong scatter-ing. Measurements were taken in the same manneras for the bead specimen. Scanning was performedover a region of approximately 50 μm square. Aswas observed for the bead specimen, the sensorresponse to Zernike modes was predominantly linearin all cases, although the gradient was dependentupon the pinhole size and is different betweenmodes.The gradients observed for all modes 5 to 11 areshown in Fig. 9. As with the bead specimen, the mea-surements showed that the even aberration modes(i � 5, 6) were detected more efficiently than oddmodes (i � 7 to 10). It was also observed that thesensitivity was in general higher than the corre-sponding values using the bead specimen.

Further measurements were taken from a speci-men of 100 nm diameter gold beads suspended ingelatin. In this case, the specimenwas sparsely popu-lated with beads, such that the wavefront measure-ments were taken by scanning over a single bead.The sensitivity and crosstalk measured for modes5 to 11 are shown in Fig. 10. The sensor responsefor this specimen shows similar characteristics forboth the even and odd modes, which should be ex-pected for scattering from point-like objects. Thisis in contrast to the measurement from the polysty-rene beads and collagen, where the response to oddmodes was lower. However, the measured coefficientswere all lower than those for the induced aberration.The lower sensitivity to mode 11 (spherical) forthe larger pinhole is attributed to an anomalousmeasurement; we note that significant defocus wasdetected in this case, indicating that an axial dis-placement of the bead had led to filtering of lightilluminating the outer regions of the SHWFS.

4. Measurements for Different Optical Configurations

The light detected by the SHWFS can be backscat-tered by various specimen structures, which may

Fig. 5. (a) Confocal reflection microscope image of the 200 nmbead specimen. (b–d) SHWFS spot patterns using light backscat-tered from the sample 200 nm beads. A subregion of the pupil isshown. (b) Stationary specimen using a 200 μm pinhole, showingsignificant intensity variation across pupil. (c) Scanning and frameaveraging with no pinhole, showing near uniform intensity butwith out-of-focus light causing secondary spots within each suba-perture domain. (d) Scanning and averaging, using a 200 μmpinhole, showing greater uniformity and no secondary spots.

(a) (b) (c)

Fig. 4. Sensitivity to the induced Zernike mode of index i (black) and the total rms crosstalk sensitivity measured in other modes (gray)for Zernike modes 5 to 11 using the mirror specimen for (a) 100 μm diameter pinhole, (b) 300 μm pinhole, and (c) no pinhole.


encompass specular reflections from locally flatinterfaces, scattering by point-like features, and scat-ter from other intermediate structures. As indicated

by theory and backed up by the measurements forthe mirror and gold bead specimen (Figs. 4 and 10),the light scattered from smaller objects is likely to

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Fig. 6. SHWFS measurements of the even aberration modes Z5 (astigmatism) and Z11 (spherical) and the the odd aberration modes Z7

(coma) and Z9 (trefoil) for the specimen of 200 nm diameter beads suspended in gelatin. Left: Measured aberration amplitude as a functionof induced aberration amplitude. Right: Maximum measured amplitude of all modes, showing modal cross talk.

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Fig. 7. Sensitivity (black) and crosstalk (gray) for Zernike modes 5 to 11 using the specimen of 200 nm diameter beads in gelatin with(a) 200 μm diameter pinhole, (b) 300 μm pinhole, and (c) 600 μm pinhole.


provide more accurate measurements than thosefrom larger structures. The separation of scatteringfrom specular reflections should therefore improvethe purity of the coefficient measurements. We haveinvestigated two optical configurations to achievethis: asymmetric illumination and detection andpolarization filtering.

A. Asymmetric Illumination and Detection

The results presented in previous sections used illu-mination that filled the full aperture of the objectivelens. If the illumination is instead restricted to asmaller NA by underfilling the objective pupil, thenspecular components of the reflection would be con-fined to a limited region of the detection pupil. In con-trast, scattered light would be distributed across thewhole of the detection pupil. Aberration measure-ments using asymmetric illumination and detection,where the NA of the illumination beam ismuch lowerthan the NA of the detection path, might therefore bemore sensitive to the scattered light from small ob-jects. This asymmetric dual pass method has beenshown in ophthalmic research of the human eye toproduce better readings of both even and odd aberra-tion modes [18,19].

Figure 11 shows results comparing symmetric andasymmetric illumination and detection. The opticalsystem was modified so that the illumination NAwas reduced to approximately 0.096, while the detec-tion NA was maintained at 1.2. The response toodd aberration modes (i � 7 to 10) is higher for theasymmetric case, whereas the response to the evenmodes is slightly lower. This corresponds to the

expectation that the asymmetric case should yieldbetter sensitivity to odd modes, although the differ-ence in sensitivity is small.

It should be noted that the asymmetric illumina-tion configuration causes the spots at the center ofthe SH pattern to be saturated. The wavefrontreconstruction scheme replaced these saturatedmeasurements with a zero-phase gradient. Thisapproximation would have the effect of reducing themeasured coefficients compared to the symmetriccase, where there was no saturation.

B. Polarization Filtering

Light reflected by a specular specimen in the focalregion of a high NA objective lens will maintainthe same polarization state as the illumination. How-ever, when light is scattered by a small object in thefocus, a degree of depolarization is introduced [20].This means, for example, that x-polarized illumina-tion will result in the detection of some y-polarizedlight upon scattering. This phenomenon suggeststhat polarization filtering will aid the separation oflight reflected from larger objects from the light scat-tered by small objects, thus avoiding the complica-tions of the double-pass nature of the optical system.

The optical system described above was modifiedto include an analyzer in the detection path beforethe pinhole of the filtered SHWFS. The extinction ra-tio for light measured at the SHWFS was measuredas approximately 70 with a mirror specimen. Mea-surements were again taken using a specimen con-sisting of 200 nm beads in gelatin. Figures 11(e)and 11(f) show example spot patterns for this con-figuration. For a stationary specimen, the pupil seesa nonuniform amplitude due to speckle arising fromscatter from multiple beads. Following scanning andaveraging, the spot pattern forms two bands of moreuniform intensity, with a dark central band. This issimilar to the averaged intensity distribution foundthrough numerical modelling of the light scatteredfrom a single point object scanned across the focus(not shown). The resulting sensitivity and crosstalkmeasurements are shown in Figs. 11(g) and 11(h).

The polarization filtered results show greatervariation in sensitivity between modes, when com-pared to the nonfiltered measurements. This iscounter to expectations, but may arise from a combi-nation of factors. First, the amount of light availablefor sensing is much lower in the filtered case

Fig. 8. Confocal reflection microscope image of the artificialcollagen specimen.

(a) (b) (c)

Fig. 9. Sensitivity (black) and crosstalk (gray) for Zernike modes 5 to 11 using the artificial collagen specimen with (a) 200 μm diameterpinhole, (b) 300 μm pinhole, and (c) 600 μm pinhole.


(<1% according to theoretical modelling). Thismeans that the measurements are more susceptibleto systematic errors due to background from else-where in the system. Another limitation arises fromthe dark pupil region, as seen in Figs. 11(e) and 11(f).In this region, the phase gradient measurements forthe missing spots were replaced by a zero value. Thishas the effect of flattening the aberration in this re-gion, leading to lower coefficient measurements. Thisreduction is worse for modes whose orientation in thepupil means that high gradients are present in thisregion (e.g., this affects the coma mode i � 8 more

than coma mode i � 7, as these modes are rotatedby 90° with respect to each other).

5. Biological Specimens

The performance of the filtered SHWFS was also in-vestigated using biological specimens. We presenthere data from a C. elegans specimen mounted inan aqueous medium. Figure 12 shows a confocal re-flection microscope image of the C. elegans sample.Aberration measurements were performed usinglight scattered back from the specimen’s internalstructure and filtered by a 600 μm pinhole. Symmet-ric and asymmetric illumination and polarization-filtered configurations were used. The compiledmeasurements of sensitivity and crosstalk for modes5 to 11 are shown in Fig. 12. The response to the twocoma modes (i � 7, 8) was found to be much lesssensitive than the other low-order modes in all con-figurations. Neither asymmetric illumination norpolarization filtering was found to produce signifi-cant improvements in performance for this speci-men. Indeed, for some modes, the measurementquality was found to decrease.

6. Discussion and Conclusion

The results show that the characteristics of thepinhole-filtered SHWFS are strongly dependentupon the specimen structure. Both the sensitivity toa given mode and the degree of crosstalk between dif-ferent modes can vary. However, in many situations,the sensor does measure predominantly the inducedmode, albeit with uncertain sensitivity. While thisuncertainty precludes the use of this sensor in open-loop measurement, the results indicate that thismethod could be applied in a closed-loop correctionroutine. If the cross sensitivities between modes (de-noted aik for induced mode i and measured mode k)

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Fig. 10. Sensitivity (black) and crosstalk (gray) for Zernikemodes5 to 11 using the gold bead specimen, with (a) 300 μm and(b) 600 μm pinhole.

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Fig. 11. Asymmetric and polarization-filtered detection. SHWFS spot patterns using asymmetric illumination and detection with thespecimen of 200 nm diameter beads in gelatin: (a) stationary specimen, (b) scanned specimen with time averaging. The bright region in thecenter corresponds to the specular reflection component. Sensitivity (black) and crosstalk (gray) using a 300 μmdiameter pinhole: (c) usingsymmetric illumination/detection, (d) using asymmetric illumination/detection. SHWFS spot patterns using polarization filtering:(e) stationary specimen, (f) scanned specimen with time averaging. Sensitivity (black) and crosstalk (gray) for: (g) no polarization filtering,(h) polarization filtering.


are known, one can calculate the actual inducedaberration. A square sensitivity matrix S is con-structed from the elements aik, and the vectorscactual and cmeasured represent the actual induced mo-dal coefficients and the measured coefficients, re-spectively. Assuming S is invertible, we can calculate

cactual � S−1cmeasured: (1)

The degree to which uncertainty in the elements of Sor in the measurements of cmeasured can affect thecalculation of actual induced aberrations is indicatedby the condition number of S. The condition numberwas calculated as

κ � ‖S‖‖S−1‖; (2)

where ‖…‖ represents the 2-norm of the matrix. Con-dition numbers calculated from the earlier resultsare compiled in Table 1. It can be seen that higher

condition numbers (corresponding to ill-conditionedcoefficient calculations) are seen for the cases forwhich some modal sensitivities were lower than themodal crosstalk, as shown in the bar charts of theearlier figures. For example, the most extreme caseis for the polystyrene beads and the 200 μm pinhole,where κ � 2000. Inspection of the data in Figure 7shows that the crosstalk exceeded the sensitivityfor five of the seven modes used.

The condition number provides a concise indicatorof the likely reliability of measurements using differ-ent sensor and specimen combinations. One possiblepractical approach would be to estimate the matrix Sthrough a sequence of measurements using the speci-men region of interest. The condition number of thisestimate would indicate the usefulness of closed-loopcorrection based upon this matrix. Further inspec-tion of the elements of S could provide informationabout which subset of modes—those for which mea-surements are well conditioned—might be effectivelymeasured and corrected. Correction of the remainingmodes could be implemented using a complementarymethod, such as indirect optimization.

In conclusion, the results outlined in this paperillustrate the complexities of direct wavefront sens-ing using spatially filtered, backscattered light inadaptive scanning laser microscopes. Experimentalmeasurements of mirror-like and point-like speci-mens has shown clearly that ambiguities can arisewith this sensing method. The data show that abso-lute aberration measurements from complex scatter-ing specimens are also ambiguous. For each of theaberration modes tested, an approximately linearrelationship between the induced and detectedaberration amplitude was obtained. However, thegradient of this relationship was dependent upon ex-perimental conditions, such as pinhole diameter andscatterer density. It was also observed that the sen-sitivity to odd-symmetry aberration modes, such ascoma, was in general lower than the sensitivity toeven modes, such as astigmatism. The use of differ-ent illumination and detection apertures and polari-zation filtering provided improvements in somemeasurements, but significant benefit of these

Fig. 12. Sensitivity (black) and crosstalk (gray) for Zernike modes 5 to 11 using the C. Elegans specimen with a 600 μm pinhole. Key: s/a,symmetric/asymmetric illumination and detection; p, polarization filtering. Image: Confocal reflection microscope image of the region ofthe C. Elegans specimen used for the aberration measurements.

Table 1. Condition Numbers of Sensitivity Matrices Shown to TwoSignificant Figuresa

Specimen Pinhole (μm) Configuration Figure Condition No.

PS beads 200 — 7 2000PS beads 300 — 7 11PS beads 600 — 7 4.5Collagen 200 — 9 11Collagen 300 — 9 8.0Collagen 600 — 9 4.9Gold beads 300 — 10 4.4Gold beads 600 — 10 20PS beads 300 symm 11 14PS beads 300 asymm 11 3.4PS beads 300 no polar 11 7.7PS beads 300 polar 11 48C. elegans 600 symm 12 3.6C. elegans 600 asymm 12 15C. elegans 600 polar 12 6.6aKey: PS, polystyrene; symm/asymm, symmetric/asymmetric

illumination and detection; polar, polarization filtering. Thepolystyrene beads were 200 nm diameter. The gold beads were100 nm diameter. The figure column relates to the correspondingfigure numbers in this paper.


configurations was not observed. Despite the varia-tion of the sensitivity with specimen structure, thepresence of a proportional response to the inducedaberration for all modes means that this method ofwavefront sensing could be used to provide a suitablefeedback signal for a closed-loop correction system.

This research was supported by the Engineeringand Physical Sciences Research Council, UnitedKingdom, grant no. EP/E055818/1.

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