Aberrations are known to severely compromise image quality in optical microscopy, especially when high numerical aperture (NA) lenses are used in confocal fluorescence microscopy (CFM) and two-photon microscopy (TPM). The method of adaptive optics may correct aberrations and restore diffraction limited operation. So far the problem of aberrations that occur in the imaging of biological specimens has not been quantified. However, this information is essential for the design of adaptive optics systems. We have therefore built an interferometer incorporating high NA objective lenses to measure the aberrations introduced by biological specimens. The measured wavefronts were decomposed into their Zernike mode content in order both to classify and quantify the aberrations. We calculated the potential benefit of correcting different numbers of Zernike modes using different NAs in an adaptive CFM by comparing the signal levels before and after correction. The results indicate that adaptive correction of low order Zernike modes can provide significant benefit for many specimens. The results also show that quantitative fluorescence microscopy may be strongly affected by specimen induced aberrations in non-adaptive systems.
©2004 Optical Society of America
Optical microscopes, including CFM and TPM, have become standard instruments in many life science laboratories. Modern microscopes are highly optimised and are designed to deliver a performance close to the physical diffraction limit. However, there is one element in the optical path that has a strong influence on the image quality but is hard to optimise: the specimen itself. A biological specimen may have, apart from absorption properties, a variation in refractive index in the range from 1.30 to 1.50 [1, 2] and in some cases up to 1.70 . If one intends to image details within a cell, the distribution of the refractive index in the obstructing layers above the focal region introduces aberrations. Aberrations have been subject to many investigations due to their detrimental effect on imaging quality [4–9]. In CFM [10, 11] and TPM  aberrations lead to reduced resolution and lower signal levels. It is clearly desirable to compensate for these effects in order to restore imaging quality and reduce phototoxicity arising from the use of unnecessarily high illumination power.
In this paper we describe the measurement and quantification of specimen induced aberrations from a number of biological specimens. Furthermore, using the examples of a CFM and TPM, we investigate the viability of low order aberration correction in high NA microscopy and the expected improvement in signal levels. In Table 1 the specimen types investigated, their approximate thickness, embedding medium and the corresponding scan ranges are listed.
2. Adaptive optics for aberration correction
Aberrations can be removed by adaptive optics, a concept which is widely used in astronomy  where aberrations are introduced by refractive index variations in the atmosphere. In microscopy one attempts to image through an obstructing volume of inhomogeneous refractive index and possibly layers of mismatched refractive index. The idea of adaptive optics is to introduce conjugate aberrations that cancel out the aberrations arising within the system, thus recovering optimum, diffraction limited imaging. An adaptive optics system consists of a wavefront sensor (e.g. a Shack Hartmann sensor , an interferometer  or a modal wavefront sensor ) and a wavefront corrector (such as a deformable mirror [16–19] or a spatial light modulator (SLM)) that are combined in closed control loop. The principle of aberration correction has been demonstrated within a two photon microscope [19–22]. A more efficient implementation was incorporated into a CFM , where a single deformable mirror was used for both wavefront sensing and correction employing a modal wavefront sensing technique. Related systems have been implemented in ophthalmology using a deformable mirror in combination with a Shack-Hartmann sensor for imaging of the retina [18, 19]. Several theoretical investigations [5–9] were primarily concerned with spherical aberration arising from refractive index mismatch. However, biological specimens can have complicated variations in refractive index. The average refractive index of tissue or the refractive index of particular regions of biological specimens has been investigated [1–3] but relatively little information is available about the form and magnitude of aberrations introduced by complex biological specimens under high NA conditions. However, this information is essential for a systematic and directed approach to the design of an adaptive optical microscope.
3. Wavefront characterisation
Aberrations can be modelled by a complex pupil function
The function ψ(r,θ) of the wavefront could then be recovered from ϕ(r,θ), for example, using a phase unwrapping technique . A convenient way to describe the aberrations is a series of Zernike polynomials, so that ψ(r,θ)=∑ iMiZi (r,θ). The Zernike polynomials Zi (r,θ) are a set of orthogonal functions defined over the unit circle and the lower orders correspond to the classical aberration terms such as spherical aberration, astigmatism and coma [29, 30]. The Zernike mode amplitudes, Mi , can be extracted from the unwrapped phase as
Throughout this paper we use the indexing and normalization scheme of the Zernike polynomials as in  where all polynomials for i>1 have a standard deviation of one radian over the unit circle. We can separate the Zernike modes into two classes: the first set consists of the Zernike modes tip (i=2), tilt (i=3) and defocus (i=4). In a CFM they simply produce a displacement of the focal spot from its nominal position within the specimen. However, the image of the focal spot always falls on the pinhole detector since a conjugate displacement is introduced when the spot is re-imaged via the aberrating medium. In a sense, these modes are self correcting in a CFM but the displacement between the actual and nominal focal position in the specimen causes geometric distortion in the acquired three-dimensional image which can influence the accuracy of spatial measurements. This issue will be addressed in a future publication. The second class of modes (i>4) changes the shape of the intensity distribution at the focal spot and influences signal level and resolution.
The quality of an optical system is often characterised by the Strehl ratio , which is defined as the ratio of the maxima of the focal intensity distributions for the aberrated and the unaberrated wavefront. We define the Strehl ratio as
This definition is slightly different from that in  and takes into account the amplitude variations. For moderate aberrations and especially if the wavefront contains only small amounts of the Zernike modes tip, tilt or defocus, equation (4) is equivalent to the ratio of the maximum focal intensity to the diffraction limited maximum intensity. Also, for small aberrations and no amplitude variations the Strehl ratio may be estimated from the variance of the wavefront  and finally the Zernike mode coefficients Mi as:
This is valid for Var(ψ(r,θ))≪1.
The Strehl ratio can be related to the effective point spread function (PSF) in CFM and TPM. If we describe the focal distribution by the intensity PSF h (x,y, z) then
if h(x,y, z) is normalized to the diffraction limited maximum intensity. In a CFM system with a vanishingly small pinhole, the effective PSF is the product of the illumination PSF, hil , and emission PSF, hem . These would normally differ in scale because of the difference between the excitation and emission wavelengths. If we make the approximation that the fluorescence has the same wavelength as the illumination, then hem =hil and the image of a point-like fluorescent object is given by
It is clear that the maximum signal is equivalent to S 2. In a TPM system that does not use a detection pinhole, the imaging depends only on the square of the illumination PSF  and the image of the fluorescent point object is described by
Again, the maximum signal is equivalent to S 2. Hence we can define the signal improvement factor, Fsig , for a CFM or TPM system to be:
Here Sini denotes the initial Strehl ratio of the uncorrected wavefront and Scorr the Strehl ratio of the corrected wavefront. The value of Fsig represents the improvement in signal when aberration correction is applied to the excitation path of a TPM or the excitation and emission paths of a CFM using an infinitely small pinhole. This model of the imaging process allows us to estimate the potential benefit of adaptive optics in a straightforward manner. For small but finite sized pinholes in CFM, Fsig is expected to be similar. We note that the Zernike coefficients have units of phase and are therefore dependent upon wavelength. If no dispersion is present, the measured Zernike coefficients scale inversely with the wavelength. In this case it would be a simple matter to recalculate the factor of improvement for other wavelengths. The results shown assume an excitation wavelength of 633nm.
4. Experimental setup
We built a Mach-Zehnder phase stepping interferometer (Figure 1) to measure the aberrations induced by various specimens. The expanded He-Ne (633 nm) laser beam was split into reference and object paths. The specimen was placed in the object path between two opposing water immersion lenses (Carl Zeiss C-Apochromat, 63x Korr, NA=1.2) that were equipped with cover glass correction collars. A rotation of the λ/2-plate in front of the polarising beam-splitter permitted the adjustment of the relative intensities of the two paths. When the object beam traversed the specimen it was made to interfere with the reference beam and the resulting interference pattern was recorded on a CCD camera. The image plane of the CCD was conjugate to the pupil plane of the second objective lens. The phase stepping unit placed in the reference beam path took the form of a flat mirror mounted on a calibrated piezo drive. This allowed us to change the relative phase between reference and object beams in well defined steps. The phase stepping was synchronised with the CCD-camera and digital interferogram images were recorded using a framegrabber. The specimen was attached to a piezo driven stage that could be positioned in three dimensions. Furthermore, for specimen navigation, the object path could be switched to operate as a conventional transmission microscope using an additional CCD camera and illumination elements (not shown in the Figure).
Because of the short working distance of the high NA lenses the specimen had to be mounted between two coverslips. The coverslip correction collars of the lenses were set to minimize the first order spherical aberration coefficient M 11 when focussing through the coverslip and mounting medium adjacent to the specimen. Ideally, one should measure the absolute change in phase introduced by the specimen. In practice, small residual aberrations might be introduced by the optical system and were removed by the following calibration step. For each specimen, calibration measurements were taken by focussing adjacent to the specimen. All subsequent wavefronts were measured relative to this calibration reference. It is possible that some specimen induced spherical aberration was removed by this step (or introduced by the correction ring collars). However, this still permitted the measurement of the variation of the spherical aberration across the specimen and the introduction of spherical aberration by a refractive index mismatch is well known [4, 7, 9]. Since some static, uncompensated spherical aberration may be present the real aberrations might in fact be worse than that measured here.
The CFM is usually implemented in an epi configuration, where aberrations are introduced in the excitation and emission optical paths between the lens and the focal spot, whereas our interferometer measured aberrations in transmission for a single pass through the specimen. Therefore, one has to be aware that the aberrations measured are strictly equivalent to the epi case only when the focus is set to the bottom of the specimen. Then no further aberrations are introduced beyond the focal spot. This limitation is not severe since the strongest aberrations may occur when the whole specimen is traversed and we are interested in this situation to estimate the maximum correction range an adaptive optics system needs to provide. For all of our measurements the focus was set to the bottom of the specimen.
5. Data acquisition
One data set consists of 256 wavefront recordings corresponding to focal positions arranged on a 16×16 square grid across the specimen. For each wavefront a set of three interferograms at different relative phase steps were recorded and the amplitude A(r,θ) and the wrapped phase ϕ(r,θ) of the wavefront were calculated . In Figure 2 the transmitted light image and the wavefront disturbance (video) are shown as the focal spot was scanned across specimen number 5 (C. elegans). In the next step, the Zernike mode components were extracted from each of the wavefronts using a procedure of phase unwrapping that gives the Zernike coefficients directly. It is based on modal wavefront sensing described in [15, 23, 32]. First a positive bias of +biZi of the particular Zernike mode Zi one intends to measure is added to the original wavefront ϕ. Then the Strehl ratio S(ϕ+biZi ) is calculated. Now a negative bias of the same magnitude -biZi is added to the original wavefront and S(ϕ-biZi ) is computed. One can show  that the amplitude of the Zernike mode Zi contained in the original wavefront is proportional to the quantity S(ϕ+biZi )-S(ϕ-biZi ). This procedure is repeated until the modal amplitude of all Zernike modes up to i=22 are measured. It is a simple matter now to remove an amount of Zernike mode Zi from the original wavefront. The procedure is iterated until no further improvement in Strehl ratio is detected. This simulates the operation of modal wavefront sensing and correction. An advantage of this method is that the Zernike data is extracted directly from the wrapped phase data ϕ(r,θ) that was measured and so we do not require an intermediate phase unwrapping process. Furthermore we found that this method performed better than other phase unwrapping methods for the analysis of this data.
Figure 3 displays examples of measured wavefronts (top) and their simulated correction (bottom) up to Zernike mode 22. Inset (1) shows an interferogram from rat brain tissue, 30 µm thick, covering half of the pupil. Two effects are visible: a rather strong spherical aberration component (note the coloured rings at the edge of the top half of the pupil) is introduced and a modulation of the phase function with high spatial frequencies is visible. The Zernike based approach can correct for the lower order terms but cannot correct the high spatial frequency components; a partial correction is achieved and can be considered optimal for the modes included in the correction. This method worked well for 30µm thick brain slices. However, for thicker brain specimens of about 90 µm (data not shown) the aberration appeared to be dominated by high spatial frequency variations that could not be removed by a small number of modes. In (2) some of the structure of the mouse oocyte is visible in the pupil. There is sudden change of phase accompanied by amplitude effects at the cell boundaries. Again partial correction was achieved which can be considered optimal within the modes included. The mouse liver wavefront shown in (3) exhibits strong but rather smooth disturbance and good correction was achieved. In (4) the sample of striated muscle is shown. A rather smooth disturbance is present that can be well corrected with a low order Zernike modal approach. In this case a large amount of spherical aberration, probably arising from the refractive index mismatch, was removed in the calibration step. Image (5) shows a heavily aberrated wavefront from the C. elegans specimen. The smooth muscle tissue wavefronts are depicted in (6); the phase disturbance contained mainly low order Zernike terms and the almost uniform color (lower part) indicates very good phase correction.
After the Zernike mode extraction, each coefficient Mi may be represented as Zernike mode pseudo image that shows the variation of Mi as one scans across the specimen. Figure 4 displays an example for the C. elegans specimen. Here modes 5 to 12 are depicted. The total Zernike figure is calculated using ( )1/2.
For the purpose of designing and implementing adaptive optics it is important to consider the variations of the aberrations across the field of view. The mean and standard deviation for each mode was calculated from maps such as shown in Figure 4; a result for the C. elegans is displayed in Figure 5. The Zernike mode standard deviation declines with rising order. This general behavior was found for all the specimens. The aberration effect that is contributed from the individual coefficients is proportional to the square of the Zernike mode amplitude, as can be inferred from equation (5). Thus the influence of the higher order modes could be considered to decline even faster than the amplitudes shown in Figure 5. The magnitude of the Zernike mode amplitudes of all specimens were within the range of up to 1.5 for the modes 5 to 11. Maximum magnitudes smaller than 1 Zernike unit were observed for modes from 12 to 22, and for modes above 22 all magnitudes were smaller than 0.5.
In addition we are also interested to assess the potential benefit of aberration correction. We simulated Zernike mode based correction and calculated the Strehl ratios of the wavefronts before (Sini ) and after the correction (Scorr ). Modes 2 to 4 do not affect the signal in an epi-CFM system. Therefore these modes were subtracted from the raw wavefront prior to the Strehl ratio calculations. In order to exclude the effect of static aberration that may be caused by imperfections of the optical system itself, the Zernike mode components for modes i≥5 obtained from the calibration wavefront recorded next to the sample were also subtracted. The Strehl ratio Sini was calculated for the resulting wavefront from equation (4). A second Strehl ratio Scorr was calculated for a wavefront where the Zernike modes obtained in the analysis were subtracted from the measured raw wavefront. This simulated the correction of the wavefront for a certain number of modes.
The results for the mean initial Strehl ratio, the mean corrected Strehl ratio, and the mean and median of the confocal Fsig are summarised in Table 2 where the specimen numbers used are defined in Table 1. In order to investigate the interplay between the correction quality and the number of corrected Zernike modes we calculated simulations up to modes 12, 18, 22 and 37 for all the specimens. In general, the correction of up to 37 modes was found to give only moderate improvement in the Strehl ratio compared to the correction of 22 modes. It appears that for these specimens the correction of 22 or even 18 Zernike modes is a good compromise between the effort required for the correction and the improvement in Strehl ratio. Maps of the Strehl parameters are shown Figure 6, the specimens are again denoted by the numbers listed in Table 1. Significant spatial variations of the initial Strehl ratio are evident in all specimens. It is important to note this phenomenon. The measured intensity displayed in a standard CFM image is normally regarded as a quantity that represents fluorophore concentration. Our results indicates that aberrations and not just the fluorophore concentration affect the measured intensity. This has important implications for quantitative fluorescence microscopy.
We note that in some cases the specimen does not fill the whole field of view and Fsig is smaller in the areas adjacent to the specimen. The calculated mean Fsig was typically in the range between 2 and 10 for a correction of the Zernike modes 5 to 22. The median value is listed as this tends to be less sensitive to extreme values.
Another aspect investigated was the influence of the NA on the size of the aberrations and the benefit of modal adaptive correction. Wavefront data for different numerical apertures can be extracted from the interferograms by analysing an aperture subregion of smaller radius. Our data sets were recorded with a physical NA of 1.2 and data analysis was performed for the numerical apertures 1.2, 0.9 and 0.6 for a correction up to Zernike mode 37; the results are listed in Table 3 (see Table 1 for specimen numbers). The magnitude of aberrations, especially higher orders, increases with NA. As an example the phase function corresponding to spherical aberration [7–9] rises sharply towards the edge of a high NA pupil. Since a lower NA objective accepts only the central portion, the effects of spherical aberration are correspondingly reduced.
The dependence of other, higher order aberration modes is conceptually similar. Therefore, in a low NA system, the aberrations tend to be smaller in amplitude and the initial Strehl ratio is correspondingly higher. This is supported by the experimental results. As a consequence the benefit of correction for lower NA was found to be smaller. For the particular case of the mouse oocyte (specimen 2), the mean of Fsig for NA=0.6 is largest but this is because of a few locations with very large values of Fsig . However, the median value of the distribution for NA=0.6 is lower than that for the higher numerical apertures.
We can conclude: specimen induced aberrations lead to reduced signal levels and a deterioration in image quality in optical microscopy, especially in CFM and TPM. For the first time, specimen induced aberrations that occur with various biological specimens have been classified and quantified for the most relevant condition of high NA. The above approach can provide detailed information about the variation of each Zernike coefficient across the scan. This could also deliver information about the temporal dynamics of the aberrations due to the scanning of the focal spot through the specimen. Our calculation of the correction benefit is based on the assumption that a correction would be applied at every position within the scanned area. The feasibility of this assumption depends on the bandwidth of the wavefront sensing and correction devices and the scan speed. In some cases it may be required to either reduce the scan speed or update the aberration correction every few pixels only.
In a practical AO system that employs a deformable mirror other sets of orthogonal modes, for example the mirror deformation eigenmodes , may be more appropriate as a basis for correction. It is generally straightforward to convert from one representation to another.
The measurements indicate significant variations of the uncorrected Strehl ratio throughout the specimen, which could influence quantitative fluorescence measurements in an uncorrected system. We showed that low order aberration correction based on Zernike modes will provide significant recovery of signal levels in CFM and TPM, even if the diffraction limit is not restored. For the six specimens examined, the mean Fsig was in the range between 2 and 10 for a correction of the Zernike modes 5 to 22 at an NA of 1.2. Note that the quoted values refer to frame averages and the factors in specific areas might be even higher. It should be pointed out that the set of biological specimens investigated is necessarily incomplete and differences between biological specimens can be large. However, the benefit found for these specimen would be highly significant in CFM and TPM since light budgets are typically tight and efficient use of available photons is crucial to minimise photobleaching and phototoxic effects .
As expected from theory, lower NA systems are less susceptible to aberrations than high NA system under otherwise similar conditions. Low order correction would still provide benefits, even though the initial aberrations are smaller. The results presented here quantify the benefit of adaptive optics for biological microscopy and provide the bounds within which these systems must operate.
We would like to thank the following for providing specimens: Richard Gardner and Tim Davis, Dept. of Zoology, Oxford; Allison Woollard, Dept. of Biochemistry, Oxford; Tom Cunnane, Keith Brain and Patricia Tynan, Dept. of Pharmacology, Oxford. This work was supported by Carl Zeiss Jena GmbH and Research Councils UK under the Basic Technology scheme. M.S. is supported by the DAAD Germany (German Academic Exchange Service). M.J.B. is a Royal Academy of Engineering/EPSRC Research Fellow.
References and links
1. F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference “Refractive index of some mammalian tissues using a fiber optic cladding method.” Appl. Opt. 28, 2279–2303 (1989). [CrossRef]
2. G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto “Determination of the refractive index of highly scattering human tissue by optical coherence tomography” Opt. Lett. 20, 2258–2260 (1995). [CrossRef] [PubMed]
3. A. Dunn. “Light scattering properties of cells”. Dissertation, University of Texas, Austin, http://www.nmr.mgh.harvard.edu/~adunn/papers/dissertation/node7.html, 1998.
4. S. Hell, G. Reiner, C. Cremer, and E.H.K. Stelzer. “Aberrations in confocal fluorescence microscopy induced by mismatches” in refractive index. J. Microscopy 169, 391–405 (1993).
5. T. Wilson and A.R. Carlini. “The effects of aberrations on the axial response of confocal imaging” systems. J. Microscopy 154, 243–256 (1989). [CrossRef]
6. M.J. Booth, M.A.A. Neil, and T. Wilson. “Aberration correction for confocal imaging in refractive-index-mismatched media.” J. Microscopy 192, 90–98 (1998). [CrossRef]
7. M.J. Booth and T. Wilson. “Strategies for the compensation of specimen induced aberration in confocal microscopy of skin.” J. Microscopy 200, 68–74 (2000). [CrossRef]
8. P. Török, P. Varga, and G. Németh. “Analytic solution of the diffraction integrals and interpretation of wave-front distortion when light is focussed through a planar interface between materials of mismatched refractive indices.” J. Opt. Soc. Am. A 12, 2660–2671 (1995). [CrossRef]
9. P. Török, P. Varga, Z. Laczik, and G. R. Booker. “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation.” J. Opt. Soc. Am. A 12, 325 (1995). [CrossRef]
10. J.B. Pawley. “Handbook of Biological Confocal Microscopy”. Plenum, New York, 1995.
11. T. Wilson and C.J.R. Sheppard. “Theory and practice of Scanning Optical Microscopy”. Academic Press, London, 1984.
13. J. W. Hardy. “Adaptive Optics for Astronomical Telescopes”. Oxford University Press, 1998.
14. B.C. Platt and R. Shack. “History and priciples of Shack-Hartmann wavefront sensing.” J. Refractive Surgery 17, 573–577 (2001).
15. M. A. A. Neil, M. J. Booth, and T. Wilson. “New modal wavefront sensor: a theoretical analysis.” J. Opt. Soc. Am. 17, 1098–1107 (2000). [CrossRef]
16. L. Zhu, P. Sun, D.U. Bartsch, W.R. Freeman, and Y. Fainman. “Adaptive control of a micromachined continuous-membrane deformable mirror for aberration compensation.” Appl. Opt. 38, 168–170 (1999). [CrossRef]
17. G. Vdovin, P.M. Sarro, and S. Middelhoek. “Technology and applications of micromachined adaptive mirrors.” J. Micromech. and Microeng. 9, R8 (1999). [CrossRef]
18. N. Doble, G. Yoon, L. Chen, P. Bierden, B. Singer, S. Olivier, and D. R. Williams. “Use of a microelectromechanical mirror for adaptive optics in the human eye.” Opt. Lett. 27, 1537–1539 (2002). [CrossRef]
19. “A. Roorda, F. Romero-Borja,W.J. Donnelly III, H. Queener, Thomas J. Hebert and M.C.W. Campbell. Adaptive optics scanning laser ophthalmoscopy.” Opt. Express10, 405–411 (2002). [PubMed]
20. M. A. A. Neil, R. Juškaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata. “Adaptive aberration correction in a two-photon microscope.” J. Microscopy 200, 105–108 (2000). [CrossRef]
21. 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. Microscopy 206, 65–71 (2002). [CrossRef]
23. M.J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson. “Adaptive aberration correction in a confocal microscope.” Proceedings of the National Academy of Science 99, 5788–5792 (2002). [CrossRef]
24. J.H. Bruning, D.R Heriott, J.E. Gallhager, D.P. Rosenfeld, A.D. White, and D.J Brangaccio. “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses.” Appl. Opt. 13, 2693–2703 (1974). [CrossRef] [PubMed]
25. J. E. Greivenkamp and J. H. Bruning. “Phase shifting interferometry.” Optical shop testing, ed. byD. Malacara, Wiley New York, 1992.
26. M. Schwertner, M. J. Booth, M.A.A. Neil, and T. Wilson. “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry.” J. Microscopy 213, 11–19 (2004). [CrossRef]
27. M. Schwertner, M.J. Booth, and T. Wilson. “Simulation of specimen induced aberrations for objects with spherical and cylindrical symmetry.” J. Microscopy 215, 271–280 (2004). [CrossRef]
28. D. C. Ghiglia and M. D. Pritt. “Two Dimensional Phase Unwrapping. Theory, Algorithms and Software.” Wiley, 1998.
29. M. Born and E. Wolf. “Principles of Optics”. Pergamon Press, 6th edition, 1983.
30. R.J. Noll. “Zernike polynomials and atmospheric turbulence.” J. Opt. Soc. Am. 66, 207–277 (1976). [CrossRef]
31. M. J. Booth and T. Wilson. “Refractive-index-mismatch induced aberrations in single-photon and two-photon microscopy and the use of aberration correction.” J. Biomed. Opt. 6, 266–272 (2001). [CrossRef] [PubMed]
32. M.J. Booth. “Direct measurement of Zernike aberration modes with a modal wave front sensor.” Proc. SPIE, Advanced Wavefront Control: Methods, Devices, and Applications 5162, 79–90, 2003.