## Image based adaptive optics through optimisation of low spatial frequencies

Optics Express, Vol. 15, Issue 13, pp. 8176-8190 (2007)

http://dx.doi.org/10.1364/OE.15.008176

Acrobat PDF (1874 KB)

### Abstract

We present a wave front sensorless adaptive optics scheme for an incoherent imaging system. Aberration correction is performed through the optimisation of an image quality metric based upon the low spatial frequency content of the image. A sequence of images is acquired, each with a different aberration bias applied and the correction aberration is estimated from the information in this image sequence. It is shown, by representing aberrations as an expansion in Lukosz modes, that the effects of different modes can be separated. The optimisation of each mode becomes independent and can be performed as the maximisation of a quadratic function, requiring only three image measurements per mode. This efficient correction scheme is demonstrated experimentally in an incoherent transmission microscope. We show that the sensitivity to different aberration magnitudes can be tuned by changing the range of spatial frequencies used in the metric. We also explain how the optimisation scheme is related to other methods that use image sharpness metrics.

© 2007 Optical Society of America

## 2. Image formation in an incoherent imaging system

*I*(

**x**) is given by the convolution of the object function

*t*(

**x**) and the intensity point spread function (IPSF),

*h*(

**x**), of the system:

**x**is the position vector in the image plane; for clarity we have omitted magnification factors. The object is, of course, independent of any aberrations in the optical system and all aberration effects are therefore manifested in the IPSF. If instead we consider the imaging process in the frequency domain, the image Fourier transform (FT),

*J*(

**m**), is given by

**m**is the spatial frequency vector,

*H*(

**m**) is the optical transfer function (OTF), which is equivalent to the FT of

*h*(

**x**), and

*T*(

**m**) is the FT of

*t*(

**x**). In general, each of the terms in Eq. 2 is a complex quantity. In order to deal solely with real quantities, we can also describe the imaging process in terms of spectral density functions. Defining the object spectral density function as

*S*(

_{T}**m**) = |

*T*(

**m**)|

^{2}and the image spectral density as

*S*(

_{J}**m**) = |

*J*(

**m**)|

^{2}, then

*H*(

**m**)| is the modulation transfer function (MTF). In Eq. 3, all aberration effects are confined to the MTF.

## 3. Image spectral density for low spatial frequencies

*S*(

_{J}**m**) at low spatial frequencies. Expressions for heavily aberrated OTFs at low spatial frequencies have been derived using the geometrical OTF [14

14. W. Lukosz, “Der Einfluβ der Aberrationen auf die optische Übertragungsfunktion bei kleinen Orts-Frequenzen,” Optica Acta **10**, 1–19, 1963. [CrossRef]

15. J. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A **4**, 643–650, 1987. [CrossRef]

*P*(

**r**):

**r**is the position vector in the pupil and

*P*

^{*}is the complex conjugate of

*P*. This is illustrated in Fig. 1. When the pupils are circular and aberration free, we define the pupil function

*P*(

**r**) = ∏(

**r**), where ∏(

**r**) = 1 for |

**r**| ≤ 1 and zero otherwise. From this we obtain the familiar expression for the OTF as

*m*= |

**m**|. A normalisation factor has been introduced so that

*H*

_{0}(

**0**) = 1. The spatial frequencies are also normalised such that the cut-off of the incoherent imaging system corresponds to |

**m**| = 2. We model phase aberrations as a function Φ(

**r**), such that the pupil function

*P*(

**r**) = ∏(

**r**)exp[

*j*Φ(

**r**)], where

*j*= √-1. The corresponding, aberrated OTF can be calculated as

**r**-

**m**) can be approximated using a Taylor series expansion and Eqn. 6 can therefore be approximated by

*m*

^{2}) represents error terms of at least the second order in

*m*. For small arguments, the exponential term can also be expanded as a Taylor series, so that Eq. 7 becomes

*H*

_{0}(

**m**). For the other integrals, the effective region of integration is defined by the overlap of the two offset pupils ∏(

**r**-

**m**)∏(

**r**), shown as A in Fig. 1(b). If we approximate this region by the circular pupil P, then the corresponding approximation error is at least second order in

*m*. Equation 8 can therefore be written

*H*(

**m**)|

^{2}, follows as

**r**) contains a component of constant phase gradient (see Appendix A). The effect of this aberration component, which corresponds to the tip and tilt modes, is simply to shift the image laterally. We assume in the rest of this paper that these aberration modes play no role, although we note that the components are readily extracted from the phase of the image FT. If these modes are not present, Eq. 10 becomes

*H*

_{0}(

**m**)

^{2}would be replaced by unity. The corresponding error in the calculation of the MTF would be of order

*m*, leading to significant differences compared to the diffraction OTF calculations.

## 4. Optimisation metric based upon low spatial frequencies

*g*(

*M*

_{1},

*M*

_{2}) be defined as the total “energy” in all image spatial frequencies lying within the annulus for which

*M*

_{1}≤ |

**m**| ≤

*M*

_{2}, where

*M*

_{2}is small:

**m**= (

*m*cos

*ξ*,

*m*sin

*ξ*), then

*S*(

_{T}**m**) must be a periodic function of the polar angle

*ξ*and can be represented by its Fourier series:

*π*as the spectral density always has even symmetry about the origin, such that

*S*(

_{T}**m**) =

*S*(-

_{T}**m**). This permits us to calculate the first integral with respect to

*ξ*in Eq. 13 as

*ξ*, we can first show that the pupil integral, by expanding the scalar product, becomes

*χ*(

**r**) is the polar angle of the vector ∇Φ(

**r**). The second integral is then calculated as

*α*

_{0}(

*m*) and the first order terms

*α*

_{1}(

*m*) and

*β*

_{1}(

*m*) contribute to the value of

*g*. Significant values of these first order coefficients will indicate that the object has noticeable periodicity in a predominant direction, such as a one dimensional grid. For other object structures more likely to be encountered in applications

*α*

_{1}(

*m*) and

*β*

_{1}(

*m*) are expected to be small so the corresponding terms in Eq. 17 can be neglected. Hence, we find that

*α*

_{0}(

*m*) ≠ 0 in the frequency range of interest. If

*M*

_{1}and

*M*

_{2}are fixed and the object contains frequencies in this range, it can be seen from Eq. 18 that

*g*is maximum if and only if |∇Φ(

**r**)| = 0 for all

**r**or equivalently when Φ(

**r**) is a constant. Although

*g*is based only on low spatial frequencies, the optimisation process will remove all phase aberrations and hence improve imaging quality for all spatial frequencies. It is therefore appropriate to use

*g*as an optimisation metric for aberration correction. It can also be seen from Eq. 18 that the variation of

*g*for different aberrations can be derived entirely from the properties of the integral

## 5. Aberration expansion in Lukosz functions

*I*

_{1}, it is useful to represent the aberration as a combination of Lukosz functions. These functions, based upon the Zernike polynomials, were first derived by Lukosz [14

14. W. Lukosz, “Der Einfluβ der Aberrationen auf die optische Übertragungsfunktion bei kleinen Orts-Frequenzen,” Optica Acta **10**, 1–19, 1963. [CrossRef]

15. J. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A **4**, 643–650, 1987. [CrossRef]

*n*and

*m*are the radial and azimuthal indices, respectively, and

*R*(

^{m}_{n}*r*) is the Zernike radial polynomial given by

*N*Lukosz modes with coefficients

*a*:

_{i}*i*= 1,2,3 respectively) have been omitted. Using this aberration expansion, we find that each mode contributes independently to

*I*

_{1}:

*B*(

^{m}_{n}*r*) to ensure that the weighting of each coefficient in Eq. 26 is independent of the coefficient’s indices. The normalisation of

*B*(

^{m}_{n}*r*) used here is also slightly different to that employed in Reference [13

13. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. **32**, 5–7, 2007. [CrossRef]

*g*in Eq. 18 can now be directly expressed in terms of the set of aberration coefficients, {

*a*}, to give

_{i}*M*

_{1}and

*M*

_{2}. This approximation is accurate for small aberration amplitudes. However, for larger amplitudes it can give inaccurate, even negative values, whereas in practice

*g*would tend to zero. A more appropriate approximation is a Lorentzian function, which is always positive, tends to zero for large aberrations, and retains an identical form to relation 27 for small aberrations:

*q*

_{2}= 1/

*q*

_{0}and

*q*

_{3}=

*q*

_{1}/

*q*

^{2}

_{0}. This Lorentzian approximation provides a close fit to empirical measurements of

*g*, as shown in the next Section.

## 6. Experimental investigation of the optimisation metric

*g*({

*a*}) were investigated experimentally using the system shown in Fig. 2(a). The system comprised an incoherent transmission microscope with a deformable mirror (DM) and a CCD camera. For the purposes of this demonstration, the DM acted as both aberration source and correction element. A light emitting diode (LED; Lumileds, Luxeon Star/O, centre wavelength 650nm) provided incoherent illumination to a transmissive specimen which was imaged using a 150mm focal length achromatic doublet as the objective lens. An iris provided the 5mm diameter limiting aperture of the imaging system at the pupil plane of the objective. This pupil plane was imaged onto the DM (Boston Micromachines Corp., Multi-DM, 140 element, 1.5

_{i}*μ*m stroke) using a 4f system. The DM was then re-imaged through the same 4f system onto the pupil plane of the tube lens, which formed an image of the specimen on the CCD camera.

*c*}, where each control signal was proportional to the square root of the corresponding electrode voltage. This was found to produce linear operation over most of the DM’s deflection range. In order to produce desired combinations of Lukosz modes, the control signals were obtained from Lukosz modal coefficients {

_{i}*a*} through a matrix multiplication:

_{i}**a**and

**c**are identical to the elements of {

*a*} and {

_{i}*c*}, respectively. The matrix

_{i}**D**provides conversion from Lukosz coefficients into Zernike coefficients (see Appendix B). The pseudo-inverse matrix

**B**

^{†}permits the calculation of the control signals from the Zernike coefficients. This matrix was obtained using an interferometric method described in Reference [17

17. M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. **44**, 5131–5139, 2005. [CrossRef] [PubMed]

*g*, we used a holographic scatterer (Physical Optics Corp.) as the transmissive specimen. An aberration-free image of the scatterer is shown in Fig. 2(b). This specimen is ideal for this characterisation as it contains all spatial frequencies within the pass band of the imaging system; this can be seen in the image spectral density (Fig. 2(c)). Figure 3(a) shows the measured variation of

*g*with the root mean square (rms) aberration amplitude using different spatial frequency ranges. The aberration consisted of eight Lukosz modes (

*i*= 4 to 11). The rms amplitude was calculated from the Lukosz coefficients as

*a*= |

**a**| (see Appendix B). Each data point shows the mean and standard deviations for an ensemble of 200 random aberrations of magnitude

*a*. Each aberration was constructed by generating random coefficients in the range (-1,1) with uniform probability; the resulting vector was then scaled to the magnitude

*a*. When only small spatial frequencies are used in the calculation of

*g*, the deviation from the mean is small and the response is predominantly quadratic, as predicted by Eq. 27. When larger frequencies are also included, so that the low frequency approximations no longer hold, the value of

*g*drops off more quickly and the deviation from the mean is more significant. However, the Lorentzian approximation is found to provide a close fit to the mean value of

*g*for all curves. In Fig. 3(b), it can be seen that the width of the experimentally determined

*g*response fits the theoretical prediction for low spatial frequencies.

*g*response can be tuned by changing the spatial frequency range. In other words, the use of smaller spatial frequencies in the metric permits the measurement of larger aberrations. This property presents the possibility of schemes where aberrations are corrected in a series of optimisations covering first the large magnitude aberrations (using the smallest spatial frequencies), progressing to correction of less significant aberrations using larger frequencies.

## 7. Optimisation scheme

*g*({

*a*}). Using relation 28, we see that this is equivalent to the minimisation of a different metric

_{i}*G*({

*a*}) defined as

_{i}*a*} as an

_{i}*N*-dimensional coordinate basis, it is clear from Eq. 30 that

*G*({

*a*}) has a uniform paraboloidal shape in the neighbourhood of its minimum. This representation is particularly advantageous for optimisation, as the minimum of a paraboloidal function is readily found from a small number of metric evaluations. Moreover, the minimisation of

_{i}*G*can be decomposed into a sequence of

*N*independent one dimensional parabolic minimisations in each of the coefficients

*a*. In order to perform a minimisation with respect to the coefficient

_{i}*a*of a particular mode

_{k}*L*, we can express

_{k}*G*as

*q*′

_{2}and

*q*

_{3}are not known, the value of

*a*that minimises

_{k}*G*(

*a*) can be calculated from a minimum of three measurements of

_{k}*G*. In practice, we took these three measurements by intentionally introducing different aberrations using the adaptive element. We refer to these aberrations as biases. The biases were chosen to be Φ = -

*bL*, Φ = 0 and Φ = +

_{k}*bL*, where

_{k}*b*is a suitable bias amplitude. An image was acquired and its FT and spectral density were calculated. The appropriate range of frequency components was summed, giving the metric measurements

*g*

^{-},

*g*

_{0}and

*g*

_{+}respectively, and the reciprocal of each result was calculated, giving

*G*

_{-},

*G*

_{0}and

*G*

_{+}. The optimum correction aberration was then estimated by parabolic minimisation as [18]

*g*. To correct this single mode, the correction aberration Φ =

*a*

_{corr}

*L*would be added to the deformable mirror. For multiple mode correction, each modal coefficient would be measured in this manner before applying the full correction aberration containing all modes.

_{k}### 7.1. Correction of a single mode

*g*was calculated. Positive and negative bias aberrations were added in turn and the corresponding values of

*g*were calculated. The correction aberration was obtained using Eq. 33 and the correction was applied to the DM. The final rms phase aberration was found to be 0.18, corresponding to a Strehl ratio of 0.97.

### 7.2. Correction of multiple modes

**a**

_{in}, introduced by the DM consisted of eight Lukosz modes (

*i*= 4 to 11) and had a total amplitude of |

**a**

_{in}| = 11.5. Each modal coefficient was estimated in turn using a bias amplitude

*b*= 9.8. Once all eight coefficients had been estimated, the full correction aberration,

**a**

_{corr}, was added to the DM. We note that the unbiased measurement was identical for each modal estimation, so was only taken once. The final Strehl ratios were found to be 0.87 for the scatterer and 0.91 for the test chart. A further cycle of correction was also performed (not shown in the figure) using bias

*b*= 4.9, giving final Strehl ratios of 0.99 for the scatterer and 0.98 for the test chart.

### 7.3. Accuracy of correction

**a**

_{in}, introduced by the DM consisted of a random combination of the eight Lukosz modes

*i*= 4 to 11. The values of optimisation metric were obtained in a similar manner and the correction aberration,

**a**

_{corr}, was determined. We define the aberration error to be

**a**

_{err}=

**a**

_{in}+

**a**

_{corr}and the error magnitude as

*i*= 12 to 19) in the initial aberration. The original eight modes (

*i*= 4 to 11) were corrected in the same manner as before and

*ε*was calculated taking into account only the modes that were corrected. The results obtained when different amounts of the additional modes were present are shown in Fig. 7. The error

*ε*shows only a small variation as the amplitude of the additional modes is increased. This illustrates that different aberration modes can be corrected independently using this procedure.

## 8. Discussion and Conclusions

19. J. P. Hamaker, J. D. O’Sullivan, and J. E. Noordam, “Image sharpness, Fourier optics, and redundant-spacing interferometry,” J. Opt. Soc. Am. **67**, 1122–1123, 1977. [CrossRef]

*σ*can also be calculated in the Fourier domain as

*g*if using the spatial frequency range (

*M*

_{1},

*M*

_{2}) = (0,2). The methods described in this paper could therefore be extended for use with image sharpness metrics, obviating the need to calculate the image FT. For example, they would be directly applicable if the object spectrum were dominated by low spatial frequency components.

## Appendix A: Evaluation of an integral

**m**, which can be removed from the integrand, so we find that

*c*is an infinitesimal section of C that has the corresponding unit normal vector

**n**. If we define

**m**= (

*m*cos

*ξ*,

*m*sin

*ξ*) and

*ϕ*(

*θ*) = Φ(

**r**) when |

**r**| = 1 then Eq. 39 can be rewritten as

*ϕ*(

*θ*) as a Fourier series:

*I*

_{2}is zero unless

*μ*

_{1}or

*v*

_{1}are non-zero. If Φ(

**r**) is expressed as a series of Lukosz modes, then the only modes that can contribute to

*μ*

_{1}or

*v*

_{1}are mode 2 (tip) and mode 3 (tilt). All other Lukosz modes of the first azimuthal order are identically zero at

*r*= 1 so do not contribute to

*ϕ*(

*θ*). The effect of the tip and tilt modes is simply to translate the image laterally. Their influence on the OTF is the introduction of a phase variation and has no effect on the OTF magnitude. The role of

*I*

_{2}in Eq. 10 is therefore to compensate for the previous second order term so that the OTF magnitude is not affected by tip and tilt.

## Appendix B: Zernike and Lukosz functions

*i*or the dual indices (

*n*,

*m*), are explained by Neil et al. [20

20. M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A **17**, 1098–1107, 2000. [CrossRef]

^{2}. The Lukosz functions are normalised such that a coefficient of value 1 corresponds to a focal spot second moment (or equivalently rms spot radius) of

*λ*/(2

*πNA*), where

*λ*is the wavelength and

*NA*is the numerical aperture of the focussing lens.

*a*} and Zernike coefficients {

^{m}_{n}*z*} can be performed using the following relationships:

^{m}_{n}**a**and

**z**respectively then we can use the matrix-vector equation

**z**=

**Da**for the conversion, where the elements of the sparse matrix

**D**are calculated from Eq. 43. The rms phase aberration can be easily calculated from the Zernike mode coefficients as |

**z**|. It follows that, in terms of Lukosz coefficients, the rms phase aberration is given by

## Acknowledgements

## References and links

1. | R. K. Tyson, |

2. | J. W. Hardy, |

3. | R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Engineering |

4. | D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical Wavefront Reconstruction: Theory and Numerical Methods,” SIAM Review |

5. | R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. |

6. | A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemin, and R. G. Smits, “Correction of atmospheric distortion with an image-sharpening telescope,” J. Opt. Soc. Am. |

7. | M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A |

8. | M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. |

9. | N. Doble, |

10. | J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A |

11. | L. Murray, J. C. Dainty, and E. Daly, “Wavefront correction through image sharpness maximisation,” in Proc. S.P.I.E. , ‘Opto-Ireland 2005: Imaging and Vision’ |

12. | M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express |

13. | M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. |

14. | W. Lukosz, “Der Einfluβ der Aberrationen auf die optische Übertragungsfunktion bei kleinen Orts-Frequenzen,” Optica Acta |

15. | J. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A |

16. | V. N. Mahajan, |

17. | M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. |

18. | W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, |

19. | J. P. Hamaker, J. D. O’Sullivan, and J. E. Noordam, “Image sharpness, Fourier optics, and redundant-spacing interferometry,” J. Opt. Soc. Am. |

20. | M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(110.4850) Imaging systems : Optical transfer functions

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: March 29, 2007

Revised Manuscript: June 11, 2007

Manuscript Accepted: June 11, 2007

Published: June 14, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Delphine Debarre, Martin J. Booth, and Tony Wilson, "Image based adaptive optics through optimisation of low spatial frequencies," Opt. Express **15**, 8176-8190 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8176

Sort: Year | Journal | Reset

### References

- R. K. Tyson, Principles of Adaptive Optics, Academic Press, London, 1991.
- J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford University Press, 1998).
- R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Engineering 21, 829-832 (1982).
- D. R. Luke, J. V. Burke and R. G. Lyon, "Optical wavefront reconstruction: theory and numerical methods," SIAM Review 44, 169-224 (2002).Q1 [CrossRef]
- R. A. Muller and A. Buffington, "Real-time correction of atmospherically degraded telescope images through image sharpening," J. Opt. Soc. Am. 64, 1200-1210 (1974). [CrossRef]
- A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemin, and R. G. Smits, "Correction of atmospheric distortion with an image-sharpening telescope," J. Opt. Soc. Am. 67, 298-303 (1977). [CrossRef]
- M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, "Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field," J. Opt. Soc. Am. A 13, 1456-1466 (1996). [CrossRef]
- M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, "Adaptive phase-distortion correction based on parallel gradient-descent optimization," Opt. Lett. 22, 907-909 (1997). [CrossRef] [PubMed]
- N. Doble, Image Sharpness Metrics and Search Strategies for Indirect Adaptive Optics. PhD thesis, University of Durham, United Kingdom, 2000.
- J. R. Fienup and J. J. Miller, "Aberration correction by maximizing generalized sharpness metrics," J. Opt. Soc. Am. A 20, 609-620, 2003. [CrossRef]
- L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," in Proc. S.P.I.E., ‘Opto-Ireland 2005: Imaging and Vision’ 5823, 40-47 (2005).Q2
- M. J. Booth, "Wave front sensor-less adaptive optics: a model-based approach using sphere packings," Opt. Express 14, 1339-1352 (2006). [CrossRef] [PubMed]
- M. J. Booth, "Wavefront sensorless adaptive optics for large aberrations," Opt. Lett. 32, 5-7 (2007). [CrossRef]
- W. Lukosz, "Der Einfluß der Aberrationen auf die optische Übertragungsfunktion bei kleinen Orts-Frequenzen," Optica Acta 10, 1-19 (1963). [CrossRef]
- J. Braat, "Polynomial expansion of severely aberrated wave fronts," J. Opt. Soc. Am. A 4, 643-650 (1987). [CrossRef]
- V. N. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, (SPIE, Bellingham, WA, 2001).
- M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, "Methods for the characterisation of deformable membrane mirrors," Appl. Opt. 44, 5131-5139 (2005). [CrossRef] [PubMed]
- W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, (Cambridge University Press, 2nd ed., 1992).
- J. P. Hamaker, J. D. O’Sullivan, and J. E. Noordam, "Image sharpness, Fourier optics, and redundant-spacing interferometry," J. Opt. Soc. Am. 67, 1122-1123 (1977). [CrossRef]
- M. A. A. Neil, M. J. Booth, and T. Wilson, "New modal wavefront sensor: a theoretical analysis," J. Opt. Soc. Am. A 17, 1098-1107 (2000). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.