Extension of the modal wave-front reconstruction algorithm to non-uniform illumination

Xiaoyu Ma; Jie Mu; ChangHui Rao; Jinsheng Yang; XueJun Rao; Yu Tian

doi:10.1364/OE.22.015589

1. Introduction

The Hartmann wave-front sensor was first proposed by Johannes Hartmann in 1900. After being perfected by Roland Shack in 1971 [1], it offers high precision, real-time detection and can be applied to the detection of wave-fronts affected by atmospheric turbulence in astronomical adaptive optics [2,3]. With the development of photoelectric technology, Hartmann wave-front sensors bestow various advantages, such as: simplicity of structure, ease of data-processing, an adjustable dynamic range, they are accessible to white light, and offer high precision wave-front detection. Owing to these advantages, they are widely used in mirror surface analysis, laser parameter diagnosis, flow field CT (Coherent Tomography) reconstruction, in the diagnosis of human eye aberrations, optical collimation [4–8], etc. Thus, researchers have devoted more and more effort to developing an understanding, and refinement of the detection precision of Hartmann wave-front sensors.

The Hartmann wave-front sensor, as a slope wave-front sensor, reconstructs the wave-front using a modal method wherein the wave-front slope of each sub-aperture is measured and multiplied by the reconstruction matrix to obtain the Zernike coefficient vector for the measured wave-front: the wave-front is then reconstructed using this vector [9–12]. Therefore, with an adequate spatial sampling rate and an array of lenses, the detection precision of the Hartmann wave-front sensor is mainly associated with the detection precision of the wave-front itself and the calculation of the reconstruction matrix. Nowadays, regarding the detection error in a Hartmann wave-front sensor, research mainly focuses on the noise in the charge-coupled-device (CCD) camera and the influences of sampling error on the detection precision of its spot centroid (wave-front) and the correction thereof [13–15].

However, the light intensity of the measured wave-front is generally not uniform in actual application due to influences from the optical source and other elements. For example, when the Hartmann wave-front sensor is recalibrated using an absolute spherical wave [16,17], since it is influenced by pinhole diffraction, the light intensity distribution of the spherical wave becomes non-uniform; in the detection of systemic aberrations in large optical systems (such as is used in lithography [18]), the light intensity distribution becomes non-uniform owing to the cumulative effect of the non-uniformly coated optical elements. It has been proved that, with the non-uniform distribution of light intensity on a measured wave-front, the wave-front slope of a single sub-aperture in the Hartmann wave-front sensor is not only associated with the mean wave-front slope of the sub-aperture, but also with the distribution of light intensity therein [19].

During quasi-static measurement, the light intensity of the measured wave-front can be deduced by carrying out simulation and then measurement. Hence, the research first deduced the theoretical formulae for calculating the relationship between the wave-front slope of a single sub-aperture and the phase and light intensity distribution of the incident wave-front; the formulae were then applied to the calculation of the reconstruction matrix. The wave-front reconstruction error induced by neglecting the variation of light intensity in traditional reconstruction matrix calculations was thereby corrected. In this way, the research improved the detection precision of the Hartmann wave-front sensor in the presence of non-uniform illumination while meeting the demands imposed by the development requirements of advanced optical measurement technology.

2. The basic principles of Hartmann wave-front sensor

The method by which wave-front measurement is accomplished using a Hartmann wave-front sensor is based on slope measurement, as illustrated in Fig. 1. The instrument is composed of a micro-lens array and a centroid detection system such as a charge-coupled device (CCD). The measurement process proceeds as follows. The detected wave is divided into multiple sampling units by the micro-lens array. The light from each sampling unit is then focused into separate spots. The spot array so formed is collected by the centroid detector. When the detected wave-front is not a plane wave, the spots generated will suffer deviations in the x- and y-directions due to the wave-front’s inclination at each sub-aperture. The deviation degree reveals the wave-front’s slope (in these two directions) at the corresponding sampling unit.

Fig. 1 A schematic diagram illustrating the working principles involved in wave-front measurement using a Hartmann wave-front sensor.

Download Full Size | PDF

The wave-front reconstruction algorithm used subsequently is one of the key technologies underlying the successful use of the Hartmann wave-front sensor. Naturally, researchers have proposed various algorithms during the development of the Hartmann wave-front sensor. Of them all, the Zernike modal wave-front reconstruction algorithm is the one which is most widely used [20]. In this algorithm, the wave-front is described over a circular region using a set of different Zernike polynomials, $Z_{k} (x, y)$ , represent different aberration modes (such as: inclination, defocusing, astigmatism, coma, and spherical aberration). Overall, the wave-front is thus written in this form,

φ (x, y) = \sum_{k = 1}^{n} a_{k} Z_{k} (x, y) .

Where,

φ (x, y)

is the phase of the wave-front,

a_{k}

is the k^th Zernike polynomial coefficient.

By multiplying the wave-front slope vector $G$ ( $G = [g_{x 1}, g_{y 1}, g_{x 2}, g_{y 2}, .... g_{x m}, g_{y m}]'$ ) and the reconstruction matrix which will be introduced in section 4, $D^{+}$ , the Zernike coefficient vector A ( $A = {[a_{1}, a_{2}, \dots, a_{n}]}^{'}$ ) of the measured wave-front can be obtained:

A = D^{+} G .

3. The relationship between wave-front slope and the phase and light intensity

In traditional modal wave-front reconstruction algorithms, calculation of the wave-front slope is based on the assumption there is a uniform distribution in the light intensity of the measured wave-front, then

{\begin{cases} g_{x i} = \frac{λ}{2 π S} \iint \frac{\partial φ (x_{0}, y_{0})}{\partial x_{0}} d x_{0} d y_{0} \\ g_{y i} = \frac{λ}{2 π S} \iint \frac{\partial φ (x_{0}, y_{0})}{\partial y_{0}} d x_{0} d y_{0} \end{cases},

where,

λ

is the wavelength of the measured wave-front,

(x_{0}, y_{0})

is the coordinate of the wave-front, and

S

refers to the normalized area of a single sub-aperture.

In any practical application, the light intensity of the measured wave-front is usually non-uniform due to the influence brought about by atmospheric turbulence and optical components. Therefore, it is necessary to analyze the relationship between the wave-front slope and the distributions in the phase and light intensity of the wave-front.

In the wave-front slope calculation, directions x and y are taken to be completely symmetrical. Therefore, in this section, the slope relationship is mainly discussed using the x-direction only — those in the y-direction are similar to those in the x-direction because of symmetry.

According to the Fourier transforms in the lens, the complex amplitude distribution on the back focal plane of the micro-lens, $U (x, y)$ , can be obtained using

U (x, y) = \iint_{\infty} T (x_{0}, y_{0}) \exp (- j k \frac{x_{0} x + y_{0} y}{f}) d x_{0} d y_{0},

where,

(x, y)

is the coordinate of the focal plane, f is the focal length of the micro-lens,

k = 2 π / λ

, and

T (x_{0}, y_{0})

is the complex amplitude distribution of the incident wave-front on the front focal plane. The latter quantity is given by:

T (x_{0}, y_{0}) = t (x_{0}, y_{0}) e^{j φ (x_{0}, y_{0})} .

Using the centroid formula, the relationship between the wave-front slope and the light intensity distribution on the back focal plane of the micro-lens and the focal length of the micro-lens can be written as

g_{x i} = \frac{X_{c}}{f} = \frac{1}{f} \cdot \frac{\iint x I_{u} (x, y) d x d y}{\iint I_{u} (x, y) d x d y},

where,

X_{c}

is the centroid of the spot on the back focal plane,

X_{c} = \frac{\iint x I_{u} (x, y) d x d y}{\iint I_{u} (x, y) d x d y}

, and

I_{u} (x, y) = {| U (x, y) |}^{2}

.

According to the Parseval's theorem, we obtain that

\iint I_{u} (x, y) d x d y = \iint I_{t} (x_{0}, y_{0}) d x_{0} d y_{0}

where,

I_{t} (x_{0}, y_{0}) = {| T (x_{0}, y_{0}) |}^{2} = {| t (x_{0}, y_{0}) |}^{2}

Let $R (x_{0}, y_{0}) = T (x_{0}, y_{0}) * T^{*} (x_{0}, y_{0})$ , then:

\begin{matrix} I_{u} (x, y) = {| U (x, y) |}^{2} = U (x, y) U^{*} (x, y) \\ = \iint R (x_{0}, y_{0}) e^{- i 2 π (u x_{0} + v y_{0})} d x_{0} d y_{0} \end{matrix}

So using the property of fourier transform:

\begin{matrix} \iint x I_{u} (x, y) d x d y = \iint [R (x_{0}, y_{0}) \int x e^{- j k x x_{0}} d x \int e^{- j k y y_{0}} d y] d x_{0} d y_{0} \\ = \iint [R (x_{0}, y_{0}) (\frac{1}{j k} δ' (- x_{0})) δ (- y_{0})] d x_{0} d y_{0} \\ = \frac{1}{j k} \int R (x_{0}, 0) δ' (- x_{0}) d x_{0} \\ = Re [\frac{1}{j k} {\frac{\partial R (x_{0}, 0)}{\partial x_{0}} |}_{x_{0} = 0}] \end{matrix}

As $R (x_{0}, y_{0}) = T (x_{0}, y_{0}) * T^{*} (x_{0}, y_{0})$ , then:

\begin{array}{l} {\frac{\partial R (x_{0}, 0)}{\partial x_{0}} |}_{x_{0} = 0} = {\frac{\partial \iint [t (x_{0} + x_{0}^{'}, y_{0}^{'}) e^{j φ (x_{0} + x_{0}^{'}, y_{0}^{'})}] {[t (x_{0}^{'}, y_{0}^{'}) e^{j φ (x_{0}^{'}, y_{0}^{'})}]}^{*} d x_{0}^{'} d y_{0}^{'}}{\partial x_{0}} |}_{x_{0} = 0} \\ = \iint t (x_{0}^{'}, y_{0}^{'}) \frac{\partial t (x_{0}^{'}, y_{0}^{'})}{\partial x_{0}^{'}} d x_{0}^{'} d y_{0}^{'} + j \iint t^{2} (x_{0}^{'}, y_{0}^{'}) \frac{\partial φ (x_{0}^{'}, y_{0}^{'})}{\partial x_{0}^{'}} d x_{0}^{'} d y_{0}^{'} \end{array}

Noted that only the real part of ${\frac{\partial R (x_{0}, 0)}{\partial x_{0}} |}_{x_{0} = 0}$ is needed in Eq. (9), so

\iint x I_{u} (x, y) d x d y = \frac{1}{k} \iint I_{t} (x_{0}, y_{0}) \frac{\partial φ (x_{0}, y_{0})}{\partial x_{0}} d x_{0} d y_{0}

Thus, the facula centroid at focal plane can be expressed as

X_{c} = \frac{λ f}{2 π} \frac{\iint I_{0} (x_{0}, y_{0}) \frac{\partial φ (x_{0}, y_{0})}{\partial x_{0}} d x_{0} d y_{0}}{\iint I_{0} (x_{0}, y_{0}) d x_{0} d y_{0}}

Therefore, in the x-direction, the relationship between the wave-front slope and the phase and light intensity distributions of the incident wave-front is:

g_{x i} = \frac{λ}{2 π} \frac{\iint I_{0} (x_{0}, y_{0}) \frac{\partial φ (x_{0}, y_{0})}{\partial x_{0}} d x_{0} d y_{0}}{\iint I_{0} (x_{0}, y_{0}) d x_{0} d y_{0}} .

where,

I_{0} (x_{0}, y_{0})

the light intensity distribution of the wave-front.

In the same way, in the y-direction, the analogous expression is:

g_{y i} = \frac{λ}{2 π} \frac{\iint I_{0} (x_{0}, y_{0}) \frac{\partial φ (x_{0}, y_{0})}{\partial y_{0}} d x_{0} d y_{0}}{\iint I_{0} (x_{0}, y_{0}) d x_{0} d y_{0}} .

Equations (13) and (14) show that when the intensity of the incident light wave has a non-uniform distribution, the wave-front slope is related to the phase and light intensity distributions of the incident wave-front. Also, Eq. (3) is merely a special case of the general expression for wave-front slope when the light intensity is uniform.

The relationship between wave-front slope and the phase and light intensity distributions is shown in Fig. 2.

Fig. 2 Schematic diagram showing a comparison of the action of the wave-front slope when illuminated by (a) is uniform, and (b) is non-uniform. (a) is uniformly distributed while that in (b) is non-uniform. It can be seen from the figure that the light intensities of the spots formed by the micro-lens, as well as the slope of the wave-front calculated by the intensities of the spots, are not the same.

Download Full Size | PDF

4. The Extension of the modal wave-front reconstruction

Due to the orthogonality of the Zernike polynomials in the circular region, the relationships between the slopes of the wave-front in the i^th sub-aperture with Zernike polynomial coefficient and the mean slope of the single-order aberration in the i^th sub-aperture are expressed as follows:

{\begin{cases} g_{x i} = \sum_{k = 1}^{n} a_{k} \cdot d_{x i} (Z_{k}) \\ g_{y i} = \sum_{k = 1}^{n} a_{k} \cdot d_{y i} (Z_{k}) \end{cases}

where,

d_{x i} (Z_{k})

and

d_{y i} (Z_{k})

denote the mean slopes of the aberrations represented by the k^th Zernike polynomial in the x and y-direction in the i^th sub-aperture.

Equation (10) can be rewritten in matrix form:

G = D A

where, G is the wave-front slope vector, A is the Zernike coefficient vector, and D is the reconstruction matrix,

\begin{array}{l} \begin{matrix} D = & [\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} d_{x 1} (Z_{1}) \\ d_{y 1} (Z_{1}) \end{matrix} \\ d_{x 1} (Z_{2}) \\ d_{y 1} (Z_{2}) \\ \dots \end{matrix} \\ d_{x 1} (Z_{m}) \\ d_{y 1} (Z_{m}) \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} d_{x 2} (Z_{1}) \\ d_{y 2} (Z_{1}) \end{matrix} \\ d_{x 2} (Z_{2}) \\ d_{y 2} (Z_{2}) \\ \dots \end{matrix} \\ d_{x 2} (Z_{m}) \\ d_{y 2} (Z_{m}) \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \dots \\ \dots \end{matrix} \\ \dots \\ \dots \\ \dots \end{matrix} \\ \dots \\ \dots \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} d_{x n} (Z_{1}) \\ d_{y n} (Z_{1}) \end{matrix} \\ d_{x n} (Z_{2}) \\ d_{y n} (Z_{2}) \\ \dots \end{matrix} \\ d_{x n} (Z_{m}) \\ d_{y n} (Z_{m}) \end{matrix} \end{matrix}] \end{matrix} . \end{array}

According to Eq. (3), when the light intensity of the incident wave-front has a uniform distribution, the elements of the reconstruction matrix D are traditionally calculated using

{\begin{cases} d_{x i} (Z_{k}) = \frac{λ}{2 π S} \iint_{S} \frac{\partial Z_{k} (x, y) d x d y}{\partial x} \\ d_{y i} (Z_{k}) = \frac{λ}{2 π S} \iint_{S} \frac{\partial Z_{k} (x, y) d x d y}{\partial y} \end{cases} .

When the light intensity of the incident wave-front is non-uniform, the wave-front slope is related to the distributions in the phase and light intensity of the incident wave-front. If Eq. (17) is employed to calculate the reconstruction matrix D, error will be induced in the wave-front reconstruction. The relationships between the wave-front slope and the distributions in the phase and light intensity of the incident wave-front can be expressed by Eqs. (13) and (14). The new method of calculating the elements of the reconstruction matrix D therefore involves finding

{\begin{cases} d_{x i} (Z_{k}) = \frac{λ}{2 π} \frac{\iint I_{0} (x, y) \frac{\partial Z_{k} (x, y)}{\partial x} d x d y}{\iint I_{0} (x, y) d x d y} \\ d_{x i} (Z_{k}) = \frac{λ}{2 π} \frac{\iint I_{0} (x, y) \frac{\partial Z_{k} (x, y)}{\partial x} d x d y}{\iint I_{0} (x, y) d x d y} \end{cases} .

5. Experimental results

The experimental light path is shown in Fig. 3. A plane wave was generated from a spherical wave from a single-mode optical fiber after collimation using a collimating lens. An aberration plate was used to disturb the plane wave to develop the incident wave-front. This wave-front was then measured using a Hartmann wave-front sensor by detecting the spot centroid displacements before and after the aberration plate was included. A Hartmann wave-front sensor with 35 × 35 sub-apertures was employed in this experiment.

Fig. 3 The experimental light path. Key: ① Laser bearing a tail fibre output, ② single-mode optical fibre, ③ collimating lens, ④ aberration plate, ⑤ down-collimator, and ⑥ Hartmann wave-front sensor.

Download Full Size | PDF

According to the theoretical description of the transmitted optical fiber mode, the light intensity distribution at the output end of the single-mode optical fiber should follow a standard Gaussian profile in the experiment [21].

The experimental results are shown in Figs. 4 and 5.

Fig. 4 A comparison of the residues obtained from 10 sets of aberrations when the light intensity distribution is considered (white bars, G₂), and when it is not (black bars, G₁).

Download Full Size | PDF

Fig. 5 A comparison of the RMS errors arising from the traditional algorithm and corrected methods of waver-front reconstruction.

Download Full Size | PDF

In Fig. 4, (G_x0, G_y0) denotes the wave-front slope vector detected by the Hartmann wave-front sensor, which is calculated by the centroid formula and the intensity distribution of each spots. (To reduce the error in subsequent calculations arising from CCD noise, the average value of many repeated measurements is used). As the incident wave-front can be measured first, the wave-front slope vector (G_x1,G_y1) can be obtained from Eq. (3), i.e., when the light intensity distribution is neglected. Finally, (G_x2,G_y2) is the wave-front slope vector obtained from Eqs. (13) and (14), i.e., when the light intensity distribution is taken into consideration. Using the two methods described above, the wave-front slopes in the x- and y-directions were calculated corresponding to 10 different sets of aberrations. The residuals compared to the wave-front slope vector detected by the Hartmann wave-front sensor. The root mean squares (RMSs) of the residues, λ, suggest that the wave-front slope vectors calculated when the light intensity distribution is considered are much closer to those obtained by the Hartmann wave-front sensor.

In Fig. 5, G refers to the wave-front slope vector measured using the Hartmann wave-front sensor (once again, to reduce the calculation error arising from CCD noise, an average of many measurements is used), is the inverse of the reconstruction matrix, calculated using Eq. (17), when the light intensity distribution is ignored, and ${D^{'}}^{+}$ is that calculated from Eq. (18), when the light intensity distribution is taken into account. The wave-front reconstruction errors for 10 sets of different aberrations are calculated using the two methods discussed above, as shown in Fig. 5. The results imply that the corrected modal wave-front reconstruction algorithm (i.e., in which the effect of light intensity variation on the wave-front slope is considered) shows a greater accuracy compared to the traditional method.

In summary, the wave-front slope vector calculated when the light intensity distribution is considered is closer to that obtained by actual measurements. That is, the modified modal wave-front reconstruction algorithm shows a greater accuracy compared to the results produced using the traditional algorithm.

6. Conclusions

Hartmann wave-front sensors are widely used in wave-front measurement due to their straightforward operating principles, simple structure, etc. Wave-front reconstruction is an important part of the application of a Hartmann wave-front sensor. Among the various wave-front reconstruction algorithms, modal wave-front reconstruction, which employs measured wave-front slope data to solve the weights of the different aberrations, is the method most widely used. However, traditional modal wave-front reconstruction algorithms neglect the influence of the light intensity distribution on the wave-front slope. Therefore, when a Hartmann wave-front sensor is used in illumination conditions that are non-uniform, error will be induced. The aim of this study was to eliminate this error. To do this we first analyzed theoretically the relationship between the wave-front slope and the distribution in the phase and light intensity of wave-front. Then, the influence exerted by the light intensity on the wave-front slope was introduced into the calculation of the reconstruction matrix used in the modal wave-front reconstruction algorithm. The wave-front measurement accuracy of the Hartmann wave-front sensor was thereby improved. This improvement is conducive to the use of the Hartmann wave-front sensor in the field of high-precision wave-front measurement.

References and links

1. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001). [PubMed]

2. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett. 17(24), 1737–1739 (1992). [CrossRef] [PubMed]

3. C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002). [CrossRef]

4. T.-L. Kelly, P. J. Veitch, A. F. Brooks, and J. Munch, “Accurate and precise optical testing with a differential Hartmann wave-front sensor,” Appl. Opt. 46, 861–866 (2007).

5. J. A. Koch, R. W. Presta, R. A. Sacks, R. A. Zacharias, E. S. Bliss, M. J. Dailey, M. Feldman, A. A. Grey, F. R. Holdener, J. T. Salmon, L. G. Seppala, J. S. Toeppen, L. Van Atta, B. M. Van Wonterghem, W. T. Whistler, S. E. Winters, and B. W. Woods, “Experimental comparison of a Shack-Hartmann sensor and a phase-shifting interferometer for large-optics metrology applications,” Appl. Opt. 39(25), 4540–4546 (2000). [CrossRef] [PubMed]

6. T. L. Bruno, A. Wirth, and A. J. Jankevics, “Applying Hartmann wavefront sensors technology to precision optical testing of the HST correctors,” Proc. SPIE 1920, 328–337 (1993). [CrossRef]

7. V. Y. Zavalova and A. V. Kudryashov, “Shack-Hartmann wave-front sensor for laser beam analyses,” Proc. SPIE 4493, 277–284 (2002).

8. N. Doble, G. Yoon, L. Chen, P. Bierden, B. Singer, S. Olivier, and D. R. Williams, “Use of a micro-electro-mechanical mirror for adaptive optics in the human eye,” Opt. Lett. 27(17), 1537–1539 (2002). [CrossRef] [PubMed]

9. R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31(32), 6902–6908 (1992). [CrossRef] [PubMed]

10. S. Rios and D. Lopez, “Modified Shack-Hartmann wave-front sensor using an array of super-resolution pupil filters,” Opt. Express 17, 9669–9679 (2009).

11. C. Canovas and E. N. Ribak, “Comparison of Hartmann analysis methods,” Appl. Opt. 46(10), 1830–1835 (2007). [CrossRef] [PubMed]

12. S. Barbero, J. Rubinstein, and L. N. Thibos, “Wavefront sensing and reconstruction from gradient and Laplacian data measured with a Hartmann-Shack sensor,” Opt. Lett. 31(12), 1845–1847 (2006). [CrossRef] [PubMed]

13. C. Genrui and Y. Xin, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2321–2335 (1994).

14. W. Jiang, H. Xian, and F. Shen, “Detecting error of Shack-Hartmann wavefront sensor,” Proc. SPIE 3126, 534–544 (1997). [CrossRef]

15. X. Ma, C. Rao, and H. Zheng, “Error analysis of CCD-based point source centroid computation under the background light,” Opt. Express 17(10), 8525–8541 (2009). [CrossRef] [PubMed]

16. J. Yang, L. Wei, H. Chen, X. Rao, and C. Rao, “Absolute calibration of Hartmann-Shack wavefront sensor by spherical wavefronts,” Opt. Commun. 283(6), 910–916 (2010). [CrossRef]

17. L. Wang, C. Rao, and X. Rao, “Analysis of wave-front error for nanometer pinhole vector diffraction,” Opt. Precision Eng. 20(3), 499–505 (2012). [CrossRef]

18. T. Fujii, J. Kougo, Y. Mizuno, H. Ooki, and M. Hamatani, “Portable phase measuring interferometer using Shack-Hartmann method,” Proc. SPIE 5038, 726–732 (2003). [CrossRef]

19. Y. Carmon Erez and R. Ribak, “Centroid distortion of a wavefront with varying amplitude due to asymmetry in lens diffraction,” J. Opt. Soc. Am. A 26(1), 85–90 (2009).

20. S. Barbero, J. Rubinstein, and L. N. Thibos, “Wave-front sensing and reconstruction from gradient and Laplacian data measured with a Hartmann-Shack sensor,” Opt. Lett. 31(12), 1845–1847 (2006). [CrossRef] [PubMed]

21. M. Chen, S. Roux Filippus, and C. Olivier Jan, “Detection of phase singularities with a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. A 24, 1994–2002 (2007).

Extension of the modal wave-front reconstruction algorithm to non-uniform illumination

Abstract

1. Introduction

2. The basic principles of Hartmann wave-front sensor

3. The relationship between wave-front slope and the phase and light intensity

4. The Extension of the modal wave-front reconstruction

5. Experimental results

6. Conclusions

References and links

Cited By

Figures (5)

Equations (19)

Optics Express