Modified Shack–Hartmann wavefront sensor using an array of superresolution pupil filters

Susana Ríos; David López

doi:10.1364/OE.17.009669

1. Introduction

The performance of Shack–Hartmann wavefront sensors is characterized by its dynamic range and sensitivity [1]. The sensor measures the displacements of the focal spots produced by a lenslet array from a reference position, being the displacements directly related to the mean local wavefront slope of the incoming wavefront over the subaperture area. The dynamic range of the sensor is defined as the maximum value of the slope that can be measured and is given by:

θ_{max} = \frac{d}{2 f'}

being d the lenslet diameter and f′ its back focal length. On the other hand, the measurement sensitivity is defined as the minimum value of the slope that can be measured with the sensor and is defined as:

θ_{min} = \frac{Δ l}{f'}

where Δl represents the minimum detectable spot displacement. If we represent the incoming wavefront as an expansion in Zernike coefficients [2], the number of lenslets needed to correctly sample the aperture is given by the number of Zernike terms needed to describe the wavefront [1] (this number is dependent upon the statistics of the application of interest). If the number of lenslets and the aperture size is fixed, then to increase the dynamic range of the sensor short focal lengths have to be used, meaning that the sensitivity is decreased. The sensitivity of the sensor depends also on the minimum value of the spot displacement that can be detected, and this factor is influenced by the pixel size of the detector, the accuracy in the estimate of the spot centroids and the measurement errors [1]. Several methods have been proposed to increase the dynamic range of the sensor without compromising its sensitivity [3, 4, 5, 6]. On the other hand, the accuracy in the estimate of the centroid position has been studied by many authors [7,8,9,10,11] and optimal centroid estimation algorithms have been developed [12,13]. It is well known that the variance in the estimate of the centroid position depends on the size of the window used in the calculations, meaning that a smaller window leads to a lower value of the variance, increasing the sensor accuracy [7, 8].

Superresolution techniques have been used in a variety of applications including astronomy [14, 15], image processing, optical data storage [16], free–space laser communication systems [17] and confocal microscopy [18,19], to enhance the resolution of an optical system. The aim of this technique is to reduce the size of the 3D focal region by locating a suitable pupil filter in the exit pupil of an optical system. Depending on the application of interest, superresolution in the axial direction (axial resolution) or in a plane orthogonal to the axis (lateral resolution) is considered. T.R.M. Sales [20] has found that increasing lateral resolution leads to a resolution loss in the axial direction, and that the minimum focal spot giving volumetric superresolution is about the half of the diffraction limit spot. When reducing the size of the Airy disc, there is also a reduction in the peak intensity from the diffraction limit case, and also an increase in the sidelobes intensity. Depending on the technique used to get superresolution, a displacement on the focus position can occur.

Different configurations of pupil filters have been developed, mainly designed to improve resolution in confocal scanning microscopy. Most of these filters are based on annular designs like the design of Sales and Morris [21] andWang et al. [22], and phase-only filters [23,24] are preferred because they give better performance than amplitude transmittance filters [25].

The goal of this work is to use an array of pupil filters and locate them behind the lenslet array to obtain an array of superresolved focal spots. Each focal spot will have a central lobe of reduced size compared with the spot produced without the filter. If we select the central lobe to calculate the centroid position, the window size is reduced, leading to a lower value of the variance in the centroid estimate.

In Section 2 the focal properties of superresolution pupil filters will be analyzed. In Section 3 we will study the variance in the estimate of the centroid position and its relation with the focal spot size. Some theoretical results for an annular aperture with different obstruction ratios will also be analyzed. In Section 4, experimental superresolved focal spots are presented, as well as theoretical and experimental values of the variance in the centroid estimate with and without pupil filters.

2. Focusing properties of annular amplitude or phase filters

Let’s consider an object with transmission function t(r)exp[iϕ(r)] located in front of a converging lens of focal length f′, representing the pupil filter. In the Fraunhoffer approximation, the normalized amplitude in the back focal plane of the lens, when illuminated with a normally incident monochromatic plane wave is given by [26]:

U (ρ) = 2 \int_{0}^{1} t (r) \exp [i ϕ (r)] J_{0} (η r) r d r

being $η = \frac{π D_{ρ}}{λ f'}$ the normalized radial coordinate in the back focal plane of the lens. The filter is characterized by an amplitude transmittance, t(r) and a phase function ϕ(r). Superresolution techniques differ in the way they optimize these functions, some of them being focused in optimizing the amplitude transmittance without affecting the phase of the incident beam, using obscurations or continuous amplitude filters [27, 28, 29]. Other techniques are based in modulating the phase function alone, like binary phase filters [21, 23, 30] or in hybrid techniques, which modify both the amplitude and the phase of the incident beam [31, 32, 33].

In order to characterize the effects of superresolution, several parameters can be introduced [27,34]: the transversal and axial gain factors, G_T and G_A, defined here as the diffraction limit spot size relative to the superresolved spot size on the transverse focal plane and along the optical axis respectively, the Strehl ratio, S, defined as the ratio of the superresolved peak intensity and the Airy pattern peak intensity at the back focal plane, and the focus displacement z_F. De Juana et al. [23] obtained analytical expressions for the parameters G_T, G_A, S and z_F as a function of binary phase pupil filter parameters, and proposed an optimization problem to get the filter parameters matching some focusing properties. Cagigal et al. [24] developed an analytical method to design binary phase filters with required values of the parameters G_T, G_A, S and z_F.

As an example, let us consider an annular amplitude transmission filter with outer diameter D_o=2 and inner diameter D_i, defining the obstruction ratio as $O = \frac{D_{i}}{D_{o}}$ . In this case the Strehl ratio is given by S=[1-O ²]² [26, 35, 36], and transversal and axial gains are can be approximated by G_T=1+O and G_A=[1-O]² respectively as calculated in Ref. [27]. The transversal intensity profiles of the focal spot are shown in Fig. 1 for different values of the obstruction ratio, and also for the unobstructed pupil (O=0). It can be seen that the peak intensity decreases as the obstruction ratio increases, leading to values of the Strehl ratio lower than 1. The size of the first maximum decreases as the obstruction ratio increases, giving values of the transversal gain greater than 1.

In most applications like confocalmicroscopy, optical data storage and free–space laser communications, reducing the size of the Airy disk in the axial or transversal directions directly gives a gain in resolution restricted by the limits of 3D resolution described by Sales [20]. In Shack–Hartman wavefront sensing, reducing the size of the focal spots changes the centroid variance. What we try to find out in this work is if a reduction in the transversal spot size produces a reduction in the variance of the centroid position, and how this variance is related to the parameters G_T, S and z_F. It is beyond the scope of this work to design the optimum filter for this application.

3. Accuracy in the estimate of the centroid position

The variance in the estimate of the centroid position of an image has been studied by many authors [8,12,10,11], and it depends on the centroiding algorithm used to estimate the position of the focal spots. We will deduce here an expression for the variance valid for windowing algorithms [37]. Detailed expressions for thresholding algorithms have been calculated by J. Ares and J. Arines [13] and a comparison of different centroiding algorithms has been done by S. Thomas et al. [7].

If I(x,y) is the intensity distribution in the back focal plane of the lenslets corresponding to one of the spots, its centroid in the x-direction, x_c, is defined by:

Fig. 1. Lateral intensity profiles for different values of the obstruction ratio

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x_{c} = \frac{\int x I (x, y) d x d y}{\int I (x, y) d x d y}

If we pixelize the image, the integrals have to be replaced by a sum over all pixels:

x_{c} = \frac{\sum_{i, j} x_{ij} I_{ij}}{\sum_{i, j} I_{ij}}

In the windowing method, only the pixels within some radius from image center are considered. The center has to be known approximately, and the coordinates of the brightest pixel can be taken as the window center in experimental measurements. To derive an analytical expression for the centroid variance, we will consider a circular window with the same radius as the first minimum of the Airy diffraction pattern and centered in the exact centroid position. In the case of negligible photon noise, the noise in the intensity can be modeled by an additive Gaussian distribution with zero mean and variance σ ² _n for all pixels. In the case of noisy measurements, the estimate of the centroid position is given by:

{\hat{x}}_{c} = \frac{\sum_{i, j} x_{ij} (I_{ij} + n_{ij})}{\sum_{i, j} (I_{ij} + n_{ij})}

The variance in the centroid position is defined as:

σ_{x_{c}}^{2} = 〈 {({\hat{x}}_{c} - x_{c})}^{2} 〉

Introducing in Eq.(7) the definitions of x_c and x̂_c given in Eq.(5) y (6), and considering that the noise is spatially uncorrelated and that there is no correlation between the noise and the signal, the following expression for the variance can be obtained [8, 9]:

σ_{x_{c}}^{2} = \frac{σ_{n}^{2}}{{〈 I_{T} 〉}^{2}} [\sum_{i, j} x_{ij}^{2} + x_{c}^{2} A_{p} - 2 x_{c} \sum_{i, j} x_{ij}]

where A_p is the area of the window and <I_T>² is the squared mean of the total intensity incident in the calculation window:

{〈 I_{T} 〉}^{2} = {〈 (\sum_{i, j} I_{ij} + n_{ij}) 〉}^{2}

If we center the calculation windowin the centroid position x_c, the expression for the variance can be written as:

σ_{x_{c}}^{2} = \frac{σ_{n}^{2}}{{〈 I_{T} 〉}^{2}} [\sum_{i, j} x_{ij}^{2}]

and considering a circular window with the same radius as the first minimum of the intensity distribution, ρ_min, in pixel units

σ_{x_{c}}^{2} = \frac{σ_{n}^{2}}{{〈 I_{T} 〉}^{2}} \frac{π}{4} ρ_{min}^{4}

We wish to compare the values of the variance with and without pupil filter. When no filter is used, the intensity equals to the diffraction pattern of the microlens aperture at its back focal plane, and ρ_min is the radius of the Airy disk in the case of microlenses with circular shape. When introducing the filter, the radius of the first minimum decreases giving a lower value of ρ ⁴ _min, but <I_T>² decreases giving a higher value of $\frac{1}{{〈 I_{T} 〉}^{2}}$ . The term <I_T>² has to be evaluated to know if any gain in the variance is obtained when introducing the filter without changing the illumination conditions. This means that reducing the size of the calculation window also modifies the term <I_T>² in the expression of the centroid variance, and that obtaining a gain in variance by reducing the size of the calculation window is not straightforward.

Let’s consider the term <I_T>²:

{〈 I_{T} 〉}^{2} = {〈 [\sum_{i, j} (I_{ij} + n_{ij})] 〉}^{2} = {(\sum_{i, j} I_{ij})}^{2} + σ_{n}^{2} A_{p}

For high values of the signal-to-noise ratio the noise term can be neglected, and the final expression for the variance of the centroid position is given by:

σ_{x_{c}}^{2} = \frac{σ_{n}^{2}}{{(\sum_{i, j} I_{ij})}^{2}} \frac{π}{4} ρ_{min}^{4}

where we can see that the variance is a function of the total energy incident in the calculation window, Σ_i,j I_ij, and the radius of the window.

If we introduce the energy ratio as the ratio of the total energies without and with pupil filter:

F = \frac{{(\sum_{i, j} I_{ij})}_{nofilter}}{{(\sum_{i, j} I_{ij})}_{filter}}

then the ratio of variances without filter and for the filter case, what we call the variance gain, is given by:

R = \frac{G_{T}^{4}}{F^{2}}

Fig. 2. Normalized lateral intensity profiles for different values of the obstruction ratio

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If we introduce the normalized energy ratio, F_N, defined as the ratio of the total energies corresponding to the normalized intensities, and considering that $F = \frac{F_{N}}{S}$ , the ratio of variances can be written in terms of the filter parameters G_T and S:

R = \frac{G_{T}^{4} S^{2}}{F_{N}^{2}}

where the transversal gain, the normalized energy gain and the Strehl ratio have to be evaluated. If a displacement z_F on the focus position is produced when introducing the pupil filter, the measurements have to be made in the new focus position, located at a distance f′+z_F from the lenslets, being f′ its back focal length.

As an example we calculated the theoretical values of the variance for an amplitude-only annular mask with different values of the obstruction ratio. We considered an array of circular microlenses of diameter D ₀=2mm and an array of annular masks of outer diameter D_o=2mm and inner diameter D_i, defining the obstruction ratio as $O = \frac{D_{i}}{D_{0}}$ . The total energy (normalized) was calculated by numerically evaluating the area of the intensity distributions under the central maximum. The Strehl ratio is given by S=[1-O ²]² and the transversal gain, approximately given by G_T=1+O, has been determined with more precision by numerically calculating the first minimum of the intensity distribution.

In Fig. 2 we present the transversal normalized intensity profiles for obstruction ratios ranging from 0 to 0.9. In Table 1 we present the values of the variance gain, R, for obstruction ratios O=0.15, 0.5, 0.6 y 0.75, obtaining values near or lower than unity, meaning that reducing the size of the calculation window gives no gain in the variance of the centroid estimate due to the low values of the Strehl ratio. If we increase the intensity of the incident beam after locating the pupil filter to get S⋍1, then we find that the variance gain increases as the obstruction ratio increases, obtaining a value of 1.59 for an obstruction ratio of 0.75, as can be seen in Table 1. This makes this technique useful only in applications where the illumination conditions can be controlled like in optical metrology. To make this technique useful in other applications, the design of pupil filters with new requirements is needed. For the superresolution filters designed in the literature, increasing the Strehl ratio gives a lower value of the transversal gain as in the example presented here [27, 34, 24].

Table 1. Filter parameters and variance gain for different obstruction ratios

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4. Experimental results

An array of 3×3 squared microlenses with dimensions 2×2mm ² and focal length f′=60cm were used to check that a gain in variance can be experimentally obtained. An array of annular masks with outer diameter D_o=2mm and inner diameter D_i=1.5mm were located behind the microlens array and properly centered. The mask geometry is shown in Fig. 3.

Fig. 3. Mask geometry

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The set-up was illuminated with a collimated He-Ne laser beam, and the intensity distribution, with and without the pupil mask, was measured with a 12 bit CCD camera Hamamatsu ORCA-285 (see reference [38] for characteristics) located at the microlenses back focal plane. The intensity of the incident beam was controlled with a pair of linear polarizers to match the maximum grey level for both images, with and without the pupil mask, and keeping this maximum below the saturation level of the CCD camera (4096 grey levels). A set of 500 images and an average image were recorded with and without the mask. The central spot of each image was selected to determine the variance in the centroid estimate. To determine the center and size of the calculation window, the center of gravity and the mean radius of the central maximum (across several directions) were estimated from the average images. The center of gravity of each individual frame was calculated using the same window and all the data was processed to estimate the variance of the centroid position.

Figure 4(a) represents a section of the measured transversal intensity profiles of one of the spots obtained with the lenslets alone, and Fig. 4(b) the intensity profile for an annular mask with O=0.75. It can be seen that the central maximum with the mask has a reduced size compared with the size obtained without the mask. Figure 5(a) represents the transversal intensity profile of one of the spots measured with the lenslets alone and Fig. 5(b) the transversal intensity distribution of the spot obtained with the same mask. A top view of the measured intensity distribution in the back focal plane of the lenses is represented in Fig. 6(a) and in Fig. 6(b) the intensity distribution obtained with the mask.

Fig. 4. Section of the transversal intensity profile without mask (a), and with an annular mask (O=0.75) (b)

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Fig. 5. Transversal intensity distribution without mask (a), and with an annular mask (O=0.75) (b)

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To compare the theoretical and experimental values of the variance gain, an expression of the variance for a square geometry is needed because we are using squared microlenses in the measurements. It can be easily shown that the variance for this geometry is given by:

σ_{x_{c}}^{2} = \frac{σ_{n}^{2}}{{(\sum_{i, j} I_{ij})}^{2}} \frac{{(2 x_{min})}^{4}}{12}

x_min being the location of the first minimum of the intensity distribution. From this equation, the ratio of variances with and without mask can be recalculated:

R = \frac{16}{3 π} \frac{G_{T}^{4} S^{2}}{F_{N}^{2}}

Fig. 6. Top view of the lenslets focal spots without mask (a), and with an annular mask (O=0.75) (b)

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Table 2. Theoretical filter parameters and variance gain

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Table 3. Experimental values of the filter parameters and variance gain

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In Table 2 we present the theoretical predictions for the ratio R of variances without mask and with the annular mask, as well as the parameters G_T and F_N for this geometry. In Table 3 the experimental results for the filter parameters, the centroid variances and the variance gain are presented. Δx represents the diameter of the calculation window in pixel units obtained from the average images and averaged over several directions. <I_T>² is the squared mean value of the total intensity incident in the calculation window in grey levels units and was obtained by averaging the total intensity in the 500 images. $σ_{x_{c}}^{2}$ is the variance of the centroid position in the x direction and in pixel units. It can be seen that the experimental parameters G_T and F_N differ from the theoretical values shown in Table 2. This experimental values have been introduced in Eq. (18) to recalculate the theoretical prediction, R_th. The mean values of the maximum intensity, I_M, have been obtained to calculate the experimental value of the Strehl ratio, S, and are given in grey levels. The experimental gain in the variance is 2.35 while the theoretical value is 2.12. This result shows that a significant gain in the variance can be experimentally obtained by using a pupil filter, but that a deeper study has to be done to predict the difference between experimental and theoretical results. W. Zhao et al. developed a theoretical model to analyze the effect of fabrication errors on the superresolution properties G_A, G_T and S of an N-zone annular phase filter, but no experimental results is presented [39]. In our application to Shack–Hartmannwavefront sensing, the influence of the fabrication errors on the total intensity <I_T> and on the size and shape of the focal spot have to be analyzed to evaluate the effect of these errors in the variance of the centroid estimate. If the experimental mask is narrower than the mask used in the calculations, the experimental gain will be higher than the theoretical gain. Another sources of error are the experimental determination of the size and centering of the calculation window, which in practice can be influenced by irregularities in the shape of the spot, and the indetermination in the back focal plane position. If the calculation window is narrower than the window used in the calculations, the gain will be higher than expected.

5. Conclusion

The use of superresolution techniques to enhance the accuracy of Shack-Hartmann wavefront sensors has been theoretically and experimentally demonstrated. The theoretical calculations show that if we can modify the illumination conditions to get a Strehl ratio close to S=1, some gain in the variance of the centroid estimate can be obtained, being the gain 1.59 for an annular amplitude mask with obstruction ratio O=0.75. Experimental results with this mask and an array of squared lenslets show a significant gain in the variance, obtaining a value of 2.35 for a Strehl ratio S=1.02, being the theoretical value of the gain 2.12. There is a difference between the experimental and theoretical values of the filter parameters G_T and F_N, and consequently in the variance gain, R, which shows that an study of the effects of the fabrication errors on the filter properties and variance gain has to be done.

Acknowledgments

This work was supported by the Spanish Ministerio de Ciencia e Innovación grants FIS2008–03884 and DPI2006–07906 and the European Regional Development Fund.

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O	G_T	F_N	S	R	R (S=1)
0.15	1.15	1.22	0.95	1.06	1.17
0.3	1.35	1.5	0.83	1.02	1.47
0.5	1.56	1.96	0.56	0.48	1.54
0.6	1.67	2.22	0.41	0.26	1.58
0.75	1.82	2.63	0.19	0.06	1.59

O	Δx	G_T	<I_T >²	F_N	$σ_{x_{c}}^{2}$	IM	S	R_exp	R_th
0	60	13.53×10¹²		7.89×10^-5	3871
0.75	52	1.15	9.18×10¹²	1.21	3.35×10^-5	3957	1.02	2.35	2.12

O	G_T	F_N	S	R	R (S=1)
0.15	1.15	1.22	0.95	1.06	1.17
0.3	1.35	1.5	0.83	1.02	1.47
0.5	1.56	1.96	0.56	0.48	1.54
0.6	1.67	2.22	0.41	0.26	1.58
0.75	1.82	2.63	0.19	0.06	1.59

O	Δx	G_T	<I_T >²	F_N	$σ_{x_{c}}^{2}$	IM	S	R_exp	R_th
0	60	13.53×10¹²		7.89×10^-5	3871
0.75	52	1.15	9.18×10¹²	1.21	3.35×10^-5	3957	1.02	2.35	2.12

Modified Shack–Hartmann wavefront sensor using an array of superresolution pupil filters

Abstract

1. Introduction

2. Focusing properties of annular amplitude or phase filters

3. Accuracy in the estimate of the centroid position

4. Experimental results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (3)

Equations (18)

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