1. Introduction

Atmospheric turbulence degrades the resolution of Earth-based telescopes. One way to increase their resolution is to measure the aberration of the wavefront and either compensate this aberration with a deformable mirror or some other device (adaptive optics), or detect those fleeting moments in which the aberration of the wavefront is tolerable [1] and only then record images (Lucky Imaging) [2–4]. In both cases a method to determine the degree of wavefront aberration is required. In adaptive optics, a common way of measuring the aberration is the use of a Shack-Hartmann sensor: a beamsplitter samples an incident wavefront and passes it to a 2-dimensional array of lenslets of equal focal length; the light focused by the array is detected by a 2-dimensional detector (CCD, CMOS, etc.) [5,6]. A plane wave produces an array of spots on the detector, as shown in Fig. 1(a) ; any aberration of the wavefront shifts their positions, as shown in Fig. 1(b). The spot displacements provide information on the degree and type of aberration. To use the sensor a reference wave is required, such as that produced by a bright star in the field of view of the telescope. If there are no bright stars in the field of view, an artificial “laser guide star” may be created, typically by illuminating sodium atoms that are present at 90 km above sea-level with a laser tuned to the 589 nm transition of these atoms [7] or by observing the Rayleigh scattering from air at lower altitudes (~10-30 km) [8]. Although originally developed for atmospheric wavefront measurements, the Shack-Hartmann sensor also is used in other fields such as ophthalmology, optical device alignment, laser wavefront characterization, semiconductor wafer quality control, to name only a few applications [9].

 

Fig. 1 Shack-Hartmann wavefront sensor. (a) No aberration. (b) With aberration.

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2. Modified sensor

The use of a Shack-Hartmann sensor in an imaging device requires two detector arrays, one for wavefront sensing and another for imaging, and a beamsplitter, which reduces the light incident on the detectors. Here we show a device that is both a Shack-Hartmann sensor and an imaging device that requires only one detector array and does not require a beamsplitter. It is based on substituting the lenslet array with an array of electrically controlled Fresnel zone plates [10,11]. The basic idea of the sensor is shown in Fig. 2 . An array of electrically-controlled zone plates is placed in front of a camera (an image-forming lens and a 2D detector); the image-forming lens is positioned such that a focused image of some observed object, in this case a distant point source, is formed on the detector. If no voltage is applied the array simply acts as a window and an image of the object is formed on the detector, as shown in Fig. 2(a). If voltage is applied the zone plates form an array of spots. By switching the voltage on and off the setup can be toggled between an imaging mode and a wavefront sensor mode. The plane where the points are focused in the sensor mode is shifted from the plane where the image is formed in the imaging mode; however, this shift can be minimized by making the focal length of the lenslets much larger than the focal length of the imaging lens, or if the reference point source and object are located in different planes.

 

Fig. 2 Modified Shack-Hartmann sensor. (a) No voltage is applied; the setup is used to obtain an image. (b) Voltage is applied; the setup is used as a wavefront sensor.

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The main component of this sensor is the 2D array of electrically controlled Fresnel zone plates made in a domain-engineered z-cut lithium niobate wafer [11]. Each zone plate consists of concentric ring-shaped ferroelectric domains, with radii $r_m$ given by

where λ is the wavelength, m is a positive integer and $f_0$ is the main focal length of the zone plate. Transparent electrodes (ITO) are deposited on both c-faces of the wafer. By applying a voltage V to these electrodes, the ordinary refractive index $n_o$ of the domains changes by

where $r_{13}$ is the appropriate Pockels coefficient and d is the thickness of the wafer. The sign of $\Delta n$ depends on the orientation of the spontaneous polarization within the domain, therefore the sign of $\Delta n$ changes from ring to ring. The maximum diffraction efficiency of the zone plates into the main focus, approximately 40.5%, is obtained when the phase difference introduced by the domains is 180°, which occurs when $V=V_{\pi /2}=\lambda /2n_o^3{r}_{13}$. For LiNbO₃ at room temperature and $\lambda =633\text{ nm}$, $V_{\pi /2}\approx 2.9\text{ kV}$. The details of the fabrication technique and performance of these zone plates have been given elsewhere [11].

The use of Fresnel zone plates in Shack-Hartmann sensors has been reported previously. Ref. [12] describes the use of a matrix of amplitude zone plates as an alternative to microlenses. Although simple and inexpensive, the diffraction efficiency of amplitude zone plates is low (< 10%), which limits their use. Liquid crystal spatial light modulators have been used to make adaptive Shack-Hartmann sensors where the number, size and focal length of the lenslets can be controlled in real time [13], and in a device that simultaneously senses and compensates wavefront aberrations [14]. There a few drawbacks of using liquid crystal spatial light modulators, mainly the noise and resolution limitation introduced by “pixelation” and the requirement of polarized light. In contrast, ferroelectric zone plates have low losses due to scattering and do not exhibit pixelation. Furthermore, they are transparent between ~400 and 4000 nm, they can withstand high intensities and they have a virtually instantaneous response time, limited in practice by the response time of the voltage supply. This is important since the aberration introduced by atmospheric turbulence varies over time spans of the order of milliseconds [15].

3. Experiment

To illustrate the modified sensor’s potential use, we used the device to enhance the resolution of images using a modified “Lucky Imaging” technique, where only frames with low distortion are averaged to create an enhanced composite image. For various reasons, in the standard implementation of this technique these frames are not selected with information gathered from a wavefront sensor, but rather by characteristics of the individual frames themselves, such as the appearance of intense spots that are interpreted as high Strehl -ratio images of stars and therefore nearly diffraction-limited [16]. Here we use the modified sensor to simultaneously capture the frames and quantify their degree of distortion. Figure 3 shows the setup used to demonstrate the concept. A camera plus a telescope were used to take a series of pictures of an object, a transparency of the fake constellation Onocrotalum Maxima back-lit with green light. An aberrator was placed between the object and the telescope to mimic atmospheric turbulence. A glass plate with a small dot of white paint was placed between the aberrator and the object and a red HeNe laser illuminated the dot, acting as a laser guide star used as a reference to determine the degree of wavefront distortion introduced by the aberrator. The use of green light was to ease the separation of the red “beacon” light from the image background in the sensor-mode pictures.

 

Fig. 3 Experimental setup.

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The camera was a commercial 6.1 megapixel Canon Digital EOS 300 with a ~70 mm focal length, f/4 lens. To mimic the effects of read-out noise and low exposure levels encountered in a real astronomical observation, the camera was set to the maximum sensitivity, ISO 1600, which is also the noisiest, and the lamp illuminating the transparency was dimmed such that even at this high sensitivity each shot was at least 5 EV underexposed; the shots taken in the sensor mode used the same camera settings. The telescope consisted of two lenses, a 250 mm singlet and a ~150 mm compound lens. The aberrator was a glass plate smeared unevenly with unflavored gelatin; some regions had a layer of uniform thickness, which only introduces an inconsequential “piston” aberration, in others the thickness was extremely non-uniform, which produces blurred images, and in others the layer was approximately wedge-shaped, which only shifts the image. This glass plate was mounted on a translation stage; by translating it the degree of aberration could be varied from virtually zero to extreme levels. The zone-plate array consisted of an 8 x 8 matrix of electrically controlled zone plates with a main focal length of approximately 40 mm at 633 nm, fabricated using the technique described in [11]. It was placed inside the telescope at a position such that the illuminated spot on the glass (the “guide star”) was imaged onto a matrix of points in the camera when voltage was applied to the wafer containing the zone plates.

A set of 500 pairs of pictures was taken, each pair consisting of a picture taken in the sensor mode (2.8 kV applied voltage and HeNe beam on) and a picture taken in the imaging mode (no applied voltage and HeNe beam blocked with the shutter). For each pair the variable aberrator was moved to a different position. The whole data-taking sequence was computer controlled. Figure 4 shows examples of the results. Figure 4 (a) shows the data obtained with virtually no aberration and Fig. 4 (b) shows the data obtained with moderate aberration. The left images are pictures taken in the sensor mode and the right images are the pictures obtained in the imaging mode. The images in the middle depict where the centroids of the left images are located; the coordinates $r_c$ of the centroids were calculated using

 

Fig. 4 Sensor-mode and imaging-mode results. Left: pictures obtained in the sensor mode; middle: centroids obtained from the pictures on the left; right: pictures obtained in the imaging mode. (a) With negligible distortion; (b) with moderate distortion (${\sigma }_x=4.1\text{ pixels}$,${\sigma }_y=7.4\text{ pixels}$).

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where N is the number of pixels in the region that is being evaluated, and $r_j$ and $I_j$ are the location and intensity measured of the jth pixel of the region, respectively. The centroids are spaced approximately 200 pixels in both the vertical and horizontal directions. Close inspection of the middle image of Fig. 4(b) reveals a clear deviation of the centroids from their positions shown in Fig. 4(a).

The 500 pairs of pictures were processed with a custom-made computer program that calculated the centroids of the spots in the pictures taken in the sensor mode and compared them with the centroids without aberration. The shifts of the individual spots, which were determined with sub-pixel resolution, could have been used to obtain an average shift of the image for image enhancement by the “shift-and–add” technique, or could have been fed as error signals to a deformable mirror to compensate the aberration. However, for simplicity we chose to use the standard deviation of the coordinates of the spots as a measure of the quality of each frame. The higher the standard deviation, the higher the distortion. Let ${\sigma }_x$ and ${\sigma }_y$, be the standard deviations of the horizontal and vertical coordinates of the 64 spots of each sensor-mode pictures, and let ${\sigma }_{\max }=\max \left({\sigma }_x,{\sigma }_y\right)$. Visual inspection of the picture pairs shows that, for the setup used, sensor-mode pictures with ${\sigma }_{\max }\approx 3$ pixels and below have associated imaging-mode pictures with very little aberration; for ${\sigma }_{\max }$ between 3 and 7 pixels the aberration is quite noticeable and for ${\sigma }_{\max }>7$ pixels the aberration is extreme. The example shown in Fig. 4(b) has a ${\sigma }_{\max }={\sigma }_y=7.4$ pixels and clearly the associated imaging-mode picture is extremely distorted. Figure 5 shows the histogram of ${\sigma }_{\max }$ of the 500 sensor-mode pictures. As can be seen, for more than half of these pictures ${\sigma }_{\max }>6$, so most of the images are severely distorted.

 

Fig. 5 Histogram of the standard deviations of the centroid locations.

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4. Results

Media 1 shows the result of adding the images according to the ${\sigma }_{\max }$ of the associated sensor-mode pictures. Each frame of the movie corresponds to a composite image obtained by adding all the images where the associated ${\sigma }_{\max }$ is below a certain value, shown on the bottom left side of the movie; the number of pictures that satisfied that criterion is given on the bottom right side. The overall brightness of each composite image was adjusted such that the brightest spots appeared bright green. Figure 6 shows four selected frames. As can be seen, the quality of the composite is low for small values of the maximum ${\sigma }_{\max }$ allowed, because few images satisfy that criterion and therefore the background noise is substantial.

 

Fig. 6 Single-frame excerpts from a video (Media 1) where each frame is the result of adding the images according to the ${\sigma }_{\max }$ of the associated sensor-mode pictures. (a) Maximum allowed ${\sigma }_{\max }$: 0.37 pixels. (b) Maximum allowed ${\sigma }_{\max }$: 1.0 pixels. (c) Maximum allowed ${\sigma }_{\max }$: 4.0 pixels. (d) Maximum allowed ${\sigma }_{\max }$: 13 pixels.

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The quality is also low for large values of the allowed ${\sigma }_{\max }$, above approximately 4 pixels, because the aberration is considerable. The best composite images occur for a maximum allowed ${\sigma }_{\max }$ of around 2 pixels, produced with around 40 images (Fig. 6(b)). The extreme case of not discarding any images, which is what one would obtain without the information provided by the Shack-Hartmann sensor, is shown in the last movie frame, Fig. 6 (d).

5. Conclusion

We have presented a modified Shack-Hartmann sensor based on an electrically controllable array of ferroelectric domain zone plates which uses the same detector employed for image acquisition. To the best of our knowledge, this is the first time these ferroelectric zone plates have been used in a Shack-Hartmann sensor. To exemplify its potential, we have shown its use in an experimental simulation of “Lucky Imaging.” As a concluding remark, aside from eliminating the need for a separate camera to perform wavefront sensing, the quasi-instantaneous response time of the ferroelectric zone plates would allow gating the sensor with microsecond precision. This could be used to avoid detecting unwanted Rayleigh scattering of the upward-going pulse used to create the laser guide star.