## Abstract

A waveband-splitting method is proposed for open-loop liquid crystal adaptive optics systems (LC AOSs). The proposed method extends the working waveband, splits energy flexibly, and improves detection capability. Simulated analysis is performed for a waveband in the range of 350 nm to 950 nm. The results show that the optimal energy split is 7:3 for the wavefront sensor (WFS) and for the imaging camera with the waveband split into 350 nm to 700 nm and 700 nm to 950 nm, respectively. A validation experiment is conducted by measuring the signal-to-noise ratio (SNR) of the WFS and the imaging camera. The results indicate that for the waveband-splitting method, the SNR of WFS is approximately equal to that of the imaging camera with a variation in the intensity. On the other hand, the SNR of the WFS is significantly different from that of the imaging camera for the polarized beam splitter energy splitting scheme. Therefore, the waveband-splitting method is more suitable for an open-loop LC AOS. An adaptive correction experiment is also performed on a 1.2-meter telescope. A star with a visual magnitude of 4.45 is observed and corrected and an angular resolution ability of 0.31″ is achieved. A double star with a combined visual magnitude of 4.3 is observed as well, and its two components are resolved after correction. The results indicate that the proposed method can significantly improve the detection capability of an open-loop LC AOS.

© 2012 OSA

## 1. Introduction

Liquid crystal adaptive optics systems (LC AOSs) have been investigated widely to mitigate the spatial resolution limit of deformable mirrors [1–3]. However, a liquid crystal wavefront corrector (LC WFC), which is used in LC AOSs, has the disadvantages of low energy utilization and slow response. To achieve fast switching frequency, liquid crystal materials, such as dual-frequency and ferroelectric materials, are utilized, and the switching frequency increased to greater than 1 kHz [4, 5]. Given that LC WFC is polarization-dependent and has a very narrow working waveband, the energy utilization ratio is very low. To extend the working waveband, two methods were demonstrated using chiral smectic ferroelectric liquid crystals [6] and multi-LC WFCs [7]. Light emitted from an object is normally unpolarized or partially polarized. However, given that the LC WFC can only correct polarized light, a polarizer needs to be used, which results in half of the energy being wasted.

To solve the polarization dependence problem, Love et al. proposed two methods [8, 9] and Mu et al. proposed another method that uses a polarized beam splitter (PBS) [10]. When a PBS is used, the natural light is split into two polarized beams: one beam travels to the wavefront sensor (WFS) directly, and the other is corrected using a LC WFC, after which the beam is captured by an imaging camera. In this configuration, the WFS is blind to the LC WFC, and the LC AOS requires operation with an open-loop control. Open-loop adaptive optics is essential for a multi-object adaptive optics technique, which has recently been developed to investigate distant galaxies and multi-object spectroscopy [11–15].

For LC AOSs, the open-loop control method also has the advantage of improving energy utilization ratio [7, 10, 16–19]. The first open-loop LC AOS that uses a PBS was demonstrated in Ref. 10. When a PBS is used, both the WFS and the imaging camera occupy half of the object light; this half/half energy splitting cannot be changed. However, to observe the weak object, the energy used for wavefront detection and imaging needs to be optimized with a different ratio. Given that the energy splitting ratio cannot be changed, the PBS method is unsuitable for optimizing energy splitting. In this paper, a novel energy splitting method that can split energy flexibly and consequently improve the detection capability of LC AOS is proposed. To describe the method conveniently, a working waveband of 350 nm to 950 nm is selected for the analysis.

## 2. Received energy for the WFS and the imaging camera

For LC AOSs, the optical energy should be split into two components, one for wavefront detection and the other for imaging. Normally, a Shack–Hartmann (S–H) WFS is used to detect the distorted wavefront, and only the energy split for this type of WFS is analyzed. To simplify the calculation and analysis, all errors, such as the detector noise, fitting error, WFS noise, etc., are ignored. To obtain a suitable energy split, the energy detecting capability of the S–H WFS and the imaging camera will be analyzed first in the next section. The two underlying assumptions are that whole received energy is *E* and that the WFS and imaging cameras occupy respective halves of the whole energy.

#### 2.1 Received energy of imaging camera

The received energy for one pixel of an imaging camera is first considered. By considering atmospheric turbulence, the radius of the Airy disk at the focal plane of the telescope is given by

where*r*is the atmospheric coherence length,

_{0}*λ*is the relevant wavelength, and

*f*is the focal length of the telescope.

According to the Nyquist–Shannon sampling theory, 2 × 2 pixels are used to resolve the diffraction limit of the telescope, and the pixel size *μ* = 0.61*λf*/*D* (*D* is the aperture of the telescope). The received energy of a pixel can be calculated as the pixel area multiplied by the received energy of the unit area. Therefore, the received energy of a pixel at a unit time can be calculated by

*μ*into Eq. (2), the received energy of the imaging camera can be rewritten as

#### 2.2 Received energy of WFS

S–H WFS comprises a microlens array and a camera. To correct atmospheric turbulence, the desired number of the microlens of the S–H WFS is (*D*/*r _{0}*)

^{2}[20]. Normally, at the image plane of the microlens array, each optical disc occupies 2 × 2 pixels to calculate the centroid accurately. Therefore, the focus radius of each microlens equals the pixel size

*a*, and the whole receiving energy area may be expressed as${\left(D/{r}_{0}\right)}^{2}\pi {a}^{2}$. Similar to the imaging camera, the received energy of a pixel can be calculated as the pixel area multiplied by the received energy of the unit area. As E/2 energy is received by the S–H WFS, the received energy for one pixel of the S–H WFS can be computed by

#### 2.3 Estimation of the energy occupancy ratio of WFS

Considering Eq. (4)/Eq. (3), *E _{WFS}* is four times greater than

*E*. However, to obtain the optimal energy split, the exposure time of the camera must also be considered. At present, the best camera for this process is a back-illuminated electron multiplying (EM) charge-coupled device (CCD) at a waveband range of 350 nm to 950 nm. Therefore, two back-illuminated EM-CCD cameras were selected for this study. Each camera has the same quantum efficiency (QE) and noise characteristics for imaging and S–H WFS. Normally, imaging and WFS cameras have different pixels. To analyze energy splitting, only the pixels that can receive light were considered. A suitable energy-splitting scheme is one wherein each used pixel receives the same amount of energy for both the imaging and the WFS camera. Furthermore, the signal on a CCD camera is assumed to be proportional to the exposure time. Under this condition, the following is achieved:

_{CCD}*t*and

_{CCD}*t*are the exposure times for the imaging camera and the S–H WFS, respectively; and

_{WFS}*R*and

_{CCD}*R*are the energy ratios of the CCD camera and the S–H WFS to the whole energy, respectively.

_{WFS}Substituting ${E}_{WFS}=4{E}_{CCD}$and ${R}_{CCD}=1-{R}_{WFS}$into Eq. (5), *R _{WFS}* can be expressed as

To achieve higher sampling frequency, the exposure time of S–H WFS should be as short as possible. Normally, high bandwidth is achieved by selecting 1 kHz to 2 kHz as the frame rate of S–H WFS. If 1 ms exposure time of S–H WFS is selected, *R _{WFS}* as a function of the exposure time of the imaging camera is shown in Fig. 1
. For

*T*of less than 30 ms,

_{CCD}*R*is drastically increased and subsequently changes slowly. Furthermore, the received energy of the S–H WFS is approximately 70% with

_{WFS}*T*= 10 ms and

_{CCD}*R*= 80%, whereas

_{WFS}*T*= 17 ms (Fig. 1).

_{CCD}## 3. Optimal design for energy splitting

#### 3.1 Waveband-splitting method

To capture the image of a star, an exposure time of tens of milliseconds is acceptable. Thus, according to the above analysis, more energy can be used to detect wavefront distortion. Assuming that *T _{CCD}* = 10 ms, 70% energy can be allocated for the S–H WFS, and 30% energy can be used to perform imaging. Therefore, using half of the energy is unnecessary to conduct imaging, and more energy can thus be allocated to conduct wavefront detection. In this study, a novel waveband-splitting method is demonstrated as a replacement for the PBS method. Similar to the deformable mirror-based closed-loop AOSs, a short waveband is used to perform distortion detection, and a long waveband is captured using an imaging camera [21]. To the best of the researchers’ knowledge, this study is the first to apply this method to open-loop LC AOSs.

The novel optical configuration for the waveband-splitting method is shown in Fig. 2 . The object light is collimated using lens L1 and is then split into two beams using a long-wave pass filter (LWPF) in such a way that the beam with the short waveband is reflected toward the WFS, and the other beam with the long waveband passes through the LWPF and is zoomed using L2 and L3. Therefore, the energy occupied by the WFS can be adjusted easily by changing the 50% cutoff point of the LWPF. The PBS is placed in front of the LC WFC to split the unpolarized light into two linearly polarized beams. Thus, each polarized beam is directed toward the corresponding LC WFC, and the energy loss attributed to polarization dependence is avoided.

The reflected light from the LC WFC is separated completely from the incident beam by tilting the incident beam at an angle of 1° to 3° [22]. The reflected beam is again reflected by M1 and then travels to the camera. For open-loop LC AOSs, measuring the response function of the LC WFC is a complicated process [10, 19]. In the proposed design, an associated optical configuration is added to measure the response function of the LC WFC. When measuring the response function, a white light source with fiber bundle output is placed at the conjugated location of the image plane, and M2 is moved into the optical layout. Thus, the beam that is emitted by the fiber bundle travels to the LC WFC and is then reflected out. To ensure that the reflected light can be measured using the WFS with the long waveband, the LWPF is replaced by a short-wave pass filter (SWPF). Therefore, the LWPF and SWPF are switchable in the optical setup. After measuring the response function, the SWPF and the mirror are removed, and the LWPF is switched into the optical configuration. For the previously described optical design, the LC AOS is operated using open-loop control because the WFS is blind to the LC WFC.

The waveband-splitting method has the following advantages.

- a) The energy can be split flexibly, and the wavefront detection capability is significantly improved. Given that the divided waveband width can be adjusted easily, the energy used for wavefront detection can be optimally designed for different observation objects. Therefore, more energy can be utilized to perform wavefront detection, such that fainter objects can be observed.
- b) The optical setup is simplified, and the shorter waveband can be utilized. For the PBS energy splitting scheme, multi-LC WFCs are needed to expand the working waveband to a range of 400 nm to 900 nm [7]. PBS cannot perfectly split the natural light into two polarized beams for such a wide waveband. At present, only two LC WFCs are used to correct atmospheric distortion, and the waveband is limited to 600 nm to 900 nm [19]. Hence, the energy of the 400 nm to 600 nm waveband is wasted. However, this shortcoming may be eliminated using the waveband-splitting method, and the system is simplified using only two LC WFCs with a working band of 400 nm to 900 nm.
- c) Polarization energy loss is avoided. Polarization dependence is a main problem for LC AOSs. In the proposed design, the energy loss attributed to polarization dependence is avoided using a PBS and two LC WFCs.

#### 3.2 Calculation of the waveband split point

To demonstrate the waveband-splitting method, the energy ratio between the S–H WFS and the imaging camera is selected as 7:3, the spectrum of the sun is utilized, and a full waveband in the range of 350 nm to 950 nm is selected. Given that the radiant energy and the QE differ at varying wavelengths, the received radiant energy of a CCD camera at a certain wavelength should be calculated by the QE times the radiant energy as follows:

where*E*is the radiant energy of the object. Then, using the QE and the radiant spectrum curves, the received radiant energy of a CCD camera may be calculated to obtain a waveband. If the waveband is confirmed, the whole received energy of a CCD camera can be calculated bywhere

_{rad}*w*is the wavelength, and

*w*and

_{min}*w*are the minimum and maximum wavelengths, respectively, which are used for the integral. Assuming the short waveband is used for wavefront detection, the received energy of S–H WFS can be calculated bywhere

_{max}*w*is the splitting wavelength. Therefore,

_{split}*R*can be calculated by

_{WFS}*R*can be calculated at different splitting wavelengths.

_{WFS}A high-performance back-illuminated EM CCD camera (DU860, Andor) was selected to perform the calculation; its quantum efficiency curve is shown in Fig. 3
. The spectrum curve of the sun on the ground is shown in Fig. 4
. Consequently, the received radiant energy of the camera can be calculated using Eq. (7), and the result is shown in Fig. 5
. In the proposed design, the short waveband is used for wavefront detection. Therefore, *R _{WFS}* is calculated according to Eq. (10) at different splitting wavelengths, and the result is shown in Fig. 6
. At the splitting point of 703 nm,

*R*is equal to 70%. Consequently, the full waveband is split into 350 nm to 700 nm and 700 nm to 950 nm. The short waveband is used to detect the wavefront, and the long waveband is captured using an imaging camera. Furthermore, based on Fig. 6, the splitting point is selected according to the designed energy ratio of the S–H WFS.

_{WFS}## 4. Experiments

#### 4.1 Comparison between waveband-splitting and PBS method

An experiment was conducted to validate the above analysis. The optical layout is shown in Fig. 7 . A xenon lamp with fiber is used as the white light source because its spectrum is similar to that of the sun. A circular variable neutral density filter (NDF) is utilized to change the light intensity. The collimated light is split into two beams by using a beam splitter (BS). The reflected light goes to the optical power meter, which then measures the light intensity. The transmitted light is again split by a PBS, with one beam going to the WFS and the other captured by using an EM CCD camera. This configuration enables the simultaneous measurement of the signal-to-noise ratio (SNR) of the WFS and that of the camera.

Normally, the SNR of the imaging camera and the WFS should be greater than 5 to guarantee that the images can be resolved well. Given that the SNR is directly related to the received optical energy, it is used to evaluate the energy distribution of the waveband-splitting and PBS methods. If the SNR of the WFS and the imaging camera have very little difference, the energy split can be considered optimal. By rotating the NDF, the light intensity can be decreased gradually, and the SNR of the WFS and imaging camera will then be diminished simultaneously. When the SNR of the WFS or the imaging camera is equal to 5, the measurement will be stopped, and the variation of the SNR is acquired. The above optical layout is used to measure the SNR for the PBS method. For the waveband-splitting method, the PBS is replaced by a LWPF with a 50% cutoff point at 700 nm.

In the test configuration, the S–H WFS contained an EM CCD camera (DU860, Andor). The imaging camera was also an EM CCD camera (DV897, Andor), the sensitivity of which is equal to that of DU860. An optical power meter (THORLABS, PM100D) was used to detect the variation in light intensity. First, the variation in SNR was measured for the PBS method, as shown in Fig. 8 . This variation indicates that the SNR changes as a linear function of the normalized intensity. Furthermore, the SNR of the WFS is always less than that of the imaging camera. When the SNR of the WFS is 5, the SNR of the imaging camera is 9. This relationship indicates that the energy used for the imaging camera is superfluous and that more energy should be distributed to the WFS when a weaker object is detected.

Upon replacing the PBS with the LWPF, the variation in the SNR was measured for the waveband-splitting method, as shown in Fig. 9 , which indicates that the SNR of the WFS is almost equal to that of the imaging camera, whereas the normalized intensity increases gradually. These results show that the energy is split reasonably by the waveband-splitting method, which indicates that the method is suitable for the open-loop LC AOS. An open-loop LC AOS with the waveband-splitting method was then designed for a 1.2-meter telescope. This design will be described in the next section.

#### 4.2 Observations of stars with adaptive correction on the 1.2-meter telescope

Using the waveband-splitting method, an open-loop LC AOS was designed and constructed for a 1.2-meter telescope. Both LC WFCs (FP256, BNS) operated with 256 × 256 pixels at a 500 Hz frame rate. The S–H WFS had a 10 × 10 microlens array and a 930 Hz acquisition frequency. The imaging camera was also an EM CCD camera (DV897, Andor). To correct the tilt aberration, a tip-tilt mirror (S330, PI) was added to the LC AOS which had a resonant frequency of 2.4 KHz. The entire waveband was split into two sub-wavebands, namely, 350 nm to 700 nm and 700 nm to 950 nm, which were used to perform wavefront detection and imaging, respectively. The LC AOS was operated using open-loop control. This control method was described in detail in our previous studies [7, 10, 16–19]. This novel LC AOS was installed on a 1.2-meter telescope in June 2011.

An adaptive correction experiment was performed at nighttime on 19 June 2011. Before performing the correction, the Greenwood frequency and atmospheric coherence length were measured as 45 Hz and 7 cm, respectively. A star, SAO 9366 (4.45 visual magnitude), was selected as the observation object. The bandwidth of the LC AOS can reach up to 37 Hz, and the system was operated using open-loop control. Figure 10 shows the image of SAO 9366 with and without adaptive correction. The blurry image changes to a small spot with the correction. The change in the normalized intensity of SAO 9366 is shown in Fig. 11 , which indicates that the peak intensity increased after correction and that the full width at half maximum is 8 pixels. Given that the effective focal length of the telescope is 84 m and the pixel size of the imaging camera is 16 μm, a pixel corresponds to the angular resolution of 0.039″. Therefore, using adaptive correction, the angular resolution of the 1.2-meter telescope can be calculated as 0.31″ or 1.7 times the diffraction limit, which is 0.18″ at 850 nm for the 1.2-meter telescope.

An adaptive correction experiment was also performed to resolve a double star. The double star Diadem (α Com) had an angular separation of 0.6″, and the visual magnitudes of the two components were 4.85 and 5.53, respectively. The images of the double star with and without open-loop adaptive correction are shown in Fig. 12 . The two components of the double star are clearly resolved after correction.

## 5. Discussions and conclusions

Although the waveband-splitting method improved energy efficiency, the angular resolution did not reach the diffraction limit of the telescope primarily because of the WFS sampling, LC WFC fitting, and system delay errors. According to our previous study, the LC WFC fitting error is approximately 0.06 λ (λ = 800 nm). The WFS sampling error is computed as 0.035 λ. The variance in the system delay error can be calculated by

*f*is the Greenwood frequency, and

_{G}*f*is the −3 dB bandwidth of the LC AOS. Given that

_{3dB}*f*and

_{3dB}*f*are 37 Hz and 45 Hz, respectively, the root mean square (RMS) of the temporal delay error may be calculated as 0.19 λ.

_{G}Therefore, the whole error can be calculated by

where ${\sigma}_{WFS}$and ${\sigma}_{WFC}$are the WFS sampling and LC WFC fitting errors, respectively. According to Eq. (12), the whole RMS error of the LC AOS is calculated as 0.2 λ, and the Strehl ratio can then be computed as 0.195. The achievable angular resolution of the telescope can be expressed approximately as the function of the Strehl ratio [23] aswhere is*λ*the wavelength,

*D*is the telescope aperture, and

*S*represents the Strehl ratio. According to Eq. (13), the achievable resolution is 0.36″ for our LC AOS. This angular resolution is close to the measured value of 0.31″.

According to this analysis, the residual error is mainly caused by temporal delay. In the future, attention will be directed toward the improvement of the bandwidth of the LC AOS. The frame rate of the LC WFC must be improved to obtain higher bandwidth. The slow response speed is mainly attributed to the liquid crystal material. Therefore, an LC material with a faster response should be considered and synthesized to achieve a frame rate of more than 1 kHz.

In summary, a novel energy-splitting scheme was demonstrated for open-loop LC AOS. Compared with the PBS energy-splitting scheme, the waveband-splitting method can split the object energy flexibly, utilize a wider working waveband, avoid the energy loss attributed to polarization dependence, and significantly improve the detection capability of open-loop LC AOS. The calculated results indicate that at an optimal energy ratio of 7:3 between wavefront detection and imaging, the waveband should be split into 350 nm to 700 nm and 700 nm to 950 nm. To validate this assumption, the SNRs of the WFS and the imaging camera are measured for both the waveband-splitting and PBS methods. The experimental results show that for the waveband-splitting method, the SNR value of the WFS is significantly closer to that of the imaging camera while the light intensity changes gradually. However, the SNR of the WFS and the imaging camera differ significantly in the PBS method.

The waveband-splitting method is proven to be an optimal energy-splitting scheme. An adaptive correction experiment was performed on a 1.2-meter telescope to validate this novel design. A star with a 4.45 visual magnitude was observed. The results indicate that the angular resolution improved to 0.31″ after adaptive correction, which is 1.7 times the diffraction limit of the telescope. Finally, the double star Diadem, which has a combined visual magnitude of 4.3 and two components with an angular distance of 0.6″, was observed and corrected in real-time. After correction, the two components were clearly resolved. All the results illustrate that the waveband-splitting method can greatly improve the detection capability of open-loop LC AOS. Therefore, this novel design is very effective for future astronomical observations.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China, with Grant Nos. 60736042, 11174274, and 11174279.

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