Issue 
A&A
Volume 530, June 2011



Article Number  A79  
Number of page(s)  5  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201116835  
Published online  13 May 2011 
Phase sensor for solar adaptiveoptics
Big Bear Solar Observatory, 40386 North Shore Lane, Big Bear City, CA 923149672, USA
email: kellerer@bbso.njit.edu
Received: 5 March 2011
Accepted: 13 April 2011
Context. Wavefront sensing in solar adaptiveoptics is currently done with correlating ShackHartmann sensors, although the spatial and temporalresolutions of the phase measurements are then limited by the extremely fast computing required to correlate the sensor signals at the frequencies of daytime atmosphericfluctuations.
Aims. To avoid this limitation, a new wavefrontsensing technique is presented, that makes use of the solar brightness and is applicable to extended sources.
Methods. The wavefront is sent through a modified MachZehnder interferometer. A small, central part of the wavefront is used as reference and is made to interfere with the rest of the wavefront.
Results. The contrast of two simultaneously measured interferencepatterns provides a direct estimate of the wavefront phase, no additional computation being required. The proposed optical layout shows precise initial alignment to be the critical point in implementing the new wavefrontsensing scheme.
Key words: instrumentation: adaptive optics / Sun: general
© ESO, 2011
1. Introduction
Adaptiveoptic corrections for solar observations are currently based on ShackHartmann (SH) wavefront sensors, see e.g. Rimmele (2004). Such sensors divide the wavefront by an array of lenslets, and each wavefront element forms a separate image on the detector. Local wavefront slopes are inferred from the image positions. When the source is a distant star, the images are small discs (either diffraction or seeinglimited) and the imagecenters are computed through a barycenter calculation. For solar adaptiveoptics, however, the image behind each lenslet is extended and the displacements must, therefore, be computed through a crosscorrelation algorithm.
With solar observations the number of lenslets is not limited by flux considerations, as with the much fainter nighttime targets, but by computational constraints. Even with the fastest available computers the calculations need to be optimized to meet the necessary correctionrates of ~2 kHz. Thus, at the New Solar Telescope (NST) in Big Bear, California, the data from the 76 subaperture SHsensor are analyzed with digital signal processors (DSP), programmed in assembly language, and the correlation is reduced to displacements within ± 3 pixels, see e.g. Rimmele et al. (2004), Richards et al. (2004, 2008) and Denker et al. (2007). Suitable algorithms are currently being developed for the planned upgrade to 308 subapertures. At the German Vacuum Tower Telescope (VTT) in Tenerife, Spain, the use of fast Fourier transforms permits correction frequencies of 2100 Hz on a total of 36 subapertures. For the GREGOR telescope, Tenerife, Spain, the same team develops algorithms to sense the wavefronts at 2500 Hz on 156 subapertures (Berkefeld et al. 2010).
A new wavefrontsensing technique is presented here. In this approach the phase is inferred without the need for a crosscorrelation algorithm, and the computation requirements are greatly relaxed. A small, central part of the wavefront is used as reference and is made to interfere with the rest of the wavefront. A phase shift of π/2 is introduced in one arm of the interferometer, and the two beams are then recombined via a beamsplitter cube. The wavefronts interfere with a phase difference from their common meanvalue which varies around π/2 in one output, and − π/2 in the other output. The normalized intensitydifference between the two outputs is a direct measure of the wavefront phase. No additional computations are required.
The method makes use of the Sun’s brightness and is applicable to extended sources. Since all phases are measured relative to the same, central phasevalue, the error propagation is independent of the number of subapertures.
2. Formalism
2.1. Quantity measured with the sensor
Let E be the electric field associated with the wavefront: (1)The wave propagates along the zaxis in the orthogonal coordinate system (x,y,z). k = 2π/λ is the wave number, λ the wavelength. ω = 2πc/λ denotes the angular frequency, c being the speed of light. φ is the wavefront phase and P the pupil transmission function. We consider a circular pupil of diameter D: (2)In the proposed method, the wavefront is sent through a modified MachZehnder interferometer: the central part of the wavefront with diameter d = D/N is propagated along path (A), the complement is propagated along path (B).

In path (A), an achromatic afocal system enlarges the centralwavefront from diameter d to D = N d. Behind the afocal system the electric field equals: (3)

Path (B) introduces an achromatic π/2 phaseshift relative to path (A), and attenuates the amplitude of the electric field by a factor N^{2} in order to make it comparable to path (A): (4)P′ is the transmission function in path (B): P′(x,y) = P(x,y) − P(x N,y N).
The two beams are recombined via a beam splitter, and the electric fields in the two outputs are then expressed by: (5)The additional phasedelay, π, for the field propagating from path (B) into output (2) is introduced upon reflection on the front of the beamsplitter mirror: the medium behind the mirror is glass, and has a higher refractive index than the air the field is traveling in. The field propagating from path (A) into output (1) reflects on the mirrorback, inside the glass of the beamsplitter and no additional phase shift is thus introduced.
The resulting intensities are: (6)These expressions do not apply inside the central disc of diameter d, where the field E_{B} has zero amplitude and where both intensities equal: . In the following, only those points (x,y) are considered where P′(x,y) = 1.
The central wavefront element – sent through path (A) – is used as reference. We therefore make the following approximation: (7)\label{eq:I12}So that, (8)Under the condition of small phase distortions sin(φ) ~ φ, and the intensitycontrast equals: (9)The phase at position (x,y) is inferred from only two intensity measurements, I_{1}(x,y) and I_{2}(x,y). The error propagation from measurement to phaseestimate is thus independent of the number of measurement points (subapertures), while it increases logarithmically with the number of subapertures when SHsensors are employed (Hudgin 1977; Fried 1977; Kellerer & Kellerer 2011).
2.2. Spectral bandwidth
Turbulenceinduced pathlength distortions, δ, are achromatic, hence phase distortions are inversely proportional to the wavelength: φ(x,y,λ) = 2π δ(x,y)/λ. If the spectral range of the phase sensor is [λ_{0},λ_{0} + Δλ ] , Eq. (9) becomes: (10)Wavefront sensing in solar adaptiveoptics is typically done inside a 50 nm wide spectralband centered around 550 nm, in this case . This band is assumed for our sensor.
3. Implementation of the phase sensor
3.1. Optical layout
Figure 1 sketches the optical layout as it would be used in the case of a point source. Along path (A) the central part of the wavefront is enlarged from its initial diameter d to the pupil diameter D = N d. Along path (B) an achromatic phaseretarder introduces a π/2 phaseshift relative to path (A). An absorption plate attenuates the fieldamplitude by a factor N^{2} in order to approximately equalize the amplitudes in the two paths. Two delay lines compensate for the increment in pathlength introduced by the achromatic afocal system in path (A).
The beams are recombined via a beamsplitter. For a flat incoming wavefront, the phase difference upon recombination is π/2 in output (1) and − π/2 in output (2). For a distorted wavefront – and if the phase variance of the reference wavefront (path A) is negligible – the phase difference upon recombination equals (φ(x,y) + π/2) in output (1) and (φ(x,y) − π/2) in output (2).
The meanphase difference introduced by a pathlength difference, dL = L_{B} − L_{A}, between paths A and B equals: dφ = ⟨ φ_{B} ⟩ − ⟨ φ_{A} ⟩ = 2π·dL/λ. A pathlength difference controlled within ~ λ/20 ensures that the instrumentally induced phase differences are well below π/2, and that the phase differences vary around π/2 and − π/2 in the two outputs.
Fig. 1 Optical layout of the phase sensor. Path (A) propagates the central part of the wavefront. The achromatic afocal system (L_{1,a},L_{2,a}) enlarges the wavefront diameter from d to D = N d. The two delaylines compensate for the pathlength increment due to the afocal system in path (A). 

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The method is not actually intended for application to point sources because, on the one hand, the intensity attenuation factor 1/N^{2} may then not be acceptable and, on the other hand, because any remaining phase fluctuations within the central wavefront segment may, with point sources, make the broadened beam in path (A) insufficiently uniform. As will be seen, both limitations are less critical in solar observations.
3.2. Adjustment for an extended object
To allow for the image extension in solar observations, the optical layout is adjusted so that the interference patterns from sources in different directions will be properly superimposed (see Fig. 2):

Mirror M_{1}, which sends the central part of the wavefront into path (A), is placed in a pupil plane, where the wavefronts from different angulardirections overlap. This ensures that the central parts of all wavefronts are picked up by M_{1}.

The afocal system (L_{1,a},L_{2,a}) is adjusted to reimage the pupil plane (M_{1}) onto the two detectors (P_{1} and P_{2}). This requires L_{1,a} and L_{2,a} to be convergent lenses, and M_{1} to be placed between L_{1,a} and the focal plane of L_{1,a}. If L_{1,a} were divergent and L_{2,a} convergent, the image of M_{1} through (L_{1,a},L_{2,a}) would be virtual, and could, thus, not be projected onto the detector planes, P_{1} and P_{2}.

Another afocal system (L_{1,b},L_{2,b}) is placed in path (B) to reimage the pupilplane, M_{1}, onto the two detectors, P_{1} and P_{2}.
With the adjusted optical layout, the measured contrast equals the intensity weighted phaseaverage over a solidangle, Ω: (11)L(θ) [m^{2} s^{1} sr^{1}] is the solar radiance from the angular direction θ.
Adaptive optics with a single deformable mirror is in principle limited to acceptance angles within the isoplanatic diameter (α = 10′′ for typical daytime observations). For larger acceptance angles, Ω > πα^{2} and the correction can merely account for the mean phase shift over Ω. Since this mean value is predominantly determined by the nearby turbulence, the resulting approximate correction is termed groundlayer adaptive optics. A second condition for exact adaptive optics is the sufficiently close spacing of actuators. Their separation should correspond to not more than the Fried distance (roughly 0.05 m for typical daytime observations), otherwise the correction accounts again merely for a mean phase shift.
With a point object, the central subaperture contains, depending on its size, a certain level of phase differences. Ideally one would wish to average out these differences in the widened reference beam (A) to make it as uniform as possible. While this can not be done, a somewhat increased angle of acceptance will have the similar effect of replacing the phase shifts at each point by a local average which makes the central beam more uniform. The proposed method is, thus, inherently suited for extended images.
Fig. 2 Adjustment of the optical layout (Fig. 1) for an extended source such as the Sun. Both afocal systems, (L_{1,a},L_{2,a}) and (L_{1,b},L_{2,b}), reimage the pupil plane, M_{1}, onto the detector planes, P_{1} and P_{2}. 

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Fig. 3 Left panel: matrix of phase values with 0.1 rad rms deviation. The phase distribution follows the Kolmogorov statistics. Right panel: quantity measured by the phase sensor: (I_{1} − I_{2})/(I_{1} + I_{2}). 

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4. Sensitivity of the phase sensor
4.1. Maximum apparent magnitude
Assume that sensor areas equal the size of the central subaperture. The broadened beam from the central subaperture is then required to interfere with the N^{2} − 1 other areas. The number, Z, of photons per detector readout must, thus, be sufficiently high: (12)I [m^{2} s^{1}] is the target’s irradiance, S the collecting surface of the central subaperture and Δt is the exposure time. Z_{0} denotes the minimum number of photons for the interference in each subarea. For Z_{0} = 100 the signal to photonnoise ratio stays below 10%.
Initial calculations (see Sect. 4.2) suggest that the rootmeansquare (rms) phasedeviation, σ_{φ}, over the whole wavefront needs to remain within ~ 1.2 rad. The central part of the wavefront can serve as a reference if its rms phasedeviation, σ_{0}, is substantially smaller than σ_{φ}, say σ_{0} < σ_{φ}/5. Accordingly, the diameter of the region on the telescope aperture that corresponds to the central subaperture should ideally not exceed a fraction of the Fried length, r_{0} (Roddier 1981): (13)with the diameter of the NST telescope, D = 1.6 m, and a Fried length r_{0} = 0.05 m, we then get: d = 0.01 m and N = D/d = 168. Daytime adaptiveoptics correction is typically done at a frequency of f = 1/Δt = 2 kHz, so that: (14)and the target’s apparentmagnitude needs to be below: (15)I_{0} = 5 × 10^{9} m^{2} s^{1} equals the irradiance from a star with zero apparent magnitude inside a 50 nm wide spectralband centered around 550 nm (Allen’s Astrophysical Quantities 2000).
When the wavefront sensing is done inside the isoplanatic cone (typically α = 10′′), the corresponding apparent magnitude of the Sun equals: m = −26.7 + 2.5log (0.5·3600/10) = −21.1. The number of photons received in the central subaperture is thus by far sufficient.
4.2. Maximum amplitude of the phasedistortions
In exploratory computations different 288 × 288 matrices of phase values have been generated with Kolmogorov statistics, and the normalized intensitydifferences, (I_{1} − I_{2})/(I_{1} + I_{2}), have been calculated in terms of Eqs. (3)–(9). The central 32 × 32 elements were taken to represent the reference wavefront that is propagated through path (A) of the interferometer. The normalized intensity differences are shown in Figs. 3–5 for different phase screens, with rms phasedeviations between σ_{φ} = 0.1 rad and 1.2 rad. σ_{φ} = 1.2 rad appears to be a limit: for larger phase distortions, sin(φ) ~ φ fails to be a valid approximation, and the intensity difference ceases to be proportional to the phase distortion. Instead (see Eq. (2.1): (16)The phase values need then to be unwrapped, and the sensing scheme looses its advantage of minimal computational requirements. For a D = 1.6 m telescope and a Fried length r_{0} = 0.05 m, the turbulenceinduced rms phasedistortions equal in fact: σ = (D/r_{0})^{5/6} = 18 rad (Roddier 1981). One must, therefore, take into account that the phase differences between beams (A) and (B) are determined only modulo 2π which requires an initial computational procedure which might start out the adaptiveoptics correctionloop on a central subset of the sensor field and progressively extend the set as the correction takes effect, this extension being either algorithmically coded or being controlled by operating a diaphragm in the collimated beam. Once an approximate phase correction is achieved the precise correction can proceed without further computational effort at the desired frequency.
Fig. 4 Same as in Fig. 3, with a different realization of phase values. The rms phase deviation equals 0.6 rad. 

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Fig. 5 Same as in Fig. 3, with a different realization of phase values. The rms phase deviation equals 1.2 rad. 

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5. Conclusion
A scheme is presented to sense wavefront distortions in observations of bright, extended sources. The central part of the wavefront is used as reference and is made to interfere with the rest of the wavefront by use of a suitably modified MachZehnder interferometer. The intensity contrast in the two outputs of the interferometer is a direct measure of the wavefrontphases. Once an approximate phase adjustment is attained the continued exact correction requires no additional calculation, i.e. the computation requirements are greatly relaxed in comparison to the current use of correlating SHsensors. The most critical step in the implementation of the new wavefrontsensing method is likely to be the initial alignment of the MachZehnder interferometer, since the path lengths in both arms of the interferometers need to be equalized within λ/20 (see Sect. 3.1). This, as well as the suitable control strategy will need to be explored in an experimental setup.
The phase differences are deduced from the contrast of two simultaneously measured intensities and the estimates are accordingly not affected by scintillation. Furthermore, all phases are measured relative to the same, central phasevalue, and the errorpropagation factor is therefore independent of the number
of sampling elements. This contrasts with the logarithmic increase of the propagation factor when increasing numbers of SH subapertures are used.
The method requires the flux in the central subaperture to be sufficiently high, and is thus especially suited for solar adaptiveoptics.
Acknowledgments
Many thanks go to Nicolas Gorceix for numerous discussions on solar wavefront sensing. The National Science Foundation is also gratefully acknowledged for funding this research through grant NSFAST0079482.
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All Figures
Fig. 1 Optical layout of the phase sensor. Path (A) propagates the central part of the wavefront. The achromatic afocal system (L_{1,a},L_{2,a}) enlarges the wavefront diameter from d to D = N d. The two delaylines compensate for the pathlength increment due to the afocal system in path (A). 

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In the text 
Fig. 2 Adjustment of the optical layout (Fig. 1) for an extended source such as the Sun. Both afocal systems, (L_{1,a},L_{2,a}) and (L_{1,b},L_{2,b}), reimage the pupil plane, M_{1}, onto the detector planes, P_{1} and P_{2}. 

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In the text 
Fig. 3 Left panel: matrix of phase values with 0.1 rad rms deviation. The phase distribution follows the Kolmogorov statistics. Right panel: quantity measured by the phase sensor: (I_{1} − I_{2})/(I_{1} + I_{2}). 

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In the text 
Fig. 4 Same as in Fig. 3, with a different realization of phase values. The rms phase deviation equals 0.6 rad. 

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In the text 
Fig. 5 Same as in Fig. 3, with a different realization of phase values. The rms phase deviation equals 1.2 rad. 

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In the text 
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