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Results of the JOSE site evaluation project for adaptive optics at the William Herschel Telescope
New Astronomy Reviews 42 (1998) 465–469
Results of the JOSE site evaluation project for adaptive optics at the William Herschel Telescope R. Wilson Royal Greenwich Observatory, Madingley Road, Cambridge, CB3 0 EZ, UK
Abstract Results are presented from a long-term study of the seeing properties at the William Herschel Telescope on La Palma. The measurements have been made over a two-year period using a Shack–Hartmann wavefront sensor equipped with high frame-rate CCD camera. The aim of the campaign is to characterize those aspects of the seeing relevant to the design and performance of astronomical adaptive optical systems for the WHT. Statistical results are presented for the value of Fried’s parameter, power spectra of Zernike mode coefficients, isoplanatism and the outer scale of turbulence. 1998 Elsevier Science B.V. All rights reserved.
1. Introduction JOSE is an acronym for the Joint Observatories Seeing Evaluation project, initiated by the United Kingdom adaptive-optics (AO) programme. Its aims are to obtain statistical data on the spatial and temporal properties of turbulence-induced wavefront distortions at the telescope focus. The JOSE seeing monitor at the WHT is a Shack–Hartmann wavefront sensor mounted in the GHRIL facility at the Nasmyth focus of the telescope. An image of the telescope pupil is projected on to an 8 3 8 grid of sub-apertures on the Shack–Hartmann lenslet array. The sensor is equipped with a fast read-out CCD allowing frame rates of 100 Hz or faster. This provides good sampling of temporal fluctuations of the wavefront on the relevant spatial scales. The design and operation of the JOSE instrument is described in detail elsewhere (St. Jacques et al., 1997). A typical JOSE observation comprises 10000 sequential CCD frames, acquired in | 100 s. More than 1200 such observations have been made on 60 nights between 1995 May and 1997 November. Observations have been made in all seasons and at
all times of the night, thereby providing good statistical sampling of the parameters of interest. Frame-by-frame analysis of the data proceeds as follows: centroids are calculated for all (unvignetted) spots in the Shack–Hartmann pattern, to define the local mean wavefront tilt over the corresponding sub-apertures on the telescope pupil. Centroids are measured relative to the mean spot location, so that static perturbations are ignored. The phase distortion over the telescope pupil is then expressed as a weighted sum of the classical Zernike aberration functions, via a least-squares fit to the set of local phase gradients. Analysis of the wavefront properties is performed in terms of the variances, power spectra and correlations of the measured mode coefficients.
2. Statistics of r 0 Noll (1976) calculated the variances of the Zernike mode coefficients for the Kolmogorov model of atmospheric turbulence, i.e. assuming a 5 / 3 power law for the phase structure function. The lowestorder modes, representing wavefront tilts over the
1387-6473 / 98 / $ – see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 98 )00054-2
466
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
Fig. 1. Estimation of r 0 via a least-squares fit of the rms Zernike mode coefficients to their theoretical values assuming Kolmogorov turbulence.
whole pupil, have the largest rms values and contribute 87% of the phase variance in atmospheric distortions. Higher-order modes carry much less phase variance. The mode variances are proportional /3 to r 25 , where r 0 is Fried’s parameter, the atmos0 pheric coherence length. By comparing measured mode variances with their theoretical (Noll) values, we can assess the validity of the 5 / 3 scaling, and determine the value of r 0 . This analysis is demonstrated in Fig. 1. Here the measured rms coefficients are plotted against their theoretical values for r 0 5 17.3 cm. This value of r 0 minimizes the least-squares difference in the logarithm of the measured and theoretical values. Note that the first order (n 5 1) modes (wavefront tilts) have been excluded from the fit, since their values may be affected by telescope tracking errors. In this case an excellent fit is obtained, with all of the data points lying close to the line y 5 x, indicating that 5 / 3 scaling applies. A good fit is also obtained for the majority of the JOSE data set, so that in general r 0 has a well defined value. This in itself is a useful result for adaptive optics, since it follows that one can predict with confidence the spatial order of correction (e.g. the number of corrected Zernike terms) which will be required to achieve a given image quality (Wilson & Jenkins, 1996). Fig. 2 is the probability distribution of r 0 for the JOSE data set. If we make the assumption of Kolmogorov turbulence, then the expected image FWHM (V ) for a large telescope is given by V 5 l /
Fig. 2. Probability distributions for r 0 (at 500 nm) from the JOSE (La Palma) and PUEO (Mauna Kea) data.
r 0 . This yields a predicted seeing distribution for the WHT, which is presented elsewhere in these proceedings (Packham, 1998), and has a median value of 0.6599 when corrected to the zenith. This is in good agreement with measurements from the ING DIMM sited close to the WHT, implying that there is no significant contribution to the seeing from within the WHT dome. The predominance of a 5 / 3 power law in the JOSE data supports this conclusion, since we would not expect local (dome or mirror) turbulence to have a Kologorov spectrum; for example Bridgeland & Jenkins (1997) predict a 2 / 3 scaling for the structure function of turbulence above a warm telescope mirror). Also plotted in Fig. 2 is the probability distribution for r 0 measurements made at the 3.6-m Canada-France-Hawaii Telescope on Mauna Kea by Rigaut et al. (1998). These values were also obtained via a fit to the Noll coefficients, for data obtained using the WFS of the PUEO adaptive optics system on 16 nights throughout 1996. The distributions and median values are similar, although there are somewhat more occurrences of poor seeing in the JOSE data.
3. Wavefront power spectra The temporal properties of atmospheric wavefront distortions are crucial in determining the performance and required closed-loop bandwidth for an astronomical adaptive-optics system. By measuring the power spectra of the Zernike modes of the phase
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
Fig. 3. Example measured (JOSE) and best-fit theoretical power spectra of atmospheric defocus.
perturbation, we can determine the AO system bandwidth required to make corrections to a given spatial order. Fig. 3 shows an example of the power spectrum of atmospherically induced defocus (Noll j 5 4), from the JOSE data. Also plotted is a theoretical power spectrum, which assumes that the turbulence can be modelled as a single phase screen, with a Kolmogorov structure function, moving over the telescope at wind velocity Vw . In this case the power spectrum is determined entirely by r 0 and Vw (Roddier et al., 1993). In general the observed power spectra cannot be adequately represented by a singlelayer model, and at least two contributions at different wind speeds are normally required. For adaptive optics the parameters of interest are the high-frequency cut-offs in the modal power spectra, since these will determine the required AO system bandwidth. The sharp cut-offs characteristic of the theoretical power spectra are rarely seen in practice, being smoothed by the addition of multiple wind-velocity contributions. Here the cut-off will be defined as the frequency for which 80 per cent of the total variance for a given mode is included. Fig. 4 shows histograms of this cut-off frequency for increasing Zernike orders, from the JOSE measurements. The required frequency increases with spatial order as expected, but is still less than 8 Hz for fifth-order corrections in most cases. This implies relatively low sampling rates ( , 100 Hz) for AO corrections to this order. This is typically the order of correction required to deliver high Strehl ratios for
467
Fig. 4. Statistics of cut-off frequency bounding 80 per cent of the mode variance, for first-order (solid lines), third-order (broken lines) and fifth-order (dotted lines) Zernikes. For the fifth-order case (n 5 5), the ‘‘effective’’ wind speed corresponding to the median value is indicated. This is the velocity of a single Kolmogorov–Taylor phase screen which would give the same cut-off frequency.
imaging in the K band on a 4-m telescope in good seeing conditions.
4. Isoplanatism The isoplanatic angle is another crucial parameter for AO. The modal power spectra determine the required system bandwidth, and hence the limiting magnitude of an AO system. Angular anisoplanatism
Fig. 5. Example of the modal correlations for binary stars of 1699 and 3599 separation, measured with the JOSE sensor.
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
468
Table 1 Measured median Zernike correlation versus radial order and binary angular separation (u ) Radial order
u (arcsec)
(n)
15–25
25–40
40–65
1 2 3 4 5
0.98 0.92 0.86 0.81 0.74
0.97 0.86 0.80 0.76 0.67
0.94 0.80 0.72 0.62 0.58
determines the search radius for guide stars. Hence, together these define the system sky coverage. For the Zernike modes, the rate of decorrelation with angle increases with their spatial order, and with the height of the turbulent layers (Chassat, 1989). The angular decorrelation of the Zernikes has been investigated by observing binary stars with the JOSE sensor. Fig. 5 shows examples of the modal correlations observed for 1699 and 3599 binary stars on the same night. An important limit is the field angle yielding 50 per cent correlation or less. If we attempt to use a guide star at this separation or greater we will add phase variance to the corrected wavefront. Table 1 shows statistics of the correlations observed for binaries of increasing separation for Zernike radial orders from 1 to 5. Observations of binary stars have so far been possible on only 12 nights, so that statistics are limited. Hence the observations have been grouped into three broad ranges of angular separation. We see that tilts (n 5 1) remain highly correlated for large angles, whereas n 5 5 corrections become marginal for separations of | 1 arcmin. The median correlations for higher-order Zernikes (n 5 3–5) have fallen to approximately 0.8 for small binary separations ( | 2099) but then decline relatively slowly as the field angle increases further. This suggests that although the bulk of the optical turbulence is at low altitudes, there is a significant contribution from higher layers. For example, the median correlations can be approximately reproduced by a crude model in which 75 per cent of the total turbulence strength is contributed by a layer at 1 km altitude, with the remaining 25 per cent in a second layer at 8 km.
5. The outer scale of turbulence The size of the outer scale of turbulence (L0 ) has a strong influence on the variance of the image motion (corresponding to Zernike n 5 1), and hence the seeing angle for an astronomical telescope (Jenkins, 1998). In contrast, the variances of higher-order Zernike modes (n . 2) remain close to their Noll (infinite outer scale) values unless the outer scale is of the order of the telescope aperture diameter (Winker, 1991). Hence in theory a measure of the outer scale is given by the variance of the n 5 1 modes relative to the higher orders. In practice it is difficult to distinguish between atmospherically-induced image motions and those due to telescope tracking errors, so that this method does not yield a reliable value for L0 . For the JOSE data the median ratio of the variance for the n 5 1 terms to their theoretical values is 0.55 (given by fitting a Kolmogorov spectrum to the higher-order modes). This suggests a median outer scale of | 100 m if telescope effects are ignored. Here L0 is as defined by Winker (1991) and Jenkins (1998) based on a modified Von-Karmen turbulence spectrum. If the outer scale is less than | 20 telescope diameters then there will also be a small but measurable decrease in the variance of the second-order Zernike functions (defocus and astigmatism) relative to higher-order modes. Measurements of the secondorder terms are not biased by telescope effects and should therefore provide an unbiased estimate of L0 . Fig. 6 shows the ratios of the rms values for n 5 2 and n 5 5 modes, for the JOSE data. These suggest a median outer scale of approximately 20 m. For this value of L0 the theoretical seeing angle for the 4.2-m WHT is significantly smaller than the value predicted by the DIMM, or by JOSE when an infinite outer scale is assumed. The predicted median seeing is reduced from 0.6599 to approximately 0.5599.
Acknowledgements The JOSE project is funded as part of the UK adaptive-optics programme of the Particle Physics and Astronomy Research Council. The JOSE monitor at the WHT was constructed by Graham Cox
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
469
Fig. 6. Ratios of the measured rms coefficients for second- and fifth-order Zernikes, for the JOSE data set. Theoretical ratios for four values of the L0 are indicated.
and David StJacques at the Mullard Radio Astronomy Observatory. I am grateful to Charles Jenkins and David StJacques for numerous discussions, and to Neil O’Mahony, Nic Walton, Chris Packham and the telescope operators at the ING for their assistance with the JOSE project. The William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Roque de los Muchachos Observatory of the Instituto ´ de Astrofısica de Canarias.
References Bridgeland, M.T. & Jenkins, C.R., 1997, MNRAS, 287, 87.
Chassat, F., 1989, J. Opt., 20, 13. Jenkins, C.R., 1998, MNRAS, in press. Noll, R.J., 1976, J. Opt. Soc. Am., 66, 207. Packham, C., 1998, NewAR, 42, these proceedings. Rigaut, F., et al., 1998, PASP, in press. Roddier, F., Northcott, M.J., Graves, J.E., Mckenna, D.L., & Roddier, D., 1993, J. Opt. Soc. Am., 10, 957. St. Jacques, D., Cox, G.C., Baldwin, J.E., Mackay, C.D., Waldram, E.M., & Wilson, R.W., 1997, MNRAS, 290, 66. Wilson, R.W. & Jenkins, C.R., 1996, MNRAS, 268, 39. Winker, D.M., 1991, J. Opt. Soc. Am., 8, 1568.
Results of the JOSE site evaluation project for adaptive optics at the William Herschel Telescope R. Wilson Royal Greenwich Observatory, Madingley Road, Cambridge, CB3 0 EZ, UK
Abstract Results are presented from a long-term study of the seeing properties at the William Herschel Telescope on La Palma. The measurements have been made over a two-year period using a Shack–Hartmann wavefront sensor equipped with high frame-rate CCD camera. The aim of the campaign is to characterize those aspects of the seeing relevant to the design and performance of astronomical adaptive optical systems for the WHT. Statistical results are presented for the value of Fried’s parameter, power spectra of Zernike mode coefficients, isoplanatism and the outer scale of turbulence. 1998 Elsevier Science B.V. All rights reserved.
1. Introduction JOSE is an acronym for the Joint Observatories Seeing Evaluation project, initiated by the United Kingdom adaptive-optics (AO) programme. Its aims are to obtain statistical data on the spatial and temporal properties of turbulence-induced wavefront distortions at the telescope focus. The JOSE seeing monitor at the WHT is a Shack–Hartmann wavefront sensor mounted in the GHRIL facility at the Nasmyth focus of the telescope. An image of the telescope pupil is projected on to an 8 3 8 grid of sub-apertures on the Shack–Hartmann lenslet array. The sensor is equipped with a fast read-out CCD allowing frame rates of 100 Hz or faster. This provides good sampling of temporal fluctuations of the wavefront on the relevant spatial scales. The design and operation of the JOSE instrument is described in detail elsewhere (St. Jacques et al., 1997). A typical JOSE observation comprises 10000 sequential CCD frames, acquired in | 100 s. More than 1200 such observations have been made on 60 nights between 1995 May and 1997 November. Observations have been made in all seasons and at
all times of the night, thereby providing good statistical sampling of the parameters of interest. Frame-by-frame analysis of the data proceeds as follows: centroids are calculated for all (unvignetted) spots in the Shack–Hartmann pattern, to define the local mean wavefront tilt over the corresponding sub-apertures on the telescope pupil. Centroids are measured relative to the mean spot location, so that static perturbations are ignored. The phase distortion over the telescope pupil is then expressed as a weighted sum of the classical Zernike aberration functions, via a least-squares fit to the set of local phase gradients. Analysis of the wavefront properties is performed in terms of the variances, power spectra and correlations of the measured mode coefficients.
2. Statistics of r 0 Noll (1976) calculated the variances of the Zernike mode coefficients for the Kolmogorov model of atmospheric turbulence, i.e. assuming a 5 / 3 power law for the phase structure function. The lowestorder modes, representing wavefront tilts over the
1387-6473 / 98 / $ – see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 98 )00054-2
466
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
Fig. 1. Estimation of r 0 via a least-squares fit of the rms Zernike mode coefficients to their theoretical values assuming Kolmogorov turbulence.
whole pupil, have the largest rms values and contribute 87% of the phase variance in atmospheric distortions. Higher-order modes carry much less phase variance. The mode variances are proportional /3 to r 25 , where r 0 is Fried’s parameter, the atmos0 pheric coherence length. By comparing measured mode variances with their theoretical (Noll) values, we can assess the validity of the 5 / 3 scaling, and determine the value of r 0 . This analysis is demonstrated in Fig. 1. Here the measured rms coefficients are plotted against their theoretical values for r 0 5 17.3 cm. This value of r 0 minimizes the least-squares difference in the logarithm of the measured and theoretical values. Note that the first order (n 5 1) modes (wavefront tilts) have been excluded from the fit, since their values may be affected by telescope tracking errors. In this case an excellent fit is obtained, with all of the data points lying close to the line y 5 x, indicating that 5 / 3 scaling applies. A good fit is also obtained for the majority of the JOSE data set, so that in general r 0 has a well defined value. This in itself is a useful result for adaptive optics, since it follows that one can predict with confidence the spatial order of correction (e.g. the number of corrected Zernike terms) which will be required to achieve a given image quality (Wilson & Jenkins, 1996). Fig. 2 is the probability distribution of r 0 for the JOSE data set. If we make the assumption of Kolmogorov turbulence, then the expected image FWHM (V ) for a large telescope is given by V 5 l /
Fig. 2. Probability distributions for r 0 (at 500 nm) from the JOSE (La Palma) and PUEO (Mauna Kea) data.
r 0 . This yields a predicted seeing distribution for the WHT, which is presented elsewhere in these proceedings (Packham, 1998), and has a median value of 0.6599 when corrected to the zenith. This is in good agreement with measurements from the ING DIMM sited close to the WHT, implying that there is no significant contribution to the seeing from within the WHT dome. The predominance of a 5 / 3 power law in the JOSE data supports this conclusion, since we would not expect local (dome or mirror) turbulence to have a Kologorov spectrum; for example Bridgeland & Jenkins (1997) predict a 2 / 3 scaling for the structure function of turbulence above a warm telescope mirror). Also plotted in Fig. 2 is the probability distribution for r 0 measurements made at the 3.6-m Canada-France-Hawaii Telescope on Mauna Kea by Rigaut et al. (1998). These values were also obtained via a fit to the Noll coefficients, for data obtained using the WFS of the PUEO adaptive optics system on 16 nights throughout 1996. The distributions and median values are similar, although there are somewhat more occurrences of poor seeing in the JOSE data.
3. Wavefront power spectra The temporal properties of atmospheric wavefront distortions are crucial in determining the performance and required closed-loop bandwidth for an astronomical adaptive-optics system. By measuring the power spectra of the Zernike modes of the phase
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
Fig. 3. Example measured (JOSE) and best-fit theoretical power spectra of atmospheric defocus.
perturbation, we can determine the AO system bandwidth required to make corrections to a given spatial order. Fig. 3 shows an example of the power spectrum of atmospherically induced defocus (Noll j 5 4), from the JOSE data. Also plotted is a theoretical power spectrum, which assumes that the turbulence can be modelled as a single phase screen, with a Kolmogorov structure function, moving over the telescope at wind velocity Vw . In this case the power spectrum is determined entirely by r 0 and Vw (Roddier et al., 1993). In general the observed power spectra cannot be adequately represented by a singlelayer model, and at least two contributions at different wind speeds are normally required. For adaptive optics the parameters of interest are the high-frequency cut-offs in the modal power spectra, since these will determine the required AO system bandwidth. The sharp cut-offs characteristic of the theoretical power spectra are rarely seen in practice, being smoothed by the addition of multiple wind-velocity contributions. Here the cut-off will be defined as the frequency for which 80 per cent of the total variance for a given mode is included. Fig. 4 shows histograms of this cut-off frequency for increasing Zernike orders, from the JOSE measurements. The required frequency increases with spatial order as expected, but is still less than 8 Hz for fifth-order corrections in most cases. This implies relatively low sampling rates ( , 100 Hz) for AO corrections to this order. This is typically the order of correction required to deliver high Strehl ratios for
467
Fig. 4. Statistics of cut-off frequency bounding 80 per cent of the mode variance, for first-order (solid lines), third-order (broken lines) and fifth-order (dotted lines) Zernikes. For the fifth-order case (n 5 5), the ‘‘effective’’ wind speed corresponding to the median value is indicated. This is the velocity of a single Kolmogorov–Taylor phase screen which would give the same cut-off frequency.
imaging in the K band on a 4-m telescope in good seeing conditions.
4. Isoplanatism The isoplanatic angle is another crucial parameter for AO. The modal power spectra determine the required system bandwidth, and hence the limiting magnitude of an AO system. Angular anisoplanatism
Fig. 5. Example of the modal correlations for binary stars of 1699 and 3599 separation, measured with the JOSE sensor.
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
468
Table 1 Measured median Zernike correlation versus radial order and binary angular separation (u ) Radial order
u (arcsec)
(n)
15–25
25–40
40–65
1 2 3 4 5
0.98 0.92 0.86 0.81 0.74
0.97 0.86 0.80 0.76 0.67
0.94 0.80 0.72 0.62 0.58
determines the search radius for guide stars. Hence, together these define the system sky coverage. For the Zernike modes, the rate of decorrelation with angle increases with their spatial order, and with the height of the turbulent layers (Chassat, 1989). The angular decorrelation of the Zernikes has been investigated by observing binary stars with the JOSE sensor. Fig. 5 shows examples of the modal correlations observed for 1699 and 3599 binary stars on the same night. An important limit is the field angle yielding 50 per cent correlation or less. If we attempt to use a guide star at this separation or greater we will add phase variance to the corrected wavefront. Table 1 shows statistics of the correlations observed for binaries of increasing separation for Zernike radial orders from 1 to 5. Observations of binary stars have so far been possible on only 12 nights, so that statistics are limited. Hence the observations have been grouped into three broad ranges of angular separation. We see that tilts (n 5 1) remain highly correlated for large angles, whereas n 5 5 corrections become marginal for separations of | 1 arcmin. The median correlations for higher-order Zernikes (n 5 3–5) have fallen to approximately 0.8 for small binary separations ( | 2099) but then decline relatively slowly as the field angle increases further. This suggests that although the bulk of the optical turbulence is at low altitudes, there is a significant contribution from higher layers. For example, the median correlations can be approximately reproduced by a crude model in which 75 per cent of the total turbulence strength is contributed by a layer at 1 km altitude, with the remaining 25 per cent in a second layer at 8 km.
5. The outer scale of turbulence The size of the outer scale of turbulence (L0 ) has a strong influence on the variance of the image motion (corresponding to Zernike n 5 1), and hence the seeing angle for an astronomical telescope (Jenkins, 1998). In contrast, the variances of higher-order Zernike modes (n . 2) remain close to their Noll (infinite outer scale) values unless the outer scale is of the order of the telescope aperture diameter (Winker, 1991). Hence in theory a measure of the outer scale is given by the variance of the n 5 1 modes relative to the higher orders. In practice it is difficult to distinguish between atmospherically-induced image motions and those due to telescope tracking errors, so that this method does not yield a reliable value for L0 . For the JOSE data the median ratio of the variance for the n 5 1 terms to their theoretical values is 0.55 (given by fitting a Kolmogorov spectrum to the higher-order modes). This suggests a median outer scale of | 100 m if telescope effects are ignored. Here L0 is as defined by Winker (1991) and Jenkins (1998) based on a modified Von-Karmen turbulence spectrum. If the outer scale is less than | 20 telescope diameters then there will also be a small but measurable decrease in the variance of the second-order Zernike functions (defocus and astigmatism) relative to higher-order modes. Measurements of the secondorder terms are not biased by telescope effects and should therefore provide an unbiased estimate of L0 . Fig. 6 shows the ratios of the rms values for n 5 2 and n 5 5 modes, for the JOSE data. These suggest a median outer scale of approximately 20 m. For this value of L0 the theoretical seeing angle for the 4.2-m WHT is significantly smaller than the value predicted by the DIMM, or by JOSE when an infinite outer scale is assumed. The predicted median seeing is reduced from 0.6599 to approximately 0.5599.
Acknowledgements The JOSE project is funded as part of the UK adaptive-optics programme of the Particle Physics and Astronomy Research Council. The JOSE monitor at the WHT was constructed by Graham Cox
R. Wilson / New Astronomy Reviews 42 (1998) 465 – 469
469
Fig. 6. Ratios of the measured rms coefficients for second- and fifth-order Zernikes, for the JOSE data set. Theoretical ratios for four values of the L0 are indicated.
and David StJacques at the Mullard Radio Astronomy Observatory. I am grateful to Charles Jenkins and David StJacques for numerous discussions, and to Neil O’Mahony, Nic Walton, Chris Packham and the telescope operators at the ING for their assistance with the JOSE project. The William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Roque de los Muchachos Observatory of the Instituto ´ de Astrofısica de Canarias.
References Bridgeland, M.T. & Jenkins, C.R., 1997, MNRAS, 287, 87.
Chassat, F., 1989, J. Opt., 20, 13. Jenkins, C.R., 1998, MNRAS, in press. Noll, R.J., 1976, J. Opt. Soc. Am., 66, 207. Packham, C., 1998, NewAR, 42, these proceedings. Rigaut, F., et al., 1998, PASP, in press. Roddier, F., Northcott, M.J., Graves, J.E., Mckenna, D.L., & Roddier, D., 1993, J. Opt. Soc. Am., 10, 957. St. Jacques, D., Cox, G.C., Baldwin, J.E., Mackay, C.D., Waldram, E.M., & Wilson, R.W., 1997, MNRAS, 290, 66. Wilson, R.W. & Jenkins, C.R., 1996, MNRAS, 268, 39. Winker, D.M., 1991, J. Opt. Soc. Am., 8, 1568.