Diagnostic Support
Nutronics, Inc. has extensive expertise in diagnosis and evaluation of the performance of AO systems. Since his early work while still in the U. S. Air Force, Nutronics, Inc.’s Barchers has utilized the concept of a Strehl or Wavefront Error Budget calculation to enable visualization and quantification of various effects on performance of an adaptive optical system. This methodology has proven useful in many cases for understanding and identifying component and assembly failures and / or problems. In addition, this methodology has been used to establish requirements for various subsystems. This methodology has recently been incorporated directly into a diagnostic instrument (Adaptive Optics Requirements Test Assembly – AORTA – shown in Figure 1) developed and delivered for assessment of a highly complex system. AORTA was successful in identifying a critical error in the use of the AO system and enabling Nutronics, Inc. and the customer to define a path to correct the error. We summarize key aspects of the AORTA diagnostic methodology here.
Figure 1: Adaptive Optics Requirements Test Assembly (AORTA) for evaluation and diagnosis of the performance of an AO system.
The AORTA methodology for evaluation of contributions to performance can be invaluable. This methodology can be applied to wave optical simulation data and / or high resolution wavefront sensor data. Nutronics, Inc. can apply this approach to evaluation of existing high resolution wavefront sensor or we can develop and deliver a high resolution diagnostic sensor tailored to the application of interest. AORTA was developed as a very high end and high performance system – however if spatial or temporal sampling can be sacrificed there are many lower cost options for the wavefront sensor that can be provided. The specifications for the AORTA system are provided in Table 1.
Table 1: AORTA Specifications
Parameter |
Units |
Value |
||
Minimum |
Nominal |
Maximum |
||
GENERAL |
||||
Nominal Input Beam Size (D) |
mm |
2.7136 |
5.4272 |
8.1408 |
Wavelength Range |
nm |
1030 |
1064 |
1100 |
DUAL HARTMANN SENSOR |
||||
Number of Samples Across Beam |
number |
32 |
64 |
96 |
Beam Sample Spacing (d) |
mm |
0.0848 |
0.0848 |
0.0848 |
Pixel Field of View (Minimum) |
λ / d @ 1064 nm |
0.25 |
0.25 |
0.25 |
Pixel Field of View (Maximum) |
λ / d @ 1064 nm |
0.50 |
0.50 |
0.50 |
Subaperture Field of View (Minimum) |
λ / d @ 1064 nm |
2.00 |
2.00 |
2.00 |
Full Aperture Field of View (Minimum) |
λ / d @ 1064 nm |
64.00 |
128.00 |
192.00 |
Calibrated Wavefront Error On-Axis (RMS) |
λ RMS @ 1064 nm |
≤ 0.05 |
≤ 0.05 |
≤ 0.05 |
Fraction of Invalid Subapertures |
Percentage |
≤ 3% |
≤ 3% |
≤ 3% |
Maximum Frame Rate |
Hz |
8000.00 |
3600.00 |
1800.00 |
PUPIL PLANE INTENSITY SENSOR |
||||
Number of Samples Across Beam |
number |
64 |
128 |
192 |
Beam Sample Spacing |
mm |
0.0424 |
0.0424 |
0.0424 |
Number of Bits |
number |
12 |
12 |
12 |
Maximum Frame Rate |
Hz |
12000.00 |
6000.00 |
3000.00 |
FOCAL PLANE INTENSITY SENSOR |
||||
Number of Far Field Samples |
number |
320 |
320 |
320 |
Field of View (Minimum) |
λ / d @ 1064 nm |
32.00 |
64.00 |
96.00 |
Pixel Field of View (Minimum) |
λ / d @ 1064 nm |
0.10 |
0.20 |
0.30 |
Pixel Field of View (Maximum) |
λ / d @ 1064 nm |
0.30 |
0.40 |
0.60 |
Number of Bits |
number |
12 |
12 |
12 |
Maximum Frame Rate |
Hz |
1000.00 |
1000.00 |
1000.00 |
REFERENCE ENCIRCLED ENERGY MONITORS |
||||
Reference Encircled Energy Diameter 1 |
λ / d @ 1064 nm |
1.70 |
3.40 |
5.10 |
Reference Encircled Energy Diameter 2 |
λ / d @ 1064 nm |
4.25 |
8.50 |
12.75 |
DATA ACQUISITION |
||||
Total Capture Storage Time |
min |
³10 |
³10 |
³10 |
Data Capture Maximum Length |
sec |
³20 |
³20 |
³20 |
AUTO-ALIGN (FSM) |
||||
Auto-Align Capture Angular Dynamic Range |
mrad |
6.27 |
6.27 |
6.27 |
ENVIRONMENT |
||||
Temperature |
deg C |
20–30 deg C ** |
||
Humidity |
Percentage |
10–50% ** |
||
Vibration PSD |
g^2 / Hz |
PSD < 5e-6 g^2/Hz 0–2 kHz ** |
||
** Environmental Specifications have not been tested and are estimates only / not guaranteed. |
AORTA Methodology
The AORTA methodology provides a means to evaluate and assess the error components by post-processing of wavefront phase and intensity patterns. These error components can then be used to construct a breakdown of components of Strehl loss terms, RMS wavefront error components, and contributions to Encircled Energy (EE) loss.
The error components are defined in the context of an assumed adaptive optical system of interest including a deformable mirror and wavefront sensor geometry. The reader will note that while the error components can largely be related to the performance elements of an adaptive optical system they can not necessarily be related to individual subsystems or components without conducting additional experiments both with and without subsystems or components of interest in the beam path and / or active.
Consideration of the components of the Strehl, Wavefront Error, or Encircled Energy begins with a breakdown of a measured wavefront into various components. This breakdown of error components is motivated by a linear algebra based interpretation of adaptive optical systems developed by Ellerbroek for covariance based analysis of adaptive optical systems [1].
We begin by breaking the measured phase into a “static” and dynamic phase. Given 2-D wavefront phase data ø_{k} where k indexes the frame number (out of K total frames in a data set), the static phase component, ø_{s}, is computed by taking the average phase over the data set,
Given the static phase, the dynamic phase, Ø_{d,k}, is given by,
Both the static and dynamic phase are now decomposed into different error components (where the k subscript and s or d subscript are dropped for convenience but clearly should be included to designate the appropriate wavefront component):
ø_{DM} : Best Fit DM Phase – Best fit DM phase estimate applied by the deformable mirror (DM)
ø_{DMFIT} : Fitting Error – Residual error after correction by best fit DM phase pattern.
The best fit DM phase,ø_{DM}, further is broken down into the following components:
ø_{DMSAT} : DM Saturation Error – Error representing failure to achieve best fit applied by a DM that is due to DM saturation due to insufficient actuator stroke.
ø_{AOS} : AOS Fitting Error – Error representing failure to achieve best fit applied by a deformable mirror (DM) that is associated with wavefront sensing and reconstruction from coarse measurements associated with a specific AO system.
ø_{DMC} : DM Correctable Error – Error representing remaining wavefront error that could be compensated by the DM but is present in the observed wavefront and not explained by either DM saturation or AO system fitting error. Typically if an AO system is operating during AORTA data collection then any wavefront error assigned to this term would be considered uncompensated servo lag error that would have been compensated by a faster AO system. There are also scenarios in which this error source could be due to unknown sources.
In order to illustrate the wavefront error component breakdown, we work with a specific example laser wavefront shown in Figure 2. This wavefront was generated using AORTA to measure the aberrations from a reflective turbulence generator developed by Nutronics, Inc.
Figure 2: Example AORTA measurement of (a) Tilt Included and (b) Tilt Removed Phase for a turbulence simulator.
Best Fit DM Phase
We now walk through illustration of computation of the wavefront error components, beginning with the best fit deformable mirror phase calculation. We define a MxN matrix, H, where M is the number of AORTA phase measurements in the active area defined by the current BASE file and N is the number of actuators, as the deformable mirror influence function matrix mapping deformable mirror commands to AORTA phase measurements. Each row of the matrix H corresponds to an AORTA phase measurement coordinate in the active area defined by the current BASE file. Each column of the matrix H corresponds to an actuator on the DM being modeled.
The matrix H is nominally generated using a hypothetical “perfect” DM with no inter-actuator coupling and cubic spline interpolation between actuator locations. Once the matrix H is computed, then evaluation of the DM fitting error is computed by solving a least squares problem to compute the best fit commands to be applied to the DM. It is given that the best fit due to the DM is given by
where c_{DM} are the best fit vector of DM actuator commands to be applied to the DM. The solution to the least squares problem,
is given by,
and thus the best fit DM phase is given by,
The best fit DM phase corresponding to our example wavefront of interest is shown in Figure 3.
Figure 3: Example AORTA measurement of (a) Tilt Removed and (b) Best Fit DM Phase for a turbulence simulator.
DM Fitting Error
Given the best fit DM phase, the DM fitting error is given trivially,
The DM fitting error corresponding to our example wavefront is shown in Figure 4.
Figure 4: Example AORTA measurement of (a) Tilt Removed and (b) DM Fitting Error for a turbulence simulator.
DM Saturation Error
Given the best fit DM phase, we can determine the phase that would not be achievable by the DM of interest due to DM saturation simply by saturating the actuator commands required to achieve the best fit,
where the function sat simply imposes an upper and lower limit to the phase commands applied (recall c_{DM} = (H^{T}H)^{–1}H^{T}øand thus sat[c_{DM}] = sat[(H^{T}H)^{–1}H^{T}ø] ). The DM Saturation Error corresponding to our example wavefront is shown in Figure 5.
Figure 5: Example AORTA measurement of (a) Best Fit DM Phase and (b) DM Saturation Error for a turbulence simulator.
AO System Fitting Error
One of the more complex and non-obvious contributions to wavefront error is the AO System fitting error term. This normally neglected and ignored error term can be an important contribution to residual wavefront error. This error arises from the fact that although a wavefront may have an idealized best fit DM pattern that leads to a small residual wavefront error, the AO system may not have an appropriate wavefront sensing and reconstruction approach to accurately achieve the best fit DM figure.
AORTA estimates this component of wavefront error by simulation of the operation of an AO system. Currently, the only AO system simulated by AORTA is a Hartmann sensor in the Fried-geometry. The measurement process follows that used by Barchers in analysis of the performance of Hartmann sensors in weak to strong scintillation [2] where the measurements are the intensity weighted gradient of the phase function over a subaperture – corresponding to the centroid measured in a focused Hartmann sensor spot. This measurement process is represented by the operator G[ø]. A standard least squares reconstructor, denoted E, is utilized and assumes that DM actuators are co-aligned with the Hartmann sensor subaperture corners. A simple AO system loop is simulated to find the approximate phase applied by the DM for this hypothetical AO system:
- Initialize c_{AOS} = 0
- Let c_{AOS} = c_{AOS} + EG[Hc_{DM} — Hc_{AOS}]
- Repeat step 2 until the loop converges or we reach a maximum number of time steps (nominally 24).
Given c_{AOS}, we note that the AOS Fitting Error is given by,
The AOS Fitting Error for our example wavefront is shown in Figure 6.
Figure 6: Example AORTA measurement of Best Fit DM Phase and AOS Fitting Error for a turbulence simulator.
DM-Correctable Error
The last remaining error term, the DM-Correctable Error, is trivially computed,
This remaining error term is often difficult to interpret. If an AO system is functioning well and implemented correctly, then this error term has two components: sensor noise that has coupled into DM commands and residual servo lag. This error term is wavefront error that could be corrected by a DM, but is not being corrected by the DM and is not attributable to either DM saturation or failure of the AO system to accurately measure the wavefront. It may be the case that this error term arises due to a non-common path error (a red flag indicator may be that the static DM-Correctable Error is large – indicative of a significant non-common path error between the AORTA beam path and the AO system path). Often multiple experiments with different combinations of system configurations may be required to sort out the source of DM-Correctable Error if it is not clearly simply due to servo lag. There are methods of estimation of the relative contributions of noise and servo lag – however we have chosen not to include these methods because there would be measurement noise in the AORTA data that would make it difficult to assess the estimated contribution of residual wavefront error due to AO system wavefront sensor noise.
The DM-Correctable component of wavefront error for our example beam is given in Figure 7. The reader will note that here we expect a large DM-Correctable component because there is no AO system in the beam path leading to AORTA for this example test data.
Figure 7: Example AORTA measurement of Best Fit DM Phase and DM-Correctable Phase for a turbulence simulator.
Strehl Budget Calculation
Given a component of wavefront phase error, ø, and an estimate of the intensity profile I (obtained by averaging the nearest subaperture intensity values), the Strehl associated with that loss term is given by,
This representation of the Strehl loss term corresponds to the focused intensity relative to the diffraction limit for that Strehl component.
Wavefront Error Budget Calculation
Given a component of wavefront phase error, ø, and an estimate of the intensity profile I (obtained simply by averaging the nearest subaperture intensity values), the wavefront error associated with that loss term is given by,
Encircled Energy Calculation
Just like the Strehl and Wavefront Error budgets, Encircled Energy can be computed from any given wavefront contribution to form an Encircled Energy Budget. One can think of the Strehl ratio as the evaluation of power in an infinitesimal-sized bucket. With this in mind, Encircled Energy budgets simply become a similar evaluation of a Strehl-like quantity: power in the bucket for a range of bucket sizes. The Encircled Energy budgets are computed for a range of values to produce a set of curves corresponding to different contributions to encircled energy. Like the Strehl ratio, the contributions to encircled energy, when normalized to the diffraction limit, can be approximately multiplied using the Marechal approximation. The computation methodology and procedure is illustrated in Figure 8.
- Brent L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-11–2-783. - Jeffrey D. Barchers, David L. Fried, and Donald J. Link, “Evaluation of the Performance of Hartmann Sensors in Strong Scintillation,” Appl. Opt. 41, 1012–1021 (2002)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-41–6-1012