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 øk where k indexes the frame num­ber (out of K total frames in a data set), the sta­tic phase com­po­nent, øs, is com­puted by tak­ing the aver­age phase over the data set,

Given the sta­tic phase, the dynamic phase, Ød,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):

øDM            : Best Fit DM Phase – Best fit DM phase esti­mate applied by the deformable mir­ror (DM)

øDMFIT        : Fit­ting Error – Resid­ual error after cor­rec­tion by best fit DM phase pattern.

The best fit DM phase,øDM, fur­ther is bro­ken down into the fol­low­ing components:

øDMSAT       : 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.

øAOS           : 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.

øDMC          : 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 cDM 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 cDM = (HTH)–1HTøand thus sat[cDM] = sat[(HTH)–1HTø] ). 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:

  1. Ini­tial­ize cAOS = 0
  2. Let cAOS = cAOS + EG[HcDM — HcAOS]
  3. Repeat step 2 until the loop con­verges or we reach a max­i­mum num­ber of time steps (nom­i­nally 24).

Given cAOS, 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.

  1. 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.
  2. 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


 

 

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