Applications with a cooperative target including FSO Communication

While the NAO Sys­tems imper­a­tive is to develop a solu­tion to the prob­lem of com­pen­sa­tion when the Rytov Num­ber exceeds 0.2 with non-cooperative tar­gets, along the way Nutron­ics, Inc. has devel­oped a sound tech­nol­ogy suite for com­pen­sa­tion when the Rytov Num­ber exceeds 0.2 with coop­er­a­tive tar­gets.  Con­ven­tional sys­tems, includ­ing Hart­mann sen­sor based sys­tems, cur­va­ture sen­sor based sys­tems, and shear­ing inter­fer­om­e­ter based sys­tems all suf­fer sig­nif­i­cant per­for­mance losses when the Rytov num­ber exceeds 0.2 even with a coop­er­a­tive point source tar­get due to branch points in the phase func­tion [1–4].

Hart­mann sen­sors suf­fer from an inabil­ity to mea­sure branch points in the phase func­tion accu­rately unless the sub­aper­tures are roughly 4 times smaller than the Fried coher­ence diam­e­ter, r0 [3].  Fur­ther, this can only be achieved when using a Com­plex Expo­nen­tial Recon­struc­tor (CER) [5] or com­pa­ra­ble esti­ma­tor to recover the branch point con­tri­bu­tion of the phase func­tion from the slope dis­crep­ancy com­po­nent of the gra­di­ent mea­sure­ments obtained from the Hart­mann sen­sor [6].  The sam­pling lim­i­ta­tion com­bined with the need to use a CER imposes sig­nif­i­cant lim­i­ta­tions on the capa­bil­ity of an AO sys­tem.  First, the CER requires lin­ear mea­sure­ment of the gra­di­ents, elim­i­nat­ing the light-efficient and min­i­mal data latency use of quad­rant mea­sure­ments for the Hart­mann sen­sor.  This leads to the use of many pix­els per sub­aper­ture, increas­ing the data latency and reduc­ing sig­nal to noise ratio.  Sec­ond, the sam­pling require­ment increases the num­ber of sub­aper­tures required, again reduc­ing sig­nal to noise ratio or impos­ing exhor­bi­tant sig­nal require­ments on the wave­front sen­sor.  Third, the CER is a non-linear esti­ma­tor and as such can have high noise gain (not a good sit­u­a­tion given the for­mer two lim­i­ta­tions).  As a counter-point to the three lim­i­ta­tions noted above, one might argue that with a point source bea­con one can have all of the sig­nal that you want or need.  This is far from the case – even with point source bea­cons – clearly astro­nom­i­cal appli­ca­tions are sig­nal lim­ited and one is con­sis­tently sur­prised that link bud­gets for free space opti­cal com­mu­ni­ca­tion appli­ca­tions become very tight, very quickly, and spar­ing light for a signal-hogging wave­front sen­sor is not very attrac­tive.  Last, the intro­duc­tion of the CER into a feed­back con­trol loop has sta­bil­ity con­se­quences that we have never man­aged to resolve in numer­i­cal sim­u­la­tion that include real­is­tic DM and WFS uncer­tainty mod­els.  Robust AO sys­tems that oper­ate in chang­ing envi­ron­ments must oper­ate in a null-seeking closed loop fash­ion to ensure robust­ness to drifts in DM and WFS cal­i­bra­tion – as such we reject solu­tions that oper­ate in an “open loop” fash­ion.  All 4 of these lim­i­ta­tions essen­tially elim­i­nate the Hart­mann sen­sor from con­sid­er­a­tion for appli­ca­tions (even with a point source bea­con) when the Rytov Num­ber exceeds 0.2.

Cur­va­ture sens­ing has a lesser known, but far more severe lim­i­ta­tion (than Hart­mann sen­sors) when the Rytov Num­ber exceeds 0.2.  The cur­va­ture sen­sor mea­sures the Lapla­cian of the phase, Δ=∇·∇, where ∇is the gra­di­ent oper­a­tor and ∇· is the diver­gence oper­a­tor.  The gra­di­ent oper­a­tor inher­ently decom­poses the mea­sured wave­front phase into a scalar gra­di­ent field and a vec­tor field defined by the curl of the vec­tor poten­tial (asso­ci­ated with the branch point phase).  How­ever, we all know and under­stand that the diver­gence of the curl of the vec­tor poten­tial is zero, thus the cur­va­ture sen­sor com­pletely anni­hi­lates the mea­sure­ment data asso­ci­ated with the branch point phase – and as such the cur­va­ture sen­sor will not be capa­ble of com­pen­sa­tion in strong scin­til­la­tion because it inher­ently has no sen­si­tiv­ity to branch points – i.e. it can not mea­sure branch points at all!  As such, we elim­i­nate cur­va­ture sen­sors from con­sid­er­a­tion (again even with a point source bea­con) when the Rytov Num­ber exceeds 0.2.

This brings us to the shear­ing inter­fer­om­e­ter based sen­sor, which at least has been shown to be capa­ble of some level of mea­sure­ment accu­racy when the Rytov Num­ber exceeds 0.2 [4].  Our first con­cern with the shear­ing inter­fer­om­e­ter is the increased num­ber of required phase-shifted image mea­sure­ments (at least 6 total mea­sure­ments – 3 for x and 3 for y).  This leads to an increase in opto-mechanical com­plex­ity.  In addi­tion, the shear­ing inter­fer­om­e­ter again requires a CER for effec­tive mea­sure­ment of the branch point con­tri­bu­tion of the phase func­tion (again extracted from the slope dis­crep­ancy as with the Hart­mann sen­sor).  We repeat the same lim­i­ta­tions asso­ci­ated with the CER: non­lin­ear / high noise gain; and the intro­duc­tion of the CER into a feed­back con­trol loop has sta­bil­ity con­se­quences that we have never man­aged to resolve in numer­i­cal sim­u­la­tion that include real­is­tic DM and WFS uncer­tainty mod­els.  Robust AO sys­tems that oper­ate in chang­ing envi­ron­ments must oper­ate in a null-seeking closed loop fash­ion to ensure robust­ness to drifts in DM and WFS cal­i­bra­tion – as such we reject solu­tions that oper­ate in an “open loop” fashion.

The lat­ter dis­cus­sion focused on mea­sure­ment tech­niques that eval­u­ate the first or sec­ond deriv­a­tive of the wave­front and then must recon­struct the phase from the mea­sure­ments.  In con­trast, NAO Sys­tems uti­lize direct com­plex field esti­ma­tion meth­ods using one of two basic tech­ni­cal approaches: a Point Dif­frac­tion Inter­fer­om­e­ter (PDI) or Self-Referencing Inter­fer­om­e­ter (SRI).  Such meth­ods have been proven (yes, in the math­e­mat­i­cal sense) to have field esti­ma­tion per­for­mance that is invari­ant with the Rytov Num­ber [7] and recent work has demon­strated the per­for­mance and effec­tive­ness of SRI-based AO sys­tems [8–10] and PDI-based AO sys­tems [11].    Our core sys­tem offer­ings for low power appli­ca­tions are shown in Table 1 below.  The sys­tem offer­ings are highly flex­i­ble and adapt­able to dif­fer­ent wave­lengths in the vis­i­ble or Near-IR (cur­rent sys­tems have been devel­oped and tested in the 1500–1600 nm band and in the 900‑1100 nm band).  Hart­mann sen­sor based sys­tems can be offered as well but we gen­er­ally only sug­gest such sys­tems for oper­a­tion in the weak scin­til­la­tion regime.  Sys­tems uti­liz­ing large for­mat DMs for high power oper­a­tion are offered as well but require fur­ther dis­cus­sion with the cus­tomer to deter­mine requirements.

Our sys­tems are opti­mized for high speed oper­a­tion to meet the chal­lenges of the most demand­ing appli­ca­tions.  There are many spe­cial con­sid­er­a­tions asso­ci­ated with high speed oper­a­tion that are not widely known in the AO com­mu­nity and our exper­tise enables us to pro­vide robust and reli­able solu­tions.  These con­sid­er­a­tions span the entire sys­tem from the opti­cal and mechan­i­cal design and devel­op­ment and the real time con­troller.  The real time con­trol algo­rithms require par­tic­u­lar atten­tion for inter­fer­o­met­ric AO sys­tems – the reader may be sur­prised that our con­trol algo­rithms have crit­i­cal sub­tle dif­fer­ences between the SRI and PDI WFS (it was a sur­prise to us that these dif­fer­ences were required until we had built both and real­ized the sub­tle issues!).

The µNAO Unit is nom­i­nally offered with a 3-bin SRI or 3-bin PDI WFS and either a 140-Ch BMC DM or 32-Ch BMC DM.  The µkNAO Unit is nom­i­nally only offered with a 3-bin SRI WFS and a 1020-Ch BMC DM (we would con­sider offer­ing a PDI WFS vari­a­tion for astro­nom­i­cal ExAO appli­ca­tions but there would be addi­tional con­sid­er­a­tions to dis­cuss with poten­tial cus­tomers).  Our PDI WFS offers the advan­tages of broad linewidth oper­a­tion (untested but the­o­ret­i­cally white light) and is insen­si­tive to input polar­iza­tion.  Our SRI WFS design requires a rel­a­tively sta­ble input polar­iza­tion and a more nar­row linewidth for oper­a­tion – how­ever our SRI WFS offers the best per­for­mance in the broad­est range of con­di­tions.  Vari­able input zoom can be included in the µkNAO Unit.  All sys­tems offer the abil­ity to trans­mit a pre-compensated beam and input a cal­i­bra­tion beam.  All sys­tems offer a com­pen­sated imag­ing beam path.

Table 1: Nutron­ics, Inc. pri­mary NAO Sys­tem con­fig­u­ra­tions for low power appli­ca­tions.  Alter­nate DM ven­dors for high power DMs are offered as well (dis­cus­sion with cus­tomer required)

µkNAO Unit

µNAO Unit


SRI 3-bin (simul­ta­ne­ous measurement)

SRI 3-bin



PDI 3-bin




1020 Ch BMC

140 Ch BMC

32 Ch BMC

140 Ch BMC

32 Ch BMC


Opti­mized for most chal­leng­ing appli­ca­tions; PDI WFS nom­i­nally not offered with 1020 Ch DM (could be con­sid­ered for astro­nom­i­cal ExAO applications)

SRI WFS will have best per­for­mance over broader range of con­di­tions (rel­a­tive to PDI WFS) but requires a more sta­ble input polarization

Our PDI WFS is polar­iza­tion insen­si­tive and enables broad linewidth operation


Com­pen­sated imag­ing beam path;

Beam path for pre-compensated trans­mit­ted beam;

Per­for­mance spec­i­fi­ca­tions based on wave­length band of interest


One of the pri­mary appli­ca­tions we have stud­ied for NAO Sys­tems is Free Space Opti­cal (FSO) Com­mu­ni­ca­tion.  Exam­ple fade prob­a­bil­ity dis­tri­b­u­tions are shown in Fig­ure 1 illus­trat­ing the types of trades that can be mod­eled for cus­tomers to assist in deter­min­ing the opti­mal sys­tem selec­tion for their appli­ca­tion of inter­est.  Bit Error Rate cal­cu­la­tions for a spe­cific data rate are com­puted from this type of sim­u­la­tion data and are con­verted into link mar­gin cal­cu­la­tions.  Sim­i­lar analy­ses can be per­formed for cus­tomer appli­ca­tions of inter­est as appro­pri­ate to assist in selec­tion of the opti­mal sys­tem to max­i­mize per­for­mance and min­i­mize cost.

Fig­ure 1: Exam­ple fade prob­a­bil­ity per­for­mance pre­dic­tions for the µkNAO Unit.  The µkNAO Unit fea­tures a 32×32 DM with vari­able input zoom to enable use with a reduced num­ber of actu­a­tors.  The µkNAO Unit design is based on the µNAO Unit but uses a 3-bin sin­gle mode fiber ref­er­ence Self-Referencing Inter­fer­om­e­ter (SRI3) for the wave­front sensor.

Sev­eral fig­ures fol­low that illus­trate the devel­op­ment process for the µNAO Unit below.  In Fig­ure 2 we illus­trated the evo­lu­tion of the µNAO Unit from opti­cal and mechan­i­cal design through inte­gra­tion.  In Fig­ure 3 pro­vides exam­ple results of our finite ele­ment mod­el­ing of the µNAO Unit for a MIL-STD input vibra­tion spec­trum.   Finally, ini­tial test data is shown in Fig­ure 4.

Please give us a call at 303−530−200 or Email at and we will be happy to dis­cuss your appli­ca­tion and needs.

Fig­ure 2: The µNAO Unit was designed for robust­ness to vibra­tion and ther­mal drift.  The µNAO Unit has the fol­low­ing fea­tures: 3-bin Point Dif­frac­tion Inter­fer­om­e­ter (PDI3) wave­front sen­sor, simul­ta­ne­ous mea­sure­ment of all 3 phase images, polar­iza­tion insen­si­tive, broad linewidth operation.

Fig­ure 3: Exam­ple design process results – the µNAO Unit is designed for high vibra­tion envi­ron­ments.  The PSD response to an input spec­trum is shown at left and the CSD at right.

Fig­ure 4: Exam­ples of com­pen­sa­tion by the µNAO Unit of DM-induced aber­ra­tions – an “F”, “Cross”, and “X”.

1. D.. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,”Appl. Opt. 31, 2865–2882, 1992.

2. D. L. Fried, “Branch point prob­lem in adap­tive optics,” J. Opt. Soc. Am. A 15, 2759–2768, 1998.

3. Barchers, J. D., Fried, D. L. and Link, D., “Eval­u­a­tion of the per­for­mance of Hart­mann sen­sors in strong scin­til­la­tion”. Feb. 2002, Applied Optics, Vol. 41, pp. 1012–1021.

4. Barchers, J. D., Fried, D. L. and Link, D., “Eval­u­a­tion of the per­for­mance of a shear­ing inter­fer­om­e­ter in strong scin­til­la­tion in the absence of addi­tive mea­sure­ment noise”. Jun. 2002, Applied Optics, Vol. 41, pp. 3674–3684.

5. D. L. Fried, “Adap­tive optics wave func­tion recon­struc­tion and phase unwrap­ping when branch points are present,” Opt. Com­mun. 200, 43–72, 2001.

6. G. A. Tyler, “Recon­struc­tion and assess­ment of the least­squares and slope dis­crep­ancy com­po­nents of the phase,” J. Opt. Soc. Am. A 17, 1828–1839, 2000.

7. J. D. Barchers and T. A. Rhoad­armer, “Eval­u­a­tion of phase-shifting approaches for a point-diffraction inter­fer­om­e­ter with the mutual coher­ence func­tion,” Applied Optics, 41:7499–7509, Dec. 2002.

8. T. A. Rhoad­armer, “Devel­op­ment of a self-referencing inter­fer­om­e­ter wave­front sen­sor,” Proc. SPIE 5553, 112 (2004).

9. T. A. Rhoad­armer and L. M. Klein, “Design of spa­tially phase shifted self-referencing inter­fer­om­e­ter wave front sen­sor,” Proc. SPIE 6306, 63060K (2006).

10. T. A. Rhoad­armer and T. A. Bren­nan, “Per­for­mance of a woofer-tweeter deformable mir­ror con­trol archi­tec­ture for high-bandwidth, high-spatial res­o­lu­tion adap­tive optics,” Proc. SPIE 6306, 63060K (2006).

11. Carl Pater­son and James Notaras, “Demon­stra­tion of closed-loop adap­tive optics with a point-diffraction inter­fer­om­e­ter in strong scin­til­la­tion with opti­cal vor­tices,” Opt. Express 15, 13745–13756 (2007)–21-13745

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