Analytic Support / Modeling and Simulation

We uti­lize two pri­mary tools for analy­sis and mod­el­ing of laser prop­a­ga­tion through tur­bu­lence with tongue-in-cheek monikers: a stan­dard wave opti­cal sim­u­la­tion code – Barchers Wave Optics Code or BWOC; and a covari­ance based mod­el­ing code – Barchers Covari­ance Code or BCC.  Both are described below.  One of the inter­est­ing points to note is that BWOC and BCC work from the same input spec­i­fi­ca­tion – enabling com­par­i­son of results.  When nec­es­sary and appro­pri­ate we develop spe­cial­ized mod­el­ing codes that exam­ine effects not nor­mally con­sid­ered or cap­tured by BWOC or BCC.  We uti­lize the Rytov the­ory exten­sively to expose under­ly­ing phe­nom­ena and key para­me­ters and are adept at apply­ing analy­sis tech­niques such as Mel­lon trans­form meth­ods to facil­i­tate accu­rate and rapid eval­u­a­tion of the inte­grals asso­ci­ated with analy­sis of the per­for­mance of laser beam con­trol sys­tems for prop­a­ga­tion through tur­bu­lence [1, 2].

BWOC Overview

When time domain mod­el­ing is required (for exam­ple either for eval­u­a­tion of time domain effects or for devel­op­ment of fad­ing prob­a­bil­ity dis­tri­b­u­tions for free space opti­cal com­mu­ni­ca­tion) or when the Rytov num­ber for the appli­ca­tion of inter­est exceeds 0.2, we uti­lize our stan­dard wave opti­cal sim­u­la­tion code, BWOC.  BWOC uti­lizes well estab­lished meth­ods for prop­a­ga­tion through tur­bu­lence [3, 4] and does not attempt to incor­po­rate spe­cial fil­ter­ing or bea­con prop­a­ga­tion meth­ods that require exten­sive jus­ti­fi­ca­tion – these types of meth­ods were heav­ily used in the 1990s to save com­pu­ta­tion time and slowly have van­ished over the last few years as PC-based com­put­ing power has con­tin­ued to steadily increase.  Instead we sim­ply accept the com­pu­ta­tional penal­ties asso­ci­ated with wave opti­cal sim­u­la­tion and per­form valid cal­cu­la­tions.  The BWOC does min­i­mize com­pu­ta­tion require­ments by includ­ing an auto­mated grid selec­tion tool that sets a diverg­ing or con­verg­ing coor­di­nate sys­tem based on rule-of-thumb require­ments for tur­bu­lence sam­pling in both prop­a­ga­tion direc­tions, sam­pling of the dif­frac­tion limit, sam­pling the focus applied by the pri­mary mir­ror, sam­pling of sub­aper­ture / actu­a­tor spac­ing, and sat­is­fy­ing the sam­pling require­ment to pre­vent alias­ing of the prop­a­ga­tion ker­nel.  BWOC inputs include des­ig­na­tion of the AO sys­tem, Tracker, AO and Track­ing sig­nal lasers (point source bea­con, active illu­mi­na­tion, pas­sive or ther­mal sig­na­ture), scor­ing beam (i.e. typ­i­cally the mod­eled High Energy Laser) char­ac­ter­is­tics, and sce­nario geom­e­try includ­ing plat­form and tar­get spec­i­fi­ca­tion and prop­a­ga­tion path char­ac­ter­is­tics (i.e. Cn2 pro­file, absorp­tion and scat­ter­ing pro­files, etc.)  3-D tar­get mod­el­ing is uti­lized and tar­get depth is mod­eled for accu­rate rep­re­sen­ta­tion of laser speckle effects.  Tar­get inter­ac­tion effects have been mod­eled if an appro­pri­ate model is defined by the cus­tomer.  Ther­mal bloom­ing is mod­eled using stan­dard advec­tion based mod­els with a vari­a­tion of mod­els defined by Barchers & Fried uti­lized to model micro-scale wind shear effects [5, 6].

BCC Overview

The Barchers Covari­ance Code or BCC is a covari­ance based mod­el­ing tool based on stan­dard analy­sis meth­ods best delin­eated by Eller­broek [7].  As illus­trated in Fig­ure 1, the BCC bridges the com­pu­ta­tional / accu­racy gap between scal­ing law codes, fre­quency domain codes [8], and wave opti­cal sim­u­la­tion.  The BCC is based on geo­met­ric ray-tracing and as such is only for­mally valid in the weak scin­til­la­tion regime (Rytov < 0.2).  The BCC does retain some accu­racy in the strong scin­til­la­tion regime assum­ing that an AO sys­tem capa­ble of mea­sure­ment and com­pen­sa­tion of the branch points in the phase func­tion is uti­lized – how­ever the BCC does not accu­rately model the severe aniso­pla­natic effects asso­ci­ated with aniso­pla­natic branch point phase in this regime.

Fig­ure 1: BCC bridges the gap between scal­ing law codes which have poor accu­racy but high speed and wave opti­cal sim­u­la­tion, which is very slow but highly accurate.

Covari­ance codes work by eval­u­a­tion of the covari­ance of resid­ual wave­front errors on a geo­met­ric prop­a­ga­tion grid of the generic form in Fig­ure 2 (shown for exam­ple for a Dual-Hartmann sen­sor [9] geom­e­try).  Covari­ances are cal­cu­lated for the scor­ing grid points and the asso­ci­ated sen­sor and con­trol sys­tem covari­ances are then expressed using lin­ear oper­a­tors (i.e. matrix oper­a­tor).  The covari­ance cal­cu­la­tions are of the form,

and are eval­u­ated using Mellin trans­form tech­niques to facil­i­tate rapid and accu­rate eval­u­a­tion.  We have incor­po­rated inner (using the Hill model) and outer scale effects and mod­eled aero-optical tur­bu­lence spec­tra as well using this method­ol­ogy but the base­line BCC uti­lizes the Kol­mogorov spectrum.

Fig­ure 2: Covari­ance code analy­sis geometry.

BCC per­for­mance cal­cu­la­tions are devel­oped using the prod­uct of Opti­cal Trans­fer Func­tions (OTFs),

This approach is sim­i­lar to the Marechal approx­i­ma­tion (prod­uct of Strehl ratios) which is used in many scal­ing law codes.  There are two advan­tages to the BCC approach.  First, the BCC accu­rately treats cor­re­la­tions between effects in the “Higher Order, Atmo­phere” (HO, ATM) term.  Sec­ond, the approach gen­er­ates an esti­mate of the beam pro­file at the focal plane.  As shown in Fig­ure 3, we have val­i­dated the BCC for a range of sys­tem types and appli­ca­tions.  As with the BWOC, arbi­trary input beam pro­files can be uti­lized and the BCC includes 3-D tar­get mod­els to pro­vide accu­rate com­pu­ta­tion of radiom­e­try for incor­po­ra­tion into noise mod­els.  Tar­get inter­ac­tion effects have been mod­eled if an appro­pri­ate model is defined by the customer.

Fig­ure 3: Val­i­da­tion of BCC against wave opti­cal sim­u­la­tion for com­pen­sa­tion with a con­ven­tional sys­tem, an alter­nate sys­tem (Inte­grated Laser Beam Con­trol or ILBC [10]), and track only compensation.

Over­all, the BCC pro­vides an inter­est­ing and alter­nate mod­el­ing capa­bil­ity that can pro­vide far more rapid, but still highly accu­rate, eval­u­a­tion of cus­tomer sce­nar­ios.  The abil­ity to pro­duce an accu­rate beam pro­file at the focal plane pro­vides the cus­tomer the abil­ity to per­form a far more accu­rate eval­u­a­tion of the poten­tial util­ity of var­i­ous com­pen­sa­tion approaches and make informed deci­sions regard­ing the appro­pri­ate com­pen­sa­tion method for the appli­ca­tion of interest.

1. Elec­tro­mag­netic Wave Prop­a­ga­tion in Tur­bu­lence: Eval­u­a­tion and Appli­ca­tion of Mellin Trans­forms, 2nd Ed. (SPIE Press Mono­graph Vol. PM171).

2. Brent L. Eller­broek, “Power series eval­u­a­tion of covari­ances for turbulence-induced phase dis­tor­tions includ­ing outer scale and servo lag effects,” J. Opt. Soc. Am. A 16, 533–548 (1999)–3-533.

3. B. L. Eller­broek. “A Wave Optics Prop­a­ga­tion Code for Multi-Conjugate Adap­tive Optics”, Gem­ini preprint #69,

4. B. L. Eller­broek, G. M. Cochran, “A Wave Optics Prop­a­ga­tion Code for Multi-Conjugate Adap­tive Optics”, SPIE Pro­ceed­ings Vol­ume 4494, “Adap­tive Optics Sys­tems and Tech­nol­ogy II”, Feb­ru­ary 2002,

5. J. D. Barchers, “Lin­ear analy­sis of ther­mal bloom­ing com­pen­sa­tion insta­bil­i­ties in laser prop­a­ga­tion,” Jour­nal of the Opti­cal Soci­ety of Amer­ica A.,26: 1638–1653, June 2009,–7-1638.

6. D. L. Fried, R. K.-H. Szeto, “Wind-shear induced sta­bi­liza­tion of PCI”, Jour­nal of the Opti­cal Soci­ety of Amer­ica A.,15: 1212–1226, 1998,–5-1212.

7. 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)–2-783.

8. Brent L. Eller­broek, “Lin­ear sys­tems mod­el­ing of adap­tive optics in the spatial-frequency domain,” J. Opt. Soc. Am. A 22, 310–322 (2005)–2-310.

9. 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)–6-1012

10. J. D. Barchers, “Inte­grated Laser Beam Con­trol Sum­mary Report,” AFRL con­tract FA9451-09-C-0009, XXXXXXX web link.


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