# 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. C_{n}^{2} 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, 2^{nd} 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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16–3-533.

3. B. L. Ellerbroek. “A Wave Optics Propagation Code for Multi-Conjugate Adaptive Optics”, Gemini preprint #69, http://www.gemini.edu/documentation/webdocs/preprints/gpre69.pdf.

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, http://www.gsmt.noao.edu/book/ch4/4_6_D.pdf.

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, http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26–7-1638.

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, http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15–5-1212.

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-11–2-783.

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

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-22–2-310.

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)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-41–6-1012

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