Determinación de planes de acción para el posicionamiento de Fortizeb SMG

A q u antity th at is of great im portance in optical com m unication links and beam steering, C haim ow icz an d Cole 1987, is the angle of arrival (AOA). It s definition depends som ew hat on the w ay it is m easured. For instance, w ith an interferom eter the AGA is related to the p hase difference m easured at tw o points sep arated by som e sm all distance, say b an d is detector dependent. A telescope will only "see" the average tilt of the w ave across its aperture. Also spatial averaging will occur if the dim ensions of the ph ase fluctuations are sm aller than the d etector ap ertu re, D. Strohbehn and Clifford 1967, have considered the definition to be the direction of the w ave norm al at the receiving point.

Chapter 3 : The Stochastic W ave Equation.

3.11.1) VARIANCE OF ANGLE-OF-ARRIVAL.

Tatarskii 1961, gives an expression for the variance of the AGA, a , for an interferom eter of separation b:

(a ^ ) = a â = - p ^ (3.68)

w here the angle is assum ed to be sm all an d DgfW is the p h ase stru ctu re function. H e also states that for a telescope of diam eter b, there is m erely a 3% difference in variance values w h en u sin g Eq. (3.68). G urvich et al 1968(a), give:

a l = VÂ% Zb (3.69)

for a spherical wave. For plane w aves the expression is identical to the above except for the n u m erical factors. If Iq « b « then 1.46 appears, how ever, if then the factor is 2.92.

U nlike the scintillation variance, there is no eq u iv alen t of the satu ratio n phenom ena for AGA fluctuations, so the above expressions give results in g o o d a g re e m e n t w ith o b se rv a tio n s re g a rd le ss of th e in te n s ity of scintillations.

A sm all n u m b er of experim ents, u n d er conditions of satu ratio n and non­ saturation, have been perform ed to verify the validity of Eq. (3.50). G urvich et al 1968(b), rep o rted m axim um values of Ogg = 15 an d 25 firad for p ath lengths of 125 an d 500 m respectively. Even over a p a th of 1750 m the m easu red variances, alth o u g h being slightly u n d e re stim a ted , com pared well w ith theory. Gurvich et al 1968(a), gave sim ilar results o f 0 ^ - 5 to 20 |irad an d 15 to 45 p rad for p a th lengths of 650 and 6500 m respectively d u rin g the daytim e. It seems then, that expressions for the phase structure function ap p ear to be valid over a greater range of turbulence regim es than

those for intensity.

A nother experim ent recently reported in the literature, Sadot an d Kopeika 1991, uses the relation Eq. (3.50) w ith a 1.46 factor to predict the strength of the stru c tu re co n stan t on the basis of the v arian ce of th e AOA. The ex p erim en t w as p erfo rm ed 4 m above the d esert floor, w h ere a 0.63 |im H eN e laser w as fired 2 km , then was reflected back across its original p ath an d finally onto a silicon vidicon TV camera. Two h u n d red m easurem ents w ere taken for each ru n an d the AOA variance com puted. A lthough the calculation of a ran d o m p aram eter such as variance of AOA from m erely 200 m easu rem en ts is statistically rather poor, good agreem ent was found b e tw e e n th e d iu rn a l v a ria tio n of C„2 a n d th a t p re d ic te d by o th e r m easurem ents of the sam e quantity from different techniques. One w ould expect th at any experim ent perform ed in these conditions w o u ld be deeply in the sa tu ratio n regim e. H ow ever, it a p p ears th at the d istrib u tio n of intensity a t the receiver was log-normal^ im plying th at w eak fluctuations are present, an d th at good agreem ent should be obtained betw een theory an d experim ent.

3.11.2) THE ANGLE POWER SPECTRUM.

Recall th at in Section (3.4) expressions for the am p litu d e an d phase spectra w ere d eriv ed . T atarskii 1961, gives an ex p ressio n for th e frequency spectrum : f A 2 (3.70)

J

^R ecall the validity range for turbulence strength fluctuations is n a rro w e r for intensity statistics than for the AOA.

Chapter 3 : The Stochastic W ave Equation.

The ex p ressio n for th e frequency sp e ctru m for p h ase flu ctu atio n s is identical to th at for the log-am plitude expect for a m inus sign. If w e define a q u a n tity Q - flfo' w h ere /q = Vj^ IdnXL)'^!^ then w e m ay w rite the sim plified expressions:

w^{f) = 3 . 2 8 - (3.72)

an d

w , { f ) = 1.64 (3.73)

The spectrum of AOA fluctuations is determ ined by the ratio betw een the tran sv erse w in d velocity an d the receiver ap ertu re diam eter. Its typical b a n d w id th should be of the ord er of tens of H ertz. For m any applications the q u a n tity of in terest is the sp e ctru m of flu ctu atio n s of th e p h ase difference m easured at two points. For this, Tatarskii gives the relation:

= / > ^ (3.74)

w h e re p is the sep aratio n of p o in ts a n d the s ta n d a rd d ev iatio n of velocity fluctuations.

3.11.3) PROBABILITY DENSITY FUNCTION OF ANGLE-OF-ARRIVAL.

A ssum ing th at there is no detector or m echanical bias in the system , it is expected th at the AOA fluctuations will be distributed as gaussian random variables. There is equal probability of receiving angles to the le ft/rig h t or b e lo w /a b o v e the optim um alignm ent position. To date, there has been little published experim ental w ork concerning the AOA p d f s .