OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

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withH 0 (x) = 1, and H 1 (x) = 2x. (3.27)A plot of the transverse amplitude of the Hermit-Gaussian beams is shownin Figure 3.2, for l = 0, 1, 2, 3 and m = 0, 1, 2, 3.3.2.3 Laguerre-Gaussian beamsThe set of Laguerre-Gaussian beams is another complete solution of theparaxial wave equation expanded in cylindrical coordinates. The complexamplitude of Laguerre-Gaussian modes take the form [67]√2p ! exp [j(2p + m + 1)(ζ(z) − ζ 0 )]f p,m (r, θ, z) =(1 + δ 0m )π(m + p)!w(z)( √ ) m ( )]2r 2r× L m 2p exp[−jk r2w(z) w 2 (z)2q(z) + imθ , (3.28)whereL m p (x) =p∑k=0(p + m)!(m + k)!(p − k)!k! (−x)k (3.29)is the generalized Laguerre polynomials and 1/q(z) = √ 2/w(z). In the abovesolution, the integer p ≥ 0 is the radial index and the integer m is theazimuthal mode index. A plot of the transverse distribution of the Laguerre-Gaussian beams is shown in Figure 3.3, for p = 0, 1, 2, 3 and m = 0, 1, 2, 3.22
Figure 3.2: Hermite Gaussian beams f l,m (x, y) with l = 0, 1, 2, 3 for columnsfrom left to right and m = 0, 1, 2, 3 for rows from top to bottom .23
Page 5 and 6: AcknowledgementFirst and foremost,
Page 7 and 8: 3.2.1 Paraxial waves and Gaussian b
Page 9 and 10: 6 Summary 996.1 Conclusions . . . .
Page 11 and 12: 4.2 Airy pattern: (a)Normalized Air
Page 13 and 14: 5.9 Monte-Carlo simulation by trans
Page 15 and 16: List of Tables4.1 Physical paramete
Page 17 and 18: Chapter 1Introduction1.1 Motivation
Page 19 and 20: eam due to turbulent media can, how
Page 21 and 22: the effects of refraction and diffr
Page 23 and 24: strate evident reduction of scintil
Page 25 and 26: are the most important causes of al
Page 27 and 28: mal blooming can be ignored and the
Page 29 and 30: centroid is caused by large eddies
Page 31 and 32: optics systems for laser beam contr
Page 33 and 34: YYouXθvXAZZ = 0Z = dFigure 3.1: Th
Page 35 and 36: When the propagation distance d is
Page 37: wherew(z) = w 0[1 +( zz 0) 2] 1/2,
Page 41 and 42: 3.3 Optical CoherenceThe optical co
Page 43 and 44: where F (⃗r, ν) is the Fourier t
Page 45 and 46: from finite surfaces with small dif
Page 47 and 48: 3.4.1 Kolmogorov turbulence modelKo
Page 49 and 50: structure function of refractive in
Page 51 and 52: The wave number k(⃗r) can be expr
Page 53 and 54: An iterative method can then be tak
Page 55 and 56: We then get( ) aψ s (⃗r) = χ +
Page 57 and 58: It can be seen that the ensemble av
Page 59 and 60: 3.6.1 Maximization of the integrate
Page 61 and 62: Chapter 4Optimal Beam to Maximize t
Page 63 and 64: A with aperture function⎧⎪⎨ 1
Page 65 and 66: 4.1 Optimal Beam in Free SpaceThe m
Page 67 and 68: −0.132.5−0.052u’ (meter)01.51
Page 69 and 70: plane as∫ z+∆zφ(⃗u) = k n(
Page 71 and 72: −0.132.5−0.052u’ (meter)00.05
Page 73 and 74: formh(⃗v, ⃗u) = 1 (jλd exp j 2
Page 75 and 76: The transverse coherence length has
Page 77 and 78: models - extended Heygens-Fresnel p
Page 79 and 80: 21.8free spaceturbulence10.91.60.81
Page 81 and 82: 2.5free spaceturbulence10.920.80.71
Page 83 and 84: Chapter 5Optimal Beam to Minimize t
Page 85 and 86: and the corresponding received inte
Page 87 and 88: following two sections, we will dis
Page 89 and 90:
where Γ(⃗u 1 , ⃗u 2 ) = 〈φ(
Page 91 and 92:
intensity can be expressed as∫∫
Page 93 and 94:
5.4 Selection of coherent modesThe
Page 95 and 96:
m = 0m = 1m = 2222000−2−2−2
Page 97 and 98:
Scintillation Index0.140.130.120.11
Page 99 and 100:
p = 0p = 1p = 2420−2420−2420−
Page 101 and 102:
N 1 2 3 4 5 6 7 8α 0 1.0000 0.1177
Page 103 and 104:
120100TheoreticalSimulation1.21Theo
Page 105 and 106:
Scintillation Index0.40.350.30.250.
Page 107 and 108:
Scintillation index of single mode0
Page 109 and 110:
0.90.80.7r 0= 0.05 mr 0= 0.1 mr 0=
Page 111 and 112:
0.6C n2 = 2x10−16C n2 = 5x10−16
Page 113 and 114:
known fact that planets scintillate
Page 115 and 116:
Chapter 6Summary6.1 ConclusionsIn t
Page 117 and 118:
showed that increasing the receiver
Page 119 and 120:
L m p (x) Laguerre polynomialn Inde
Page 121 and 122:
Bibliography[1] L. C. Andrews and R
Page 123 and 124:
[18] D. deWolf. Strong irradiance f
Page 125 and 126:
[36] A. Ishimaru. Flucturations of
Page 127 and 128:
[54] C. Primmerman and D. Fouche. T
Page 129 and 130:
[73] V. I. Tatarski. Waves Propagat
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