Abstract

I consider the far-field (or focal plane) irradiance distribution of a Gaussian beam that is truncated by a circular aperture in the presence of random angular jitter. First, for absence of jitter, I derive an accurate analytic approximation for the irradiance distribution within the main lobe of the beam for the case in which the beam diameter is less than the aperture diameter. Then I obtain the corresponding mean irradiance distribution in the presence of circularly symmetric normally distributed jitter. By maximizing the on-axis intensity I obtain the optimum ratio of the beam diameter to the aperture diameter in the presence of jitter and present results for the corresponding maximum on-axis intensity and encircled power as a function of the jitter statistics.

© 1995 Optical Society of America

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References

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  1. V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, Norwood, Mass., 1987), Chap. 3.
  2. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1989), Sec. 18.4.
  3. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
    [CrossRef]
  4. D. L. Fried, “Statistics of laser beam fade induced by pointing jitter,” Appl. Opt. 12, 422–423 (1973).
    [CrossRef] [PubMed]
  5. M. H. Leed, J. F. Holmes, “Effect of the turbulent atmosphere on the autocovariance function for a speckle field generated by a laser beam with random pointing error,”J. Opt. Soc. Am. 71, 559–565 (1981).
    [CrossRef]
  6. H. T. Yura, “Ladar detection statistics in the presence of pointing errors,” Appl. Opt. 33, 6482–8498 (1994).
    [CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Far-field beam pattern of a truncated Gaussian beam as function of normalized angular coordinate u[= θ/(λ/πD)] for d/D = 0.9. The dashed and solid curves are based on the exact [i.e., Eq. (2.5)] and the approximate [i.e., Eq. (2.9)] expressions for the beam pattern, respectively.

Fig. 2
Fig. 2

Same as Fig. 1, except that d/D = 0.8.

Fig. 3
Fig. 3

Same as Fig. 1, except that d/D = 0.7.

Fig. 4
Fig. 4

Same as Fig. 1, except that d/D = 0.6.

Fig. 5
Fig. 5

Far-field on-axis irradiance as a function of d/D for various values of the normalized-jitter standard deviation ρ [= σj/(λ/πD), where σj is the 1-axis 1-σ jitter standard deviation].

Fig. 6
Fig. 6

Optimum ratio of the beam diameter in the aperture to the truncation diameter as a function of normalized jitter.

Fig. 7
Fig. 7

Maximum mean far-field on-axis irradiance (i.e., the value for the optimum ratio of d/D) as a function of normalized jitter.

Fig. 8
Fig. 8

Encircled power distribution ΔP/PT as a function of normalized angular coordinate u0 [=θ0/(λ/πD)] for various values of normalized jitter ρ. For each value of ρ the beam is truncated at its optimum value.

Equations (21)

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I ( θ ) = I ( 0 ) G ( θ ) ,
I ( 0 ) = I U F trunc ,
I U = P A ( λ R ) 2 ,
F trunc = 2 [ 1 - exp ( - μ 2 ) ] 2 μ 2 ;
G ( θ ) = [ 2 0 μ d x x exp ( - x 2 ) J 0 ( u x / μ ) 1 - exp ( - μ 2 ) ] 2 ,
u = θ ( λ / π D ) ,
μ = D d ,
I A ( θ ) = I ( 0 ) G A ( θ ) ,
G A ( θ ) = exp [ - 1 2 ( θ 2 / θ B 2 ) ] ,
θ B 2 = ( λ π d ) 2 + ( λ b π D ) 2 ,
P T = [ 1 - exp ( - 2 μ 2 ) ] P ,
b = 2 μ [ exp ( μ 2 ) - 1 ] 1 / 2 .
θ 1 e 2 = 2 λ π D eff ,
D eff = d [ tanh ( D 2 / 2 d 2 ) ] 1 / 2 .
I ( 0 ) = I ( 0 ) F jitter = I U F trunc F jitter ,
F jitter = 1 1 + σ j 2 / θ B 2
G A j ( 0 ) = exp [ - 1 2 ( θ 2 θ B 2 + σ j 2 ) ] .
2 μ o 2 - [ exp ( μ o 2 ) - 1 ] { 1 - 2 ρ 2 [ exp ( μ o 2 ) + 1 ] 2 } = 0 ,
ρ = σ j / ( λ / π D )
( d / D ) opt = { 0.892 - 0.0536 ρ 2 + 0.0089 ρ 4 for 0 ρ 1.3 0.468 + 0.592 ρ - 0.085 ln ρ for 1.3 < ρ 10 .
Δ P ( θ 0 ) P T = 0 θ 0 I A ( θ ) 2 π θ d θ , = 1 - exp ( - 1 2 θ 0 2 θ B 2 ) .

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