Abstract

We obtained an exact analytical solution of the annular-aperture averaged angle-of-arrival (AOA) variance for plane waves. The difference between the circular- and annular-aperture averaged AOA variance with the same values of the outer diameter is calculated, and the results show that the difference is no more than 3.12%, if the inner diameter is not larger than half the outer diameter.

© 2010 Optical Society of America

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References

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  1. W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
    [CrossRef]
  2. Y. Cheon and A. Muschinski, J. Opt. Soc. Am. A 24, 415 (2007).
    [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
  4. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).
  5. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).
  6. G. Fikioris, Mellin-Transform Method for Integral Evaluation (Morgan & Claypool, 2007).
  7. R. J. Sasiela and J. D. Shelton, J. Math. Phys. 34, 2572 (1993).
    [CrossRef]

2009 (1)

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

2007 (1)

1993 (1)

R. J. Sasiela and J. D. Shelton, J. Math. Phys. 34, 2572 (1993).
[CrossRef]

Cheon, Y.

Du, W.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

Fikioris, G.

G. Fikioris, Mellin-Transform Method for Integral Evaluation (Morgan & Claypool, 2007).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Jiang, Y.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

Ma, J.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

Muschinski, A.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Sasiela, R. J.

R. J. Sasiela and J. D. Shelton, J. Math. Phys. 34, 2572 (1993).
[CrossRef]

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

Shelton, J. D.

R. J. Sasiela and J. D. Shelton, J. Math. Phys. 34, 2572 (1993).
[CrossRef]

Tan, L.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

Xie, W.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

Yu, S.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

J. Math. Phys. (1)

R. J. Sasiela and J. D. Shelton, J. Math. Phys. 34, 2572 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, Opt. Commun. 282, 705 (2009).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

G. Fikioris, Mellin-Transform Method for Integral Evaluation (Morgan & Claypool, 2007).

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Figures (2)

Fig. 1
Fig. 1

Dependence relation between γ ( q , ε ) and the ratio of the inner to outer diameter.

Fig. 2
Fig. 2

Difference (%) between the circular- and annular-aperture averaged AOA variance.

Equations (16)

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( Δ α ) 2 = lim ρ 0 D s ( ρ ) k 2 ρ 2 ,
D s ( ρ ) = 4 π 2 k 2 L 0 κ d κ [ 1 J 0 ( κ ρ ) ] f s ( κ ) Φ n ( κ ) ,
f s ( κ ) = 1 + 2 π ( κ f ) 2 sin ( κ f ) 2 2 π ,
σ α 2 = ( Δ α ) 2 = π 2 L 0 d κ κ 3 Φ n ( κ ) [ 1 + 2 π ( κ f ) 2 sin ( κ f ) 2 2 π ] F ( D κ / 2 , ε ) ,
F ( x , ε ) = ( 2 1 ε 2 ) 2 [ J 1 ( x ) x ε 2 J 1 ( ε x ) ε x ] 2 ,
σ α 2 = C 0 0 [ Q 1 ( κ ) + Q 2 ( κ ) Q 3 ( κ ) + Q 4 ( κ ) + Q 5 ( κ ) Q 6 ( κ ) ] d κ ,
C 0 = 0.033 × 16 × C n 2 π 2 L ( 1 ε 2 ) 2 , Q 1 ( κ ) = D 2 κ 8 / 3 [ J 1 ( κ D / 2 ) ] 2 , Q 2 ( κ ) = ε 2 D 2 κ 8 / 3 [ J 1 ( ε κ D / 2 ) ] 2 , Q 3 ( κ ) = 2 ε D 2 κ 8 / 3 J 1 ( κ D / 2 ) J 1 ( ε κ D / 2 ) , Q 4 ( κ ) = 2 π f 2 D 2 κ 14 / 3 [ J 1 ( κ D / 2 ) ] 2 sin ( κ f ) 2 2 π , Q 5 ( κ ) = 2 π f 2 ε 2 D 2 κ 14 / 3 [ J 1 ( ε κ D / 2 ) ] 2 sin ( κ f ) 2 2 π , Q 6 ( κ ) = 4 π f 2 ε D 2 κ 14 / 3 J 1 ( κ D / 2 ) J 1 ( ε κ D / 2 ) sin ( κ f ) 2 2 π .
0 Q 1 ( κ ) d κ = C 1 D 1 / 3 ,
0 Q 2 ( κ ) d κ = C 2 ε 11 / 3 D 1 / 3 ,
0 Q 3 ( κ ) d κ = C 3 ε 2 F 1 2 ( 1 / 6 , 5 / 6 ; 2 ; ε 2 ) D 1 / 3 ,
0 Q 4 ( κ ) d κ = C 41 q 1 / 3 F 5 4 ( 1 / 12 , 3 / 4 , 5 / 4 , 5 / 12 ; 1 / 2 , 3 / 2 , 2 , 1 , 3 / 2 ; π 2 q 4 / 16 ) D 1 / 3 C 42 q 7 / 3 F 5 4 ( 7 / 12 , 5 / 4 , 7 / 4 , 1 / 12 ; 5 / 2 , 3 / 2 , 2 , 2 , 3 / 2 ; π 2 q 4 / 16 ) D 1 / 3 ,
0 Q 5 ( κ ) d κ = C 51 ε 4 q 1 / 3 F 5 4 ( 1 / 12 , 3 / 4 , 5 / 4 , 5 / 12 ; 1 / 2 , 3 / 2 , 2 , 1 , 3 / 2 ; π 2 ε 4 q 4 / 16 ) D 1 / 3 C 52 ε 6 q 7 / 3 F 5 4 ( 7 / 12 , 5 / 4 , 7 / 4 , 1 / 12 ; 5 / 2 , 3 / 2 , 2 , 2 , 3 / 2 ; π 2 ε 4 q 4 / 16 ) D 1 / 3 ,
0 Q 6 ( κ ) d κ = C 6 ε 2 q 7 / 3 D 1 / 3 ,
C 6 = 2 35 / 3 π 5 / 3 m = 0 ( 1 ) m m ! ( π ε 2 q 2 16 ) 2 m × { C 61 Γ ( 1 / 2 m ) Γ ( m + 1 / 12 ) Γ ( 3 / 2 + m ) Γ ( 1 + m ) Γ ( 17 / 12 m ) · F 3 2 ( m + 1 12 , 5 12 + m ; 1 2 , 1 , 3 2 ; π 2 q 4 256 ) + C 62 Γ ( 1 / 2 m ) Γ ( m + 7 / 12 ) Γ ( 11 / 12 m ) Γ ( 3 / 2 + m ) Γ ( 2 + m ) · F 3 2 ( m + 7 12 , 1 12 + m ; 1 2 , 1 , 3 2 ; π 2 q 4 256 ) + C 63 Γ ( 1 / 2 m ) Γ ( m + 7 / 12 ) Γ ( 1 + m ) Γ ( 3 / 2 + m ) Γ ( 11 / 12 m ) · F 3 2 ( m + 7 12 , 1 12 + m ; 3 2 , 2 , 3 2 ; π 2 q 4 256 ) + C 64 Γ ( 1 / 2 m ) Γ ( 13 / 12 + m ) Γ ( 3 / 2 + m ) Γ ( 2 + m ) Γ ( 5 / 12 m ) · F 3 2 ( m + 13 12 , 7 12 + m ; 3 2 , 2 , 3 2 ; π 2 q 4 256 ) } ,
σ α 2 = γ ( q , ε ) C n 2 L D 1 / 3 ,
γ ( q , ε ) = 0.033 × 16 × π 2 ( 1 ε 2 ) 2 [ C 1 + C 2 ε 11 / 3 C 3 ε 2 F 1 2 ( 1 / 6 , 5 / 6 ; 2 ; ε 2 ) + C 41 q 1 / 3 F 5 4 ( 1 / 12 , 3 / 4 , 5 / 4 , 5 / 12 ; 1 / 2 , 3 / 2 , 2 , 1 , 3 / 2 ; π 2 q 4 / 16 ) C 42 q 7 / 3 F 5 4 ( 7 / 12 , 5 / 4 , 7 / 4 , 1 / 12 ; 5 / 2 , 3 / 2 , 2 , 2 , 3 / 2 ; π 2 q 4 / 16 ) + C 51 ε 4 q 1 / 3 F 5 4 ( 1 / 12 , 3 / 4 , 5 / 4 , 5 / 12 ; 1 / 2 , 3 / 2 , 2 , 1 , 3 / 2 ; π 2 ε 4 q 4 / 16 ) C 52 ε 6 q 7 / 3 F 5 4 ( 7 / 12 , 5 / 4 , 7 / 4 , 1 / 12 ; 5 / 2 , 3 / 2 , 2 , 2 , 3 / 2 ; π 2 ε 4 q 4 / 16 ) C 6 ε 2 q 7 / 3 ] .

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