Abstract

In studies of scalar diffraction theory and experimental practice, the basic geometric shape of a circle is widely used as an aperture. Its Fraunhofer diffraction pattern has a simple mathematical expression easily determined by use of the Fourier–Bessel transform. However, it may require considerable mathematical effort to determine the far-field diffraction patterns of aperture shapes related to the circular geometry. From a computational point of view, the mathematical difficulties posed by other aperture geometries as well as more-general apertures with irregular shapes can be surpassed by means of optical setups or fast numerical algorithms. Unfortunately, no additional insight or information can be obtained from their exclusive application, as would be the case if mathematical formulas were available. The research presented here describes the far-field diffraction patterns of single-sector apertures as well as their extension to double symmetrical sectors and to sector wheels formed by interleaved transparent sectors of equal angular size; in each case, full or annular sectors are considered. The analytic solutions of their far-field amplitude distribution are given here in terms of a series of Bessel functions, some interesting properties are deduced from these solutions, and several examples are provided to illustrate graphically the results obtained from approximate numerical computations. Our results have been verified numerically with the fast-Fourier-transform algorithm and experimentally by means of a spherical wavefront–single-lens Fourier-transform architecture.

© 2005 Optical Society of America

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References

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2005

E. W. Weisstein, “Jinc function,” MathWorld, a Wolfram Web resource (2005), http://mathworld.wolfram.com/JincFunction.html .

2004

2003

1998

1996

D. W. Lozier, “Software needs in special functions,” J. Comput. Appl. Math. 66, 345–358 (1996).
[CrossRef]

1995

1986

1983

J. Komrska, “Fraunhofer diffraction from sector stars,” J. Mod. Opt. 30, 887–925 (1983).

1977

1972

1964

Agrest, M. M.

M. M. Agrest, M. S. Maksimov, Theory of Incomplete Cylindrical Functions and Their Applications (Springer-Verlag, 1971).
[CrossRef]

Andrews, G. E.

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, 1999).
[CrossRef]

Askey, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, 1999).
[CrossRef]

Barna, A.

Becklund, O. A.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (SPIE Press, 2002).

Bitterli, C. V.

Bolognini, N.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, 1994).

Campbell, S.

Cannon, S. M.

Cao, Q.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 2002).

Garavaglia, M. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, 1994).

Hirschhorn, M. D.

Jin, J.

S. Zhang, J. Jin, Computation of Special Functions (Wiley, 1996).

Komrska, J.

J. Komrska, “Fraunhofer diffraction from sector stars,” J. Mod. Opt. 30, 887–925 (1983).

Kowalcyzk, M.

Lessard, R. A.

Lohmann, A. W.

Lozier, D. W.

D. W. Lozier, “Software needs in special functions,” J. Comput. Appl. Math. 66, 345–358 (1996).
[CrossRef]

Luke, Y. L.

Y. L. Luke, “Integrals of Bessel functions,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramovitz, I. A. Stegun, eds. (Dover, 1970), pp. 479–494.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Part II, Wave Diffraction Optics, Vol. PM103 of SPIE Press Monographs (SPIE Press, 2001).
[CrossRef]

Mahan, A. I.

Maksimov, M. S.

M. M. Agrest, M. S. Maksimov, Theory of Incomplete Cylindrical Functions and Their Applications (Springer-Verlag, 1971).
[CrossRef]

Martinez-C., M.

Ojeda-C., J.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 2002).

Rabal, H. J.

Roy, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, 1999).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, 1994).

Serrano-H., A.

Sheppard, C. J. R.

Sicre, E. E.

Sorn, S. C.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 2002).

Trivi, M.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 2002).

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1966).

Weisstein, E. W.

E. W. Weisstein, “Jinc function,” MathWorld, a Wolfram Web resource (2005), http://mathworld.wolfram.com/JincFunction.html .

Williams, C. S.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (SPIE Press, 2002).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
[CrossRef]

Zapata-R., C. J.

Zhang, S.

S. Zhang, J. Jin, Computation of Special Functions (Wiley, 1996).

Appl. Opt.

J. Comput. Appl. Math.

D. W. Lozier, “Software needs in special functions,” J. Comput. Appl. Math. 66, 345–358 (1996).
[CrossRef]

J. Mod. Opt.

J. Komrska, “Fraunhofer diffraction from sector stars,” J. Mod. Opt. 30, 887–925 (1983).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

MathWorld, a Wolfram Web resource

E. W. Weisstein, “Jinc function,” MathWorld, a Wolfram Web resource (2005), http://mathworld.wolfram.com/JincFunction.html .

Other

G. N. Watson, Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1966).

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, 1999).
[CrossRef]

A. W. Lohmann, Optical Information Processing (University California Press, 1978), Vol. 1.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

Y. L. Luke, “Integrals of Bessel functions,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramovitz, I. A. Stegun, eds. (Dover, 1970), pp. 479–494.

S. Zhang, J. Jin, Computation of Special Functions (Wiley, 1996).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 2002).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. (Academic, 1994).

M. M. Agrest, M. S. Maksimov, Theory of Incomplete Cylindrical Functions and Their Applications (Springer-Verlag, 1971).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (SPIE Press, 2002).

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, 1994).

V. N. Mahajan, Optical Imaging and Aberrations: Part II, Wave Diffraction Optics, Vol. PM103 of SPIE Press Monographs (SPIE Press, 2001).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Single complete sector of radius R and single annular sector of radii R1 < R2: Angular size, θ ∊ (0, 2π], Θ1,2= ∓Θ/2, initial and final angles. The mirror line is the x axis.

Fig. 2
Fig. 2

Double symmetrical complete sector of radius R and double symmetrical annular sector of radii R1 < R2: Angular size, θ ∊ (0, π]; Θ1 = −Θ/2, initial angle; Θ4 = π + Θ/2 final angle. The mirror lines are the x and y axes.

Fig. 3
Fig. 3

Complete four-sector wheel of radius R and annular four-sector wheel of radii R1 < R2: Angular size, Θ = π/4 = 45°; Θ1 = −π/8, initial angle; Θ8 = 13π/8, final angle. The mirror lines are the x and y axes and the two diagonal dashed–dotted lines.

Fig. 4
Fig. 4

Single complete sectors for angular sizes Θ; = 60°, 120°, 180°, 240°, 300° and their respective far-field diffraction intensity patterns.

Fig. 5
Fig. 5

Single annular sectors for angular sizes Θ = 60°, 120°, 180°, 240°, 300° and their respective far-field diffraction intensity patterns.

Fig. 6
Fig. 6

Fraunhofer diffraction patterns for the semicircle and the semiannulus and their contour map representations normalized intensity from inner to outer contours, 1, 0.8, 0.6, 0.4, and 0.2.

Fig. 7
Fig. 7

Double symmetrical complete sectors for angular sizes Θ = 30°, 60°, 90°, 120°, 150°, and their respective far-field diffraction intensity patterns.

Fig. 8
Fig. 8

Double symmetrical annular sectors for angular sizes Θ = 30°, 60°, 90°, 120°, 150° and their respective far-field diffraction intensity patterns.

Fig. 9
Fig. 9

Fraunhofer diffraction patterns for double symmetrical complete and annular sectors with Θ = π/2, and their contour map representations, normalized intensity from inner to outer contours, 1, 0.8, 0.6, 0.4, and 0.2.

Fig. 10
Fig. 10

Complete sector wheels for an even number of sectors with σ = 2, 4, 6, 8, 10 and their respective far-field diffraction intensity patterns.

Fig. 11
Fig. 11

Annular sector wheels for an even number of sectors with σ = 2, 4, 6, 8, 10 and their respective far-field diffraction intensity patterns.

Fig. 12
Fig. 12

Fraunhofer diffraction patterns for the complete and annular four-sector wheel and their contour map representations; normalized intensity from inner to outer contours, 1, 0.8, 0.6, 0.4, and 0.2.

Fig. 13
Fig. 13

Complete sector wheels for an odd number of sectors with σ = 3, 5, 7, 9, 11 and their respective far-field diffraction intensity patterns.

Fig. 14
Fig. 14

Annular sector wheels for an odd number of sectors with σ = 3, 5, 7, 9, 11 and their respective far-field diffraction intensity patterns.

Fig. 15
Fig. 15

Fraunhofer diffraction patterns for the complete and annular three-wheel sector and their contour map representations; normalized intensity from inner to outer contours, 1, 0.8, 0.6, 0.4, and 0.2.

Fig. 16
Fig. 16

Complete and annular sector wheels for σ = 16, 24, 36 and their respective far-field diffraction intensity patterns.

Fig. 17
Fig. 17

Pattern intensity profiles along mirror line. M0 = 0 (solid curves) and its orthogonal direction M 0 = π / 2 (dashed curves). Top row, semicircle and semiannulus; bottom row, double symmetrical complete and annular sectors with Θ = 90°. The Vertical scale is logarithmic.

Fig. 18
Fig. 18

Pattern intensity profiles along mirror line M0 = 0 (solid curves) and its orthogonal direction M 0 = π / 2 (dashed curves). Top row, complete and annular three wheel; bottom row, complete and annular four wheel. The vertical scale is logarithmic.

Fig. 19
Fig. 19

Optical–digital Fourier transformer. Lens L has a focal length of 500 mm, and the LCTV can be displaced to the right of L to scale the transform.

Tables (1)

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Table 1 Circular Sector Related Apertures

Equations (69)

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f ( r , θ ) = { 1 r [ R 1 , R 2 ] , θ t = 1 σ [ Θ 2 t 1 , Θ 2 t ] 0 otherwise ,
f ( r , θ ) = f ( r , θ ± 2 π σ ) .
f ( r , θ ) = f ( r , 2 M μ θ ) .
exp [ x ( t t 1 ) / 2 ] = n = t n J n ( x ) .
exp ( i x cos θ ) = n = i n exp ( i n θ ) J n ( x ) .
n = i n exp ( i n θ ) J n ( x ) = J 0 ( x ) + n = 1 i n [ exp ( i n θ ) + exp ( i n θ ) ] J n ( x ) ,
exp ( i x cos θ ) = J 0 ( x ) + 2 n = 1 ( i ) n J n ( x ) cos n θ = n = 0 n ( i ) n J n ( x ) cos n θ ,
[ exp ( i x cos θ ) ] = n = 0 ( 1 ) n 2 n J 2 n ( x ) cos 2 n θ ,
J [ exp ( i x cos θ ) ] = n = 1 ( 1 ) n 2 n 1 J 2 n 1 ( x ) × cos ( 2 n 1 ) θ .
{ f ( r , θ ) } = F ( ρ , φ ) = 0 0 2 π f ( r , θ ) exp [ i 2 π r ρ × cos ( θ φ ) ] r d r d θ .
F ( ρ , φ ) = R 1 R 2 { t = 1 σ Θ 2 t 1 Θ 2 t exp [ i 2 π r ρ cos ( θ φ ) ] d θ } r d r .
F ( ρ , φ ) = R 1 R 2 ( t = 1 σ Θ 2 t 1 Θ 2 t [ n = 0 n ( i ) n J n ( ξ r ) cos n ( θ φ ) ] d θ ) r d r , = n = 0 n ( i ) n R 1 R 2 r J n ( ξ r ) d r × [ t = 1 σ Θ 2 t 1 Θ 2 t cos n ( θ φ ) d θ ] .
Ξ n ( ρ ) = R 1 R 2 r J n ( 2 π ρ r ) d r ,
Φ n ( φ ) = t = 1 σ Θ 2 t 1 Θ 2 t cos n ( θ φ ) d θ .
F ( ρ , φ ) = n = 0 n ( i ) n Φ n ( φ ) Ξ n ( ρ ) ,
R { F ( ρ , φ ) } = n = 0 ( 1 ) n 2 n Φ 2 n ( φ ) Ξ 2 n ( ρ ) ,
J { F ( ρ , φ ) } = n = 1 ( 1 ) n 2 n 1 Φ 2 n 1 ( φ ) × Ξ 2 n 1 ( ρ ) .
Ξ n ( ρ ) = 1 ( 2 π ρ ) 2 2 π R 1 ρ 2 π R 2 ρ r J n ( r ) d r = 1 2 π n [ rect ( r r 0 ) ] ( ρ ) ,
Φ n ( φ ) = 1 n t = 1 σ [ sin n ( φ Θ 2 t 1 ) sin n ( φ Θ 2 t ) ] , n 0 .
n ( i ) n Φ n ( φ ) Ξ n ( ρ ) | n = o = σ Θ [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] .
Φ n ( φ ) = 1 n [ sin n ( φ Θ 1 ) sin n ( φ Θ 2 ) ] , sin n ( φ Θ 1 ) = sin n ( φ + 0.5 Θ ) , sin n ( φ Θ 2 ) = sin n ( φ 0.5 Θ ) .
Φ n ( φ ) = 2 n sin n Θ 2 cos n φ .
F ss ( ρ , φ ; Θ ) = Θ [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 1 ( π ρ ) 2 n = 1 ( i ) n n sin n Θ 2 cos n φ × 2 π R 1 ρ 2 π R 2 ρ r J n ( r ) d r ,
F ss ( ρ , φ ; 2 π ) = 2 π [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] .
Φ n ( φ ) = 1 n t = 1 2 [ sin n ( φ Θ 2 t 1 ) sin n ( φ Θ 2 t ) ] , sin n ( φ Θ 1 , 2 ) = sin n ( φ ± 0.5 Θ ) , sin n ( φ Θ 3 , 4 ) = sin n ( φ π ± 0.5 Θ ) .
Φ n ( φ ) = 2 n sin n Θ 2 [ cos n φ + cos n ( φ π ) ] , = 2 n sin n Θ 2 cos n φ [ 1 + ( 1 ) n ] .
F ds ( ρ , φ ; Θ ) = 2 Θ [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 1 ( π ρ ) 2 n = 1 ( i ) n [ 1 + ( 1 ) n ] n sin n Θ 2 × cos n φ 2 π R 1 ρ 2 π R 2 ρ r J n ( r ) d r .
Φ n ( φ ) = 1 n t = 1 σ [ sin n ( φ Θ 2 t 1 ) sin n ( φ Θ 2 t ) ] , sin n ( φ Θ 2 t 1 , 2 t ) = sin n [ ( φ 2 Θ ( t 1 ) ± 0.5 Θ ) ] .
Φ n ( φ ) = 2 n sin ( n π 2 σ ) S n σ ( φ ) ,
S n σ ( φ ) = t = 1 σ cos n [ φ 2 π σ ( t 1 ) ] = { σ cos n φ n 0 ( mod σ ) 0 n 0 ( mod σ )
F ws ( ρ , φ ; σ ) = π [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 1 ( π ρ ) 2 n = 1 ( i ) n n sin ( n π 2 σ ) S n σ ( φ ) × 2 π R 1 ρ 2 π R 2 ρ r J n ( r ) d r .
F ws ( ρ , φ ; σ ) = π [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 1 2 π σ ρ 2 n = 1 ( i ) n sinc ( n 2 σ ) S n σ ( φ ) × 2 π R 1 ρ 2 π R 2 ρ r J n ( r ) d r .
0 r t μ J ν ( t ) d t = z μ Γ [ ½ ( ν + μ + 1 ) ] Γ [ ½ ( ν μ + 1 ) ] k = 0 ( ν + 2 k + 1 ) Γ [ ½ ( ν μ + 1 ) + k ] Γ [ ½ ( ν + μ + 3 ) + k ] J ν + 2 k + 1 ( r ) .
0 r t J n ( t ) d t = r Γ [ 1 + ( n / 2 ) ] Γ ( n / 2 ) k = 0 ( n + 2 k + 1 ) Γ [ ( n / 2 ) + k ] Γ [ ( n / 2 ) + k + 2 ] J n + 2 k + 1 ( r ) .
0 r t J n ( t ) d t = 2 r n k = 0 ( n + 2 k + 1 ) ( n + 2 k + 2 ) ( n + 2 k ) × J n + 2 k + 1 ( r ) ,
= 2 r n k = 0 ξ n , k ξ n , k 2 1 J ξ ( n , k ) ( r ) ,
2 π R 1 ρ 2 π R 2 ρ t J n ( t ) d t = 4 π n ρ k = 0 ξ n , k ξ n , k 2 1 [ R 2 J ξ ( n , k ) × ( 2 π R 2 ρ ) R 1 J ξ ( n , k ) ( 2 π R 1 ρ ) ] .
F ss ( ρ , φ ; Θ ) = Θ R 2 Jinc ( R ρ ) + 4 R π ρ n = 1 ( i ) n × sin n Θ 2 cos n ϕ k = 0 ξ n , k ξ n , k 2 1 × J ξ ( n , k ) ( 2 π R ρ ) .
F dS ( ρ , φ ; Θ ) = 2 Θ R 2 Jinc ( R ρ ) + 4 R π ρ n = 1 ( i ) n [ 1 + ( 1 ) n ] × sin n Θ 2 cos n φ k = 0 ξ n , k ξ n , k 2 1 J ξ ( n , k ) ( 2 π R ρ ) .
F ws ( ρ , φ ; σ ) = π R 2 Jinc ( R ρ ) + 4 R π ρ n = 1 ( i ) n × sin ( n π 2 σ ) S n σ ( φ ) k = 0 ξ n , k ξ n , k 2 1 × J ξ ( n , k ) ( 2 π R ρ ) .
{ F ss } = ( · ) + ( · ) n = 1 ( i ) n sin n Θ × cos 2 n ( · ) k = 0 ξ 2 n , k J ξ ( n , k ) ( · ) ξ 2 n , k 2 1 ,
J { F ss } = ( · ) + ( · ) n = 1 ( 1 ) n sin ( 2 n 1 ) Θ 2 cos ( 2 n 1 ) ( · ) k = 0 ξ 2 n 1 , k J ξ ( 2 n 1 , k ) ( · ) ξ 2 n 1 , k 2 1 .
( n = 1 a n ) ( k = 0 b n , k ) = n = 1 c n = n = 1 k = 1 n a k b k , n k .
b k , n k = ξ 2 k , n k ξ 2 k , n k 2 1 J ξ ( 2 k , n k ) ( · ) = 2 n + 1 4 n ( n + 1 ) J 2 n + 1 ( · ) .
b k , n k = ξ 2 k 1 , n k ξ 2 k 1 , n k 2 1 J ξ ( 2 k 1 , n k ) ( · ) = 2 n 4 n 2 1 J 2 n ( · ) .
( n = 1 a n ) ( k = 0 b n , k ) = n = 1 c n = n = 1 b n * k = 1 n a k ,
{ F SS ( ρ , φ ; Θ ) } = Θ [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 1 π ρ n = 1 2 n + 1 n ( n + 1 ) [ R 2 J 2 n + 1 ( 2 π R 2 ρ ) R 1 J 2 n + 1 ( 2 π R 1 ρ ) ] k = 1 n ( 1 ) k × sin k Θ cos 2 k φ ,
J { F SS ( ρ , φ ; Θ ) } = 8 π ρ n = 1 n 4 n 2 1 [ R 2 J 2 n ( 2 π R 2 ρ ) R 1 J 2 n ( 2 π R 1 ρ ) ] k = 1 n ( 1 ) k sin ( 2 k 1 ) Θ 2 cos ( 2 k 1 ) φ .
{ F ds ( ρ , φ ; Θ ) } = 2 Θ [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 2 π ρ n = 1 2 n + 1 n ( n + 1 ) [ R 2 J 2 n + 1 ( 2 π R 2 ρ ) R 1 J 2 n + 1 ( 2 π R 1 ρ ) ] k = 1 n ( 1 ) k × sin k Θ cos 2 k φ ,
J { F ds ( ρ , φ ; Θ ) } = 0 .
{ F ws ( ρ , φ ; σ ) } = π [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] + 1 π ρ n = 1 2 n + 1 n ( n + 1 ) [ R 2 J 2 n + 1 ( 2 π R 2 ρ ) R 1 J 2 n + 1 ( 2 π R 1 ρ ) ] k = 1 n ( 1 ) k × sin ( k π σ ) S 2 k σ ( φ ) ,
J { F ws ( ρ , φ ; σ ) } = 8 π ρ n = 1 n 4 n 2 1 [ R 2 J 2 n ( 2 π R 2 ρ ) R 1 J 2 n ( 2 π R 1 ρ ) ] k = 1 n ( 1 ) k × sin ( k π σ π 2 σ ) S 2 k 1 σ ( φ ) .
k = 1 n ( 1 ) k sin ( k π σ ) S 2 k σ ( φ ) = σ p = 1 Q ( - 1 ) 1 / 2 σ ( 2 p 1 ) + p + 1 × cos ( 2 p 1 ) σ φ ,
k = 1 n ( 1 ) k sin ( k π σ π 2 σ ) S 2 k 1 σ ( φ ) = σ p = 1 Q * ( - 1 ) ½ [ σ ( 2 p 1 ) + 1 ] + p + 1 cos ( 2 p 1 ) σ φ ,
F * ( ρ , φ + π ) = F ( ρ , φ ) , F ( ρ , φ ± 2 π σ ) = F ( ρ , φ ) , σ even ;
F ( ρ , φ ± π σ ) = F ( ρ , φ ) , σ odd ;
F ( ρ , 2 M μ φ ) = F ( ρ , φ ) , μ = 0 , , σ 1 .
F ss ( 0 , φ ; Θ ) = Θ 2 ( R 2 2 R 1 2 ) , F ds ( 0 , φ ; Θ ) = Θ ( R 2 2 R 1 2 ) , F ws ( 0 , φ ; σ ) = π 2 ( R 2 2 R 1 2 ) ,
I ss ( 0 , φ ; Θ ) I ss ( 0 , φ ; 2 π ) = ( Θ 2 π ) 2 , I ds ( 0 , φ ; Θ ) I ds ( 0 , φ ; π ) = ( Θ π ) 2 , I ws ( 0 , φ ; σ ) 4 I ws ( 0 , φ ; ) = 1 4 .
{ F ss ( ρ , 0 ; Θ ) } = Θ [ · ] + 1 π ρ n = 1 ( · ) k = 1 n ( 1 ) k sin k Θ , J { F ss ( ρ , 0 ; Θ ) } = 8 π ρ n = 1 ( · ) k = 1 ( 1 ) k × ( 2 k 1 ) Θ 2 ,
{ F ss ( ρ , π 2 ; Θ ) } = Θ [ · ] + 1 π ρ n = 1 ( · ) k = 1 n sin k Θ , J { F ss ( ρ , π 2 ; Θ ) } = 0 .
{ F ds ( ρ , 0 ; Θ ) } = 2 Θ [ · ] + 2 π ρ n = 1 ( · ) k = 1 n ( 1 ) k sin k Θ , J { F ds ( ρ , π 2 ; Θ ) } = 2 Θ [ · ] + 2 π ρ n = 1 ( · ) k = 1 n sin k Θ ,
k = 1 n ( 1 ) k sin k Θ = sin n + 1 2 ( Θ + π ) sin n 2 ( Θ + π ) sec Θ 2 ,
k = 1 n ( 1 ) k sin ( 2 k 1 ) Θ 2 = 1 2 ( 1 ) n sin n Θ sec Θ 2 ,
k = 1 n sin k Θ = sin n + 1 2 Θ sin n 2 Θ csc Θ 2
lim Θ π ( 1 ) n sin n Θ 2 cos ( Θ / 2 ) = n .
I ss ( ρ , 0 ; π ) = 2 { F ss ( ρ , 0 ; π ) } + J 2 { F ss ( ρ , 0 ; π ) } , I ss ( ρ , 0 ; π ) = π 2 [ R 2 2 Jinc ( R 2 ρ ) R 1 2 Jinc ( R 1 ρ ) ] 2 + 256 π 2 ρ 2 [ n = 1 n 2 4 n 2 1 [ R 2 J 2 n ( 2 π R 2 ρ ) R 1 J 2 n ( 2 π R 1 ρ ) ] ] 2 .
σ even : { F ws ( ρ , π σ μ ; σ ) } = π [ · ] ± σ π ρ n = 1 ( · ) × p = 1 Q ( 1 ) ½ σ ( 2 p 1 ) + p + 1 J { F ws ( ρ , π σ μ ; σ ) } = 0 ;
σ odd { F ws ( ρ , π σ μ ; σ ) } = π [ · ] , J { F ws ( ρ , π σ μ ; σ ) } = ± σ π ρ n = 1 ( · ) × p = 1 Q * ( 1 ) ½ [ σ ( 2 p 1 ) + 1 ] + p + 1 .

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