Abstract

We introduce a method to analyze the diffraction integral for evaluating the point-spread function. Our method is based on the use of higher-order Airy functions along with Zernike and Taylor expansions. Our approach is applicable when we are considering a finite, arbitrary number of aberrations and arbitrarily large defocus simultaneously. We present an upper bound for the complexity and the convergence rate of this method. We also compare the cost and accuracy of this method with those of traditional ones and show the efficiency of our method through these comparisons. In particular, we rigorously show that this method is constructed in a way that the complexity of the analysis (i.e., the number of terms needed for expressing the light disturbance) does not increase as either defocus or resolution of interest increases. This has applications in several fields such as biological microscopy, lithography, and multidomain optimization in optical systems.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1992).
  2. B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
    [CrossRef]
  3. H. P. Urbach and D. A. Bernard, "Modeling latent-image formation in photolithography, using the Helmholtz equation," J. Opt. Soc. Am. A 6, 1343-1356 (1989).
    [CrossRef]
  4. W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
    [CrossRef] [PubMed]
  5. R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
    [CrossRef] [PubMed]
  6. E. R. Dowski and G. E. Johnson, "Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
    [CrossRef]
  7. J. Braat, P. Dirksen, and A. J. E. M. Janssen, "Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread-functions," J. Opt. Soc. Am. A 19, 858-870 (2002).
    [CrossRef]
  8. A. J. E. M. Janssen, "Extended Nijboer-Zernike approach for the computation of optical point-spread functions," J. Opt. Soc. Am. A 19, 849-857 (2002).
    [CrossRef]
  9. J. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, "Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system," J. Opt. Soc. Am. A 20, 2281-2292 (2003).
    [CrossRef]
  10. A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, "On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus," J. Mod. Opt. 51, 687-703 (2004).
    [CrossRef]
  11. H. A. Buchdahl, Optical Aberration Coefficients (Oxford U. Press, 1958).
  12. S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D, P. de Farias, "Analytical optimal solution of the extension of the depth of field using cubic phase wayefront coding" (in preparation; bagheri@mit.edu).
  13. C. L. Tranter, Bessel Functions with Some Physical Applications (Hart, 1969).
  14. A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, 1966).
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1944).
  16. L. Landau, "Monotonicity and bounds on Bessel functions," in Proceedings of Mathematical Physics and Quantum Field Theory, H.Warchall, ed. (2000), Vol. 4, pp. 147-154.

2005

2004

A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, "On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus," J. Mod. Opt. 51, 687-703 (2004).
[CrossRef]

2003

2002

2001

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
[CrossRef]

1999

E. R. Dowski and G. E. Johnson, "Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

1989

Bagheri, S.

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D, P. de Farias, "Analytical optimal solution of the extension of the depth of field using cubic phase wayefront coding" (in preparation; bagheri@mit.edu).

Bernard, D. A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1992).

Braat, J.

Braat, J. J. M.

A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, "On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus," J. Mod. Opt. 51, 687-703 (2004).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Oxford U. Press, 1958).

Cathey, W. T.

de Farias, D P.

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D, P. de Farias, "Analytical optimal solution of the extension of the depth of field using cubic phase wayefront coding" (in preparation; bagheri@mit.edu).

Dirksen, P.

Dowski, E. R.

W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
[CrossRef] [PubMed]

E. R. Dowski and G. E. Johnson, "Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Gray, A.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, 1966).

Hendriks, B. H. W.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
[CrossRef]

Janssen, A. J. E. M.

Johnson, G. E.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

E. R. Dowski and G. E. Johnson, "Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Landau, L.

L. Landau, "Monotonicity and bounds on Bessel functions," in Proceedings of Mathematical Physics and Quantum Field Theory, H.Warchall, ed. (2000), Vol. 4, pp. 147-154.

Mathews, G. B.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, 1966).

Narayanswamy, R.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D, P. de Farias, "Analytical optimal solution of the extension of the depth of field using cubic phase wayefront coding" (in preparation; bagheri@mit.edu).

Schleipen, J. J. H. B.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
[CrossRef]

Silveira, P. E. X.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D, P. de Farias, "Analytical optimal solution of the extension of the depth of field using cubic phase wayefront coding" (in preparation; bagheri@mit.edu).

Stallinga, S.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
[CrossRef]

Tranter, C. L.

C. L. Tranter, Bessel Functions with Some Physical Applications (Hart, 1969).

Urbach, H. P.

van de Nes, A. S.

van Houten, H.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
[CrossRef]

Wach, H. B.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1944).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1992).

Appl. Opt.

J. Mod. Opt.

A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, "On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus," J. Mod. Opt. 51, 687-703 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Rev.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, "Optical pickup for blue optical recording at NA=0.85," Opt. Rev. 6, 211-213 (2001).
[CrossRef]

Proc. SPIE

E. R. Dowski and G. E. Johnson, "Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems," in Current Developments in Optical Design and Optical Engineering VIII, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1992).

H. A. Buchdahl, Optical Aberration Coefficients (Oxford U. Press, 1958).

S. Bagheri, P. E. X. Silveira, R. Narayanswamy, and D, P. de Farias, "Analytical optimal solution of the extension of the depth of field using cubic phase wayefront coding" (in preparation; bagheri@mit.edu).

C. L. Tranter, Bessel Functions with Some Physical Applications (Hart, 1969).

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, 1966).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1944).

L. Landau, "Monotonicity and bounds on Bessel functions," in Proceedings of Mathematical Physics and Quantum Field Theory, H.Warchall, ed. (2000), Vol. 4, pp. 147-154.

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Figures (6)

Fig. 1
Fig. 1

Schematic view of the optical system under consideration.

Fig. 2
Fig. 2

Contour plots of the modulus of the PSF, h , in the presence of aberrations and defocus (normalized to 100).

Fig. 3
Fig. 3

Variation of the partial number of terms necessary with β L , M for ϵ = 0.001 and R * = 20 .

Fig. 4
Fig. 4

Radial variation of the modulus of the PSF with and without distortion (normalized to 2 π ).

Fig. 5
Fig. 5

Time required for evaluating the PSF at 400 different points versus defocus ( ϵ = 0.1 % ) .

Fig. 6
Fig. 6

Time required for evaluating the PSF versus resolution ( ϵ = 10 % ) .

Equations (131)

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h ( x , y ; x 0 , y 0 ) = n , m A n m J n + 1 ( R ) R cos [ m ( Θ + ϕ 0 ) ] ,
Q ( x , y ) = A h ( x , y ; x 0 , y 0 ) P ( x 0 , y 0 ) d x 0 d y 0 ,
h ( x , y ; x 0 , y 0 ) = C 0 2 π 0 1 e i k w ( ρ , θ , r 0 , ϕ 0 ) e i R ρ cos ( θ Θ ) ρ d ρ d θ .
u = k a ( x 0 r + x s ) ,
v = k a ( y 0 r + y s ) ,
r 2 = x 0 2 + y 0 2 + S P 2 ,
s 2 = x 2 + y 2 + S Q 2 ,
C = i k cos ( δ ) 2 π r s ,
tan ( δ ) = ( x 0 + x ) 2 + ( y 0 + y ) 2 S P + S Q .
C k 2 π S P S Q .
h ̂ ( x , y ; x 0 , y 0 ) = 1 2 π 0 2 π 0 1 e i k w ( ρ , θ , r 0 , ϕ 0 ) e i R ρ cos ( 0 Θ ) ρ d ρ d θ .
w ( ρ , θ , r 0 , ϕ 0 ) = l , m = 0 f l , m ( r 0 ) ρ 2 l + m cos m ( θ ϕ 0 ) .
w ( ρ , θ , r 0 , ϕ 0 ) = j = 1 n a b { f L j , M j ( r 0 ) ( a ρ ) 2 L j [ a ρ cos ( θ ϕ 0 ) ] M j } .
f L 1 , M 1 ( r 0 ) = f 1 , 0 ( r 0 ) = D F = 1 2 ( 1 S Q 1 S G ) .
n th - order Airy function = J n + 1 ( R ) R ,
h ̂ ( x , y ; x 0 , y 0 ) = n = 0 m = 0 n { δ ̃ n m δ m A n m cos [ m ( Θ + ϕ 0 ) ] J n + 1 ( R ) R } ,
δ ̃ i = { 1 if i is even 0 otherwise ,
δ i = { 1 if i = 0 2 otherwise ,
A n m = n + 1 2 m 1 e i m ϕ 0 ( N , D ) m D β N S n , k N m ( β ) .
m = { ( N , D ) j = 2 n a b ( M j N j ) = m + 2 k ,
D = { ( m + 2 k ) ! 2 2 k k ! ( m + k ) ! ; k , N j N } ,
S n , k N m ( β ) = 0 1 j χ 1 e β j ρ 2 L j R n m ( ρ ) ρ k N + 1 d ρ ,
β j = i k f L j , M j ( r 0 ) a 2 L j + M j ,
β N = j χ 2 ( β j ) N j N j ! ,
k N = j χ 2 ( 2 L j + M j ) N j ,
χ 1 = { j M j = 0 , j = 1 , , n a b } ,
χ 2 = { j M j 0 , j = 1 , , n a b } ,
χ 3 = { j M j = 0 , j = 2 , , n a b } ,
i k a 2 L + M f L , M ( r 0 ) = { γ 1 if ( L , M ) = ( 1 , 0 ) defocus and field curvature γ 2 if ( L , M ) = ( 2 , 0 ) spherical aberration γ 3 if ( L , M ) = ( 0 , 1 ) distortion γ 4 if ( L , M ) = ( 0 , 2 ) astigmatism γ 5 if ( L , M ) = ( 1 , 1 ) coma ,
A n m = n + 1 2 m 1 i n × ( N , D ) m D [ γ 3 N 3 γ 4 N 4 γ 5 N 5 N 3 ! N 4 ! N 5 ! S n , N 3 + 2 N 4 + 3 N 5 m ( γ 1 , γ 2 ) ] ,
m = { ( N , D ) = ( N 3 , N 4 , N 5 , D ) N 3 + 2 N 4 + N 5 = m + 2 k , D = ( m + 2 k ) ! 2 2 k k ! ( m + k ) ! ; k , N 3 , N 4 , N 5 N } ,
S n , N 3 + 2 N 4 + 3 N 5 m ( γ 1 , γ 2 ) = 0 1 e γ 1 ρ 2 + γ 2 ρ 4 R n m ( ρ ) ρ N 3 + 2 N 4 + 3 N 5 + 1 d ρ ,
h ̂ ( r ϕ ) = n = 0 m = 0 n { δ ̃ n m δ m A n m cos [ m ( ϕ + π ) ] J n + 1 ( B r ) B r } ,
0 = { ( 0 , 0 , 1 ) , ( 2 , 0 , 1 2 ) , ( 0 , 1 , 1 2 ) , ( 4 , 0 , 3 8 ) , ( 2 , 1 , 3 8 ) , ( 0 , 2 , 3 8 ) , ( 4 , 1 , 15 16 ) , ( 2 , 2 , 5 16 ) , ( 0 , 3 , 5 16 ) , ( 4 , 2 , 35 128 ) , ( 2 , 3 , 35 128 ) , ( 0 , 4 , 35 128 ) , ( 4 , 3 , 63 256 ) , ( 2 , 4 , 63 256 ) , ( 4 , 4 , 231 1024 ) } ,
11 = { ( 3 , 4 , 1 ) } ,
12 = { ( 4 , 4 , 1 ) } ,
S n , k m ( γ 1 ) = l = 0 ( n m ) 2 { C n , l m 2 ( γ 1 ) [ ( 2 + n 2 l + k ) ] 2 ( n 2 l + k 2 ) ! [ 1 e γ 1 j = 0 ( n 2 l + k ) 2 ( γ 1 ) j j ! ] } ,
h ̂ n * ( x , y ; x 0 , y 0 ) = n = 0 n * m = 0 n δ ̃ n m δ m A n m * cos [ ( Θ + ϕ 0 ) ] J n + 1 ( R ) R ,
A n m * = n + 1 2 m 1 e i m ϕ 0 ( N , D ) m * D β N S n , k N m ( β ) *
m * = { ( N , D ) j = 2 n a b ( M j N j ) = m + 2 k , D = ( m + 2 k ) ! 2 2 k k ! ( m + k ) ! ; N j N j * , k , N j N } ,
S n , k N m ( β ) * = l = 0 ( n m ) 2 C n , l m 0 1 e β 1 ρ 2 j χ 3 [ N j = 0 N j * ( β j ρ 2 L j ) N j N j ! ] ρ n 2 l + k N + 1 d ρ .
n * max [ 5 , e R * + 1 , 2 log 2 1 e ( 2 e 1 ) π ϵ ] ,
N j * max [ 4 , 2 e β j + 1 , log 2 6 e e 3 n a b ( 1 + R * 4 3 ) π ( 2 e 1 ) ϵ ] ,
h ̂ ( x , y ; x 0 , y 0 ) h ̂ n * ( x , y ; x 0 , y 0 ) ϵ
N * = n * + 2 2 n * + 2 2 .
χ 4 = { j L j 0 , j = 2 , , n a b } .
N * = R * + 7 2 R * + 7 2 .
V n m ( ρ , θ ) = R n m ( ρ ) e i m θ .
R n m ( ρ ) = l = 0 ( n m ) 2 C n , l m ρ n 2 l ,
C n , l m = ( 1 ) l ( n l ) ! l ! [ ( n + m ) 2 l ] ! [ ( n m ) 2 l ] ! .
e i k w ( ρ , θ , r 0 , ϕ 0 ) = n = 0 m = 0 n [ δ ̃ n m A n m R n m ( ρ ) e i m θ ] ,
A n m = n + 1 π 0 2 π 0 1 e i k w ( ρ , θ , r 0 , ϕ 0 ) V n m ( ρ , θ ) ρ d ρ d θ .
h ̂ ( x , y ; x 0 , y 0 ) = 1 2 π 0 2 π 0 1 { n = 0 m = 0 n [ δ ̃ n m A n m R n m ( ρ ) e i m θ ] } e i R ρ cos ( θ Θ ) ρ d ρ d θ .
0 2 π e i m θ e i R ρ cos ( θ Θ ) d θ = 2 π e i m θ i m J m ( R ρ ) ;
h ̂ ( x , y ; x 0 , y 0 ) = 0 1 { n = 0 m = 0 n [ δ ̃ n m A n m R n m ( ρ ) e i m Θ i m J m ( R ρ ) ] } ρ d ρ .
0 1 R n m ( ρ ) J m ( R ρ ) ρ d ρ = ( 1 ) ( n m ) 2 J n + 1 ( R ) R ,
h ̂ ( x , y ; x 0 , y 0 ) = n = 0 m = 0 n [ δ ̃ n m A n m e i m Θ i m ( 1 ) ( n m ) 2 J n + 1 ( R ) R ] .
e i k w ( ρ , θ , r 0 , ϕ 0 ) = e i k j = 1 n a b [ f L j , M j ( r 0 ) ( a ρ ) 2 L j ( a ρ cos ( θ ϕ 0 ) ) M j ] = e j = 1 n a b [ β j ρ 2 L j + M j cos M j ( θ ϕ 0 ) ] = e j χ 1 [ β j ρ 2 L j ] e j χ 2 [ β j ρ 2 L j + M j cos M j ( θ ϕ 0 ) ] = e j χ 1 [ β j ρ 2 L j ] j χ 2 e β j ρ 2 L j + M j cos M j ( θ ϕ 0 ) = e j χ 1 [ β j ρ 2 L j ] × j χ 2 { N j = 0 [ β j ρ 2 L j + M j cos M j ( θ ϕ 0 ) ] N j N j ! } ,
β j = i k f L j , M j ( r 0 ) a 2 L j + M j .
A n m = n + 1 π 0 2 π 0 1 e j χ 1 [ β j ρ 2 L j ] × j χ 2 { N j = 0 [ β j ρ 2 L j + M j cos M j ( θ ϕ 0 ) ] N j N j ! } × V n m ( ρ , θ ) ρ d ρ d θ .
A n m = n + 1 2 m 1 e i m ϕ 0 0 1 e j χ 1 [ β j ρ 2 L j ] R n m ( ρ ) × { ( N , D ) m D j χ 2 [ β j ρ 2 L j + M j ] N j N j ! } ρ d ρ ,
m m = m + 2 k : 0 2 π cos ( m θ ) ( cos ( θ ) ) m + 2 k d θ = π ( m + 2 k ) ! 2 2 k + m 1 k ! ( m + k ) ! , m < m = m + 2 k + 1 : 0 2 π cos ( m θ ) ( cos ( θ ) ) m + 2 k + 1 d θ = 0 , m > m : 0 2 π cos ( m θ ) ( cos ( θ ) ) m d θ = 0 , m , m : 0 2 π sin ( m θ ) ( cos ( θ ) ) m d θ = 0 ,
m = { ( N , D ) j = 2 n a b ( M j N j ) = m + 2 k ,
D = { ( m + 2 k ) ! 2 2 k k ! ( m + k ) ! ; k , N j N } .
h ̂ ( x , y ; x 0 , y 0 ) = n = 0 m = 0 n { δ ̃ n m δ m A n m cos [ m ( Θ + ϕ 0 ) ] J n + 1 ( R ) R } ,
A n m = n + 1 2 m 1 i n ( N , D ) m D β N S n , k N m ( β )
S n , k N m ( β ) = 0 1 j χ 1 e β j ρ 2 L j R n m ( ρ ) ρ k N + 1 d ρ ,
β N = j χ 2 ( β j ) N j N j ! .
k N = j χ 2 ( 2 L j + M j ) N j .
S n , k N m ( β ) = 0 1 j χ 1 e β j ρ 2 L j l = 0 ( n m ) 2 C n , l m ρ n 2 l + k N + 1 d ρ .
S n , k N m ( β ) = l = 0 ( n m ) 2 C n , l m 0 1 j χ 1 e β j ρ 2 L j ρ n 2 l + k N + 1 d ρ .
S n , k N m ( β ) = l = 0 ( n m ) 2 C n , l m 0 1 e β 1 ρ 2 j χ 3 [ N j = 0 ( β j ρ 2 L j ) N j N j ! ] ρ n 2 l + k N + 1 d ρ .
0 1 e β 1 ρ 2 ρ 2 τ + 1 d ρ ,
( β 1 ) ( τ + 1 ) 2 τ ! [ 1 e β 1 k = 0 τ ( β 1 ) k k ! ] .
S n , k N m ( β ) = l = 0 ( n m ) 2 { C n , l m 2 ( β 1 ) [ ( 2 + n 2 l + k N ) ] 2 ( n 2 l + k N 2 ) ! [ 1 e β 1 j = 0 ( n 2 l + k N ) 2 ( β 1 ) j j ! ] } .
S n , k N m ( β 1 ) = l = 0 ( n m ) 2 C n , l m 0 1 e i β ́ ρ ρ [ ( n + m ) 2 l + k ́ ] d ρ l = 0 ( n m ) 2 C n , l m 0 1 [ cos ( β ́ ρ ) + i ρ sin ( β ́ ρ ) ] d ρ = l = 0 ( n m ) 2 C n , l m e i β ́ i β ́ = e i β ́ i β ́ l = 0 ( n m ) 2 C n , l m = e i β ́ i β ́ = e β 1 β 1 .
S n , k N m ( β 1 ) = e β 1 + ( 1 ) ( n + 1 ) 2 β 1 .
S n , k N m ( β ) = { 0 if n m n ! 2 ( β 1 ) n + 1 [ 1 e β 1 k = 0 n ( β 1 ) k k ! ] if n = m .
ω l ω r ϵ .
Ω l Ω r < ϵ .
A n m n + 1 .
2 π A 00 2 + n = 1 m = 1 n [ δ ̃ n m 2 A n m 2 π n + 1 ] = 2 π ,
2 π A 00 2 + n = 1 m = 1 n [ 2 δ ̃ n m A n m 2 π n + 1 ] = 2 π .
A 00 2 + n = 1 m = 1 n [ δ ̃ n m A n m 2 1 n + 1 ] = 1 .
A n m n + 1 .
f ( x ) = e b x m ,
p * = max ( 4 , 2 e b + 1 , log 2 e ( 2 e 1 ) 2 π ϵ ) ,
f ( x ) n = 0 m p * T n f ( x ) ϵ f ( x ) .
f ( x ) n = 0 m p * T n f ( x ) = n = m p * + 1 T n f ( x ) = n = m p * + 1 f n ( 0 ) n ! x n .
f n ( 0 ) = { 0 if n m p n ! b p p ! if n = m p ,
f ( x ) n = 0 m p * T n f ( x ) = p = p * b p p ! x m p p = p * b p p ! p = p * b p p ! p = p * b p p * ! p * p p * = b p * p * ! p = p * ( b p * ) p b p * p * ! p = p * ( 1 2 e ) p = b p * p * ! 2 e 2 e 1 b p * 2 π p * p * + 0.5 e p * 2 e 2 e 1 = ( b e p * ) p * 2 e ( 2 e 1 ) 2 π p * ( 1 2 ) p * 2 e ( 2 e 1 ) 2 π p * ( 1 2 ) p * e ( 2 e 1 ) 2 π ϵ f ( x ) ϵ .
2 π p * p * + 0.5 exp ( p * ) p * ! .
n = 0 ( n + 1 ) 3 2 J n + 1 ( R ) 3 π e 2 ( 1 + R 4 3 ) R .
J n ( R ) 2 π 1 R 1 3 ,
J n ( R ) ( R 2 ) n n ! ,
J n ( R ) f n ( R ) = { ( R 2 ) n n ! for 0 R < n e 2 π 1 R 1 3 for R n e .
n = 0 ( n + 1 ) 3 2 J n + 1 ( R ) n = 1 e R n 3 2 2 π 1 R 1 3 + n = e R + 1 n 3 2 ( R 2 ) n n ! = I 1 + I 2 .
I 1 = 0 .
I 1 = 2 π 1 R 1 3 n = 1 e R n 3 2 2 π 1 R 1 3 1 e R + 1 x 3 2 d x = 2 2 5 π ( e R + 1 ) 5 2 1 R 1 3 2 2 5 π 1 R 1 3 e e ( R ) 4 3 [ 1 + ( e R + 2 ) 4 3 ] = 2 e 2 e 5 π R [ 1 + ( e R + 2 ) 4 3 ] .
I 1 2 e 2 e 5 π R [ 1 + ( e R + 2 ) 4 3 ] ,
I 2 = n = e R + 1 n 3 2 ( R 2 ) n n ! n = e R + 1 ( R 2 ) n ( n 2 ) ! = R 2 4 n = e R 1 ( R 2 ) n n ! R 2 4 n = e R 1 ( R 2 ) n ( e R 1 ) ! ( e R 1 ) n ( e R 1 ) = R 2 ( R 2 ) ( e R 1 ) 4 ( e R 1 ) ! n = 0 [ R 2 ( e R 1 ) ] n R 2 ( R 2 ) ( e R 1 ) 4 ( e R 1 ) ! n = 0 ( 1 e 1 ) n = ( e 1 ) R 2 ( R 2 ) ( e R 1 ) 4 ( e 2 ) ( e R 1 ) ! ( e 1 ) R 2 [ R 2 ( e R 1 ) ] e R 1 4 ( e 2 ) 2 π ( e R 1 ) ( e 1 ) R 3 2 ( 1 e 1 ) e R 1 4 ( e 2 ) 2 π 2 e 1 = e 1 R 3 2 ( 1 e 1 ) e R 1 4 ( e 2 ) π e 1 R 3 2 ( 1 e 1 ) e R 1 4 ( e 2 ) π ( e 1 ) e 1 R 4 ( e 2 ) π e 1 2 π ( e 2 ) R .
2 π ( e R 1 ) e R 1 + 0.5 exp [ ( e R 1 ) ] ( e R 1 ) ! .
n = 0 ( n + 1 ) 3 2 J n + 1 ( R ) 2 e 2 e 5 π R [ 1 + ( e R + 2 ) 4 3 ] + e 1 2 π ( e 2 ) R = R π { 2 e 2 e 5 [ 1 + ( e R + 2 ) 4 3 ] + e 1 2 ( e 2 ) } R π [ 3 e 2 ( 1 + R 4 3 ) ] = 3 π e 2 ( 1 + R 4 3 ) R .
A n m = n + 1 π 0 2 π 0 1 e β 1 ρ 2 j = 2 n a b f L j , M j ( ρ , θ ) V n m ( ρ , θ ) ρ d ρ d θ ,
A n m * = n + 1 π 0 2 π 0 1 e β 1 ρ 2 j = 2 n a b T L j , M j ( ρ , θ ) V n m ( ρ , θ ) ρ d ρ d θ .
h ̂ ( x , y ; x 0 , y 0 ) = n = 0 m = 1 n δ ̃ n m δ m A n m cos [ ( Θ + ϕ 0 ) ] J n + 1 ( R ) R ,
h ̂ n * ( x , y ; x 0 , y 0 ) = n = 0 n * m = 1 n δ ̃ n m δ m A n m * cos [ ( Θ + ϕ 0 ) ] J n + 1 ( R ) R
f L j , M j ( ρ , θ ) T L j , M j ( ρ , θ ) = ϵ L j , M j ( ρ , θ ) f L j , M j ( ρ , θ ) ,
N j * = max ( 4 , 2 e β j + 1 , log 2 e ( 2 e 1 ) 2 π ϵ ) ,
ϵ L j , M j ϵ ,
p * = max ( 4 , 2 e b + 1 , log 2 e ( 2 e 1 ) 2 π ϵ ) .
f L j , M j ( ρ , θ ) T L j , M j ( ρ , θ ) ϵ f L j , M j ( ρ , θ ) .
ϵ L j , M j ( ρ , θ ) f L j , M j ( ρ , θ ) ϵ f L j , M j ( ρ , θ ) .
ϵ L j , M j ϵ .
ϵ L j , M j ( ρ , θ ) = T L j , M j ( ρ , θ ) f L j , M j ( ρ , θ ) f L j , M j ( ρ , θ ) .
A n m A n m * = n + 1 π 0 2 π 0 1 ( j = 2 n a b f L j , M j j = 2 n a b T L j , M j ) e β 1 ρ 2 V n m ρ d ρ d θ = n + 1 π 0 2 π 0 1 [ j = 2 n a b f L j , M j j = 2 n a b f L j , M j ( 1 + ϵ L j , M j ) ] e β 1 ρ 2 V n m ρ d ρ d θ = n + 1 π 0 2 π 0 1 [ ( j = 2 n a b f L j , M j ) g ] e β 1 ρ 2 V n m ρ d ρ d θ .
g = 1 j = 2 n a b ( 1 + ϵ L j , M j ) = j = 2 n a b ϵ L j , M j + j , k = 2 n a b ϵ L j , M j ϵ L k , M k + + j = 2 n a b ϵ L j , M j .
g j = 2 n a b ϵ L j , M j + j , k = 2 n a b ϵ L j , M j ϵ L k , M k + + j = 2 n a b ϵ L j , M j ( n a b 1 1 ) ϵ + ( n a b 1 2 ) ϵ 2 + + ( n a b 1 n a b 1 ) ϵ n a b 1 ( n a b 1 ) ϵ + ( n a b 2 ) ϵ 2 + + ( n a b n a b ) ϵ n a b n a b ϵ ( 1 + ( 1 2 ) 2 2 ! + + ( 1 2 ) n a b n a b ! ) n a b ϵ j = 0 ( 1 2 ) j j ! = n a b ϵ e .
ϵ = π 2 3 e e 2 n a b ( 1 + R * 4 3 ) ϵ
A n m A n m * n + 1 π 0 2 π 0 1 g R n m ( ρ ) ρ d ρ d θ n a b ϵ e ( n + 1 ) π 0 2 π 0 1 R n m ( ρ ) ρ d ρ d θ = 2 n a b ϵ e ( n + 1 ) 0 1 R n m ( ρ ) ρ d ρ d θ 2 n a b ϵ e ( n + 1 ) 1 2 n + 1 = n a b ϵ e n + 1 .
h ̂ ( x , y ; x 0 , y 0 ) h ̂ n * ( x , y ; x 0 , y 0 ) = n = 0 n * m = 0 n δ m δ ̃ n m ( A n m A n m * ) cos [ m ( Θ + ϕ 0 ) ] J n + 1 ( R ) R + n = n * + 1 m = 0 n δ m δ ̃ n m A n m cos [ m ( Θ + ϕ 0 ) ] J n + 1 ( R ) R .
h ̂ ( x , y ; x 0 , y 0 ) h ̂ n * ( x , y ; x 0 , y 0 ) n = 0 n * m = 0 n δ m δ ̃ n m ( A n m A n m * ) cos [ m ( Θ + ϕ 0 ) ] J n + 1 ( R ) R + n = n * + 1 m = 0 n δ m δ ̃ n m A n m cos [ m ( Θ + ϕ 0 ) ] J n + 1 ( R ) R = I 1 + I 2 .
I 1 n = 0 n * m = 0 n δ m δ ̃ n m A n m A n m * J n + 1 ( R ) R n = 0 n * m = 0 n δ m δ ̃ n m n a b ϵ e n + 1 J n + 1 ( R ) R = n a b ϵ e n = 0 n * ( n + 1 ) 3 2 J n + 1 ( R ) R = n a b ϵ e R n = 0 n * ( n + 1 ) 3 2 J n + 1 ( R ) n a b ϵ e R 3 π e 2 ( 1 + R 4 3 ) R = n a b ϵ e 2 3 e π ( 1 + R 4 3 ) n a b ϵ e 2 3 e π ( 1 + R * 4 3 ) = ϵ 2 .
I 2 n = n * + 1 ( n + 1 J n + 1 ( R ) R { δ ̃ n + 2 m = 1 n δ ̃ n m cos [ m ( Θ + ϕ 0 ) ] } ) .
I 2 n = n * + 1 [ ( n + 1 ) 3 2 J n + 1 ( R ) R ] .
I 2 n = n * + 1 [ ( n + 1 ) 3 2 1 2 n ! ( R 2 ) n 1 ] .
I 2 R 4 n = n * 1 [ 1 n ! ( R 2 ) n ] .
I 2 R * 4 n = n * 1 [ 1 n ! ( R * 2 ) n ] R * 4 n = n * 1 ( R * 2 ) n ( n * 1 ) ! ( n * 1 ) n ( n * 1 ) = R * 4 ( R * 2 ) n * 1 ( n * 1 ) ! n = 0 ( R * 2 ( n * 1 ) ) n R * 4 ( R * 2 ) n * 1 ( n * 1 ) ! n = 0 ( 1 2 e ) n = e R * 2 ( 2 e 1 ) ( R * 2 ) n * 1 ( n * 1 ) ! n * 1 2 e ( 2 e 1 ) ( R * 2 ) n * 1 ( n * 1 ) ! n * 1 2 e ( 2 e 1 ) [ e R * 2 ( n * 1 ) ] n * 1 2 π ( n * 1 ) n * 1 2 e ( 2 e 1 ) 1 2 n * 1 2 π 1 2 e ( 2 e 1 ) 2 π 1 2 ( n * 1 ) 2 ϵ 2 .
2 π ( n * 1 ) n * 1 + 0.5 exp ( n * + 1 ) < ( n * 1 ) ! .
h ̂ ( x , y ; x 0 , y 0 ) h ̂ n * ( x , y ; x 0 , y 0 ) ϵ .

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