Abstract

The sampling theorem is applied to optical diffraction theory as a computational tool. Formulas are developed in terms of sampled values of the point spread function for the transfer function, total illuminance, line spread function and cumulative line spread function. Typical numerical examples are presented. The theory is presented for general point spread functions for slit and square apertures, but only for rotationally symmetric point spread functions for circular apertures. In the case of the slit aperture, the following question is answered: Given the real part of the incoherent transfer function, determine its imaginary part and vice versa. The theory of conjugate Fourier series and finite Hilbert transforms is introduced in order to answer this question.

© 1964 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, Progr. Opt. 1, 109 (1961).
    [Crossref]
  2. H. Gamo, J. Appl. Phys. Japan 25, 431 (1956);J. Appl. Phys. Japan 26, 102 (1957);J. Appl. Phys. Japan 26, 414 (1957);J. Appl. Phys. Japan 27, 577 (1958).
  3. T. di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 483.
  5. R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.
  6. E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).
  7. E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
    [Crossref]
  8. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
    [Crossref]
  9. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.
  10. The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.
  11. E. L. O’Neill and A. Walther, Opt. Acta 10, 33 (1962).
    [Crossref]
  12. No Hilbert transform equivalent to conjugate series in two dimensions is known to the author. However, some progress along these lines is contained in the recent paper: S. Goldman, J. Opt. Soc. Am. 52, 1131 (1962).
    [Crossref]
  13. The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 71.
  14. Footnote added in proof: The problem of determining t in terms of the sampled value of τ has been solved and will appear in a sequel to the present paper.
  15. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 7681964).
    [Crossref]

1964 (1)

1963 (1)

1962 (2)

1961 (1)

D. Gabor, Progr. Opt. 1, 109 (1961).
[Crossref]

1956 (2)

H. Gamo, J. Appl. Phys. Japan 25, 431 (1956);J. Appl. Phys. Japan 26, 102 (1957);J. Appl. Phys. Japan 26, 414 (1957);J. Appl. Phys. Japan 27, 577 (1958).

E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
[Crossref]

1955 (1)

1915 (1)

E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 483.

di Francia, T.

Gabor, D.

D. Gabor, Progr. Opt. 1, 109 (1961).
[Crossref]

Gamo, H.

H. Gamo, J. Appl. Phys. Japan 25, 431 (1956);J. Appl. Phys. Japan 26, 102 (1957);J. Appl. Phys. Japan 26, 414 (1957);J. Appl. Phys. Japan 27, 577 (1958).

Goldman, S.

Houston, A.

Laurman, J.

The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.

Linfoot, E. H.

O’Neill, E. L.

E. L. O’Neill and A. Walther, Opt. Acta 10, 33 (1962).
[Crossref]

Paley, R.

R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.

Robinson, A.

The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.

Walther, A.

E. L. O’Neill and A. Walther, Opt. Acta 10, 33 (1962).
[Crossref]

Whittaker, E. T.

E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).

Whittaker, J. M.

The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 71.

Wiener, N.

R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 483.

J. Appl. Phys. Japan (1)

H. Gamo, J. Appl. Phys. Japan 25, 431 (1956);J. Appl. Phys. Japan 26, 102 (1957);J. Appl. Phys. Japan 26, 414 (1957);J. Appl. Phys. Japan 27, 577 (1958).

J. Opt. Soc. Am. (5)

Opt. Acta (1)

E. L. O’Neill and A. Walther, Opt. Acta 10, 33 (1962).
[Crossref]

Proc. Roy. Soc. (Edinburgh) (1)

E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).

Progr. Opt. (1)

D. Gabor, Progr. Opt. 1, 109 (1961).
[Crossref]

Other (6)

The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 71.

Footnote added in proof: The problem of determining t in terms of the sampled value of τ has been solved and will appear in a sequel to the present paper.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.

The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 483.

R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (8)

Tables Icon

Table I Sampled values of point spread function t(υ) for slit aperture having W2 = 0.5λ.

Tables Icon

Table II Comparison of transfer function of slit aperture having W2 = 0.5λ as computed by exact integration and by sampling method using 33, 27, and 21 sampling points.

Tables Icon

Table III Comparison of transfer function of perfect circular aperture as computed by exact integration and by sampling method using 10 sampling points.

Tables Icon

Table IV Selected values of F(αn0).

Tables Icon

Table V Comparison of tolal illuminance L(υ0) of perfect circular aperture as computed by exact integration and by sampling method using eight sampling points.

Tables Icon

Table VI Selected values of $1(αn0).

Tables Icon

Table VII Comparison of line spread function τ(υ) of perfect circular aperture as computed by exact solution (7.13) and by sampling method (7.5) using eight sampling points.

Tables Icon

Table VIII Selected values of $2(αn0).

Equations (122)

Equations on this page are rendered with MathJax. Learn more.

T ( ω ) 0 ( for ω ω 0 = 2 ) .
Ω = ω / 2 λ F ( 0 ω ω 0 ) ,
( A ) frequency domain 0 | T ( ω ) | 1 , ( B ) spatial domain 0 | A ( x , y ) | 1 ,
A ( x , y ) = amplitude distribution over exit pupil amplitude distribution over entrance pupil .
T ( ω ) ~ 0 ( | ω | ) > 0 ,
t ( υ ) = 1 2 2 2 T ( ω ) e i υ ω d ω ,
T ( ω ) = n = B n e i n π ω / 2 ,
B n 1 4 2 2 T ( ω ) e i n π ω / 2 d ω = 1 4 t ( n π 2 ) .
T ( ω ) = T 1 ( ω ) + i T 2 ( ω ) ,
T 1 ( ω ) = 1 2 t ( 0 ) + 1 2 n = 1 [ t ( n π 2 ) + t ( n π 2 ) ] cos n π 2 ω ,
T 2 ( ω ) = 1 2 n = 1 [ t ( n π 2 ) t ( n π 2 ) ] sin n π 2 ω .
t ( υ ) = n = t ( n π 2 ) sinc ( 2 υ n π ) .
t ( υ ) = ( sinc υ ) 2 ,
t ( n π / 2 ) { = 4 / π 2 n 2 ( n = odd integer ) , = 0 ( n = even integer ) .
T 1 ( ω ) = 1 2 + 4 π 2 n = 0 cos [ ( 2 n + 1 ) ( π / 2 ) ω ] ( 2 n + 1 ) 2 .
n = 0 cos [ ( 2 n + 1 ) ( π / 2 ) ω ] ( 2 n + 1 ) 2 = π 2 8 [ 1 | ω | ] .
T 1 ( ω ) = 1 2 ( 2 | ω | ) .
t ( υ ) = | 1 1 e i ( υ ρ + 2 π W 2 ρ 2 ) d ρ | 2 .
T ( ω ) = 1 2 ( 2 | ω | ) sinc [ 8 π W 2 | ω | ( 1 | ω | ) ] .
m n = 0 2 T ( ω ) ω n d ω ,
m n = t ( n ) ( 0 ) 0 , n = odd integer , m n = t ( n ) ( 0 ) 0 , n = even integer .
m 2 = 3 2 t ( 0 ) + 16 π 2 n = 1 ( 1 ) n n 2 t ( n π 2 ) ,
m 4 = ( 16 / 5 ) t ( 0 ) + 128 π 4 n = 1 ( 1 ) n ( n 2 π 2 6 ) n 4 t ( n π 2 ) ,
m 6 = ( 64 / 7 ) t ( 0 ) + 1536 π 6 n = 1 ( 1 ) n n 6 × ( n 4 π 4 20 n 2 π 2 + 120 ) t ( n π 2 ) .
F = K 2 2 | T ( ω ) | 2 φ ( ω ) d ω ,
F = m = n = t ( m π 2 ) t ( n π 2 ) × 2 2 φ ( ω ) cos [ π 2 ( m n ) ω ] d ω .
F = K n = [ t ( n π 2 ) ] 2 ,
L ( υ 0 ) = 1 π υ 0 υ 0 t ( υ ) d υ ,
L ( υ 0 ) = 1 2 π n = t ( n π 2 ) 2 υ 0 n π 2 υ 0 n π sinc xdx = 1 2 π n = t ( n π 2 ) [ S i ( 2 υ 0 n π ) + S i ( 2 υ 0 + n π ) ] ,
L ( υ 0 ) = 1 π t ( 0 ) S i ( 2 υ 0 ) + 1 π n = 1 t ( n π 2 ) [ S i ( 2 υ 0 + n π ) + S i ( 2 υ 0 n π ) ] .
S i ( 2 υ 0 + n π ) + S i ( 2 υ 0 n π ) ~ S i ( n π ) + S i ( n π ) = 0 ,
S i ( 2 υ 0 ) ~ 2 υ 0 ,
L ( υ 0 ) ~ ( 2 / π ) υ 0 t ( 0 ) + .
L ( υ 0 ) = 2 π 0 2 T ( ω ) sin υ 0 ω ω d ω ,
L ( υ 0 ) = ( 2 / π ) S i ( 2 υ 0 ) ( 2 / π υ 0 ) sin 2 υ 0 ,
S i ( 2 υ 0 ) 2 υ 0 sin 2 υ 0 = 4 π 2 n = 0 1 ( 2 n + 1 ) 2 × [ S i ( 2 υ 0 + 2 n π + π ) S i ( 2 υ 0 2 n π π ) ] .
a ( υ ) = 1 1 A ( ρ ) e i υ ρ d ρ ,
A ( ρ ) = n = C n e i n π ω ,
C n = a ( n π ) .
t ( υ ) = n = n = a ( m π ) a * ( n π ) sinc ( m π υ ) × sinc ( n π υ ) .
T ( ω ) = N ω 1 1 A ( ρ ) A * ( ρ ω ) d ω ,
T ( ω ) = N n = m = a ( n π ) a * ( m π ) e i m π ω N ω m = | a ( m π ) | 2 e i m π ω ,
[ sin μ ( υ y ) ] / ( υ y ) ,
t ( υ ) sin μ ( υ y ) ( υ y ) d υ = 1 2 n = t ( n π 2 ) sin 2 ( υ n π / 2 ) [ υ ( n π / 2 ) ] sin μ ( υ y ) ( υ y ) d υ , = π 2 n = t ( n π 2 ) sin μ [ y ( n π / 2 ) ] [ y ( n π / 2 ) ] ,
sin m ( y x ) y x sin μ ( y z ) ( y z ) d y = π sin μ ( z x ) ( z x ) , ( 0 < μ m ) .
t ( υ ) = 2 π t ( y ) sin 2 ( υ y ) ( υ y ) d y .
a ( υ ) = 1 π a ( y ) sin ( υ y ) ( υ y ) d y .
f ( θ ) = a 0 2 + n = 1 a n cos n θ , ( 0 θ π ) ,
f ¯ ( θ ) = n = 1 a n sin n θ , ( 0 θ π ) ,
g ( θ ) = sin θ π 0 π G ( φ ) d φ cos φ cos θ ,
G ( θ ) = 1 π 0 π g ( φ ) sin φ d φ cos φ cos θ + 1 π 0 π G ( φ ) d φ .
g = H [ G ] ; G = H 1 ( g ) .
f ¯ ( θ ) = sin θ 2 π a 0 0 π d φ cos φ cos θ sin θ π × n = 1 a n 0 π cos n φ d φ cos φ cos θ .
0 cos n φ d φ cos φ cos θ { = 0 ( n = 0 ) = π sin n θ sin θ ( n 1 ) .
f ¯ ( θ ) = n = 1 a n sin n θ , ( 0 θ π ) .
1 π 0 π f ( θ ) d θ = 1 2 a 0 .
T 1 + ( ω ) = 1 4 t ( 0 ) + 1 2 n = 1 t ( n π 2 ) cos n π 2 ω ,
T 2 + ( ω ) = 1 2 n = 1 t ( n π 2 ) sin n π 2 ω ,
T 1 ( ω ) = 1 4 t ( 0 ) + 1 2 n = 1 t ( n π 2 ) cos n π 2 ω ,
T 2 ( ω ) = 1 2 n = 1 t ( n π 2 ) sin n π 2 ω ,
T 1 ( ω ) = T 1 + ( ω ) + T 1 ( ω ) , T 2 ( ω ) = H [ T 1 + ( ω ) ] + H [ T 1 ( ω ) ] ,
T 2 ( ω ) = T 2 + ( ω ) + T 2 ( ω ) , T 1 ( ω ) = H 1 [ T 2 + ( ω ) ] + H 1 [ T 2 1 ( ω ) ] ,
t ( υ x , υ y ) = 1 4 2 2 2 2 T ( ω x , ω y ) e i ( ω x υ x + ω y υ y ) d ω x d ω y ,
T ( ω x , ω y ) = m = n = B m n e ( i π / 2 ) ( m ω x + n ω y ) ,
B m n = 1 16 t [ ( m π / 2 ) , ( n π / 2 ) ] .
T ( ω x , ω y ) = T 1 ( ω x , ω y ) + i T 2 ( ω x , ω y ) ,
T 1 ( ω x , ω y ) = 1 4 t ( 0 , 0 ) + 1 4 n = 1 n = 1 t [ ( m π 2 , n π 2 ) + t ( m π 2 , n π 2 ) ] cos ( m π 2 ω x + n π 2 ω y ) ,
T 2 ( ω x , ω y ) = 1 4 n = 1 n = 1 [ t ( m π 2 , n π 2 ) t ( m π 2 , n π 2 ) ] sin ( m π 2 ω x + n π 2 ω y ) .
t ( m π 2 , n π 2 ) = t ( m π 2 , n π 2 ) , t ( m π 2 , n π 2 ) = t ( m π 2 , n π 2 ) .
T 1 + + ( ω x , ω y ) = 1 8 t ( 0 , 0 ) + 1 4 n = 1 m = 1 t ( m π 2 , n π 2 ) × cos ( m π 2 ω x + n π 2 ω y ) ,
T 2 + + ( ω x , ω y ) = 1 4 n = 1 m = 1 t ( m π 2 , n π 2 ) × sin ( m π 2 ω x + n π 2 ω y ) ,
T 1 ( ω x , ω y ) = 1 8 t ( 0 , 0 ) + 1 4 n = 1 m = 1 t ( m π 2 , n π 2 ) × cos ( m π 2 ω x + n π 2 ω y ) ,
T 2 ( ω x , ω y ) = 1 4 n = 1 m = 1 t ( m π 2 , n π 2 ) × sin ( m π 2 ω x + n π 2 ω y ) ,
T 2 + + ( ω x , ω y ) = H 2 [ T 1 + + ( ω x , ω y ) ] , T 2 ( ω x , ω y ) = H 2 [ T 2 ( ω x , ω y ) ] ,
T 2 ( ω x , ω y ) = H 2 [ T 1 + + ( ω x , ω y ) ] + H 2 [ T 1 ( ω x , ω y ) ] .
T 1 ( ω x , ω y ) = T 2 1 [ T 2 + + ( ω x , ω y ) ] + H 2 1 [ T 2 ( ω x , ω y ) ] + C ,
t ( υ x , υ y ) = 4 π 2 t ( x , y ) sin 2 ( υ x x ) ( υ x x ) × sin 2 ( υ y y ) ( υ y y ) dxdy ,
t ( υ ) = 0 2 T ( ω ) J 0 ( υ ω ) ω d ω .
T ( ω ) = n = 1 A n J 0 ( k n ω ) ,
A n = 1 J 1 2 ( 2 k n ) 0 2 T ( ω ) J 0 ( k n ω ) ω d ω = 1 J 1 2 ( 2 k n ) t ( k n ) .
J 0 ( 2 k n ) = J 0 ( α n ) = 0 ,
A n = [ 1 / J 1 2 ( α n ) ] t ( α n / 2 ) .
T ( ω ) = n = 1 1 J 1 2 ( α n ) t ( α n 2 ) J 0 ( α n ω 2 ) .
T ( ω ) = 2 π [ cos 1 ω 2 ω 2 ( 1 ω 2 4 ) 1 2 ] ,
cos 1 ω 2 ω 2 ( 1 ω 2 4 ) 1 2 = 8 π n = 1 J 1 2 ( α n / 2 ) J 0 ( α n ω / 2 ) α n 2 J 1 2 ( α n ) .
0 2 J 0 ( α n ω 2 ) J 0 ( υ ω ) ω d ω = 4 α n J 1 ( α n ) α n 2 4 υ 2 J 0 ( 2 υ ) , ( υ α n 2 ) , = 2 J 1 2 ( α n ) , υ = α n / 2 .
t ( υ ) = 4 n = 1 α n t ( α n / 2 ) ( α n 2 4 υ 2 ) J 1 ( α n ) J 0 ( 2 υ ) .
t ( 0 ) = 4 n = 1 t ( α n / 2 ) α n J 1 ( α n ) .
L ( υ 0 ) = N 0 υ 0 t ( υ ) υ d υ ,
L ( υ 0 ) = υ 0 0 2 T ( ω ) J 1 ( υ 0 ω ) d ω ,
L ( υ 0 ) = υ 0 n = 1 t ( α n / 2 ) J 1 2 ( α n ) F ( α n , υ 0 ) ,
F ( α n , υ 0 ) = 0 2 J 0 ( α n ω 2 ) J 1 ( υ 0 ω ) d ω .
F ( α n , υ 0 ) υ 0 2 0 2 J 0 ( α n ω 2 ) ω d ω 2 υ 0 α n J 1 ( α n ) .
L ( υ 0 ) 2 υ 0 2 n = 1 t ( α n / 2 ) α n J 1 ( α n ) ( υ 0 2 / 2 ) t ( 0 ) ,
L ( υ 0 ) = 1 J 0 2 ( υ 0 ) J 1 2 ( υ 0 ) .
F = K 0 2 | T ( ω ) | 2 ϕ ( ω ) ω d ω ,
F = K n = 1 n = 1 t ( m π / 2 ) t ( n π / 2 ) J 1 2 ( α m ) J 1 2 ( α n ) × 0 2 J 0 ( α m ω 2 ) J 0 ( α n ω 2 ) ϕ ( ω ) ω d ω .
F = 2 K n = 1 1 J 1 2 ( α n ) [ t ( n π 2 ) ] 2 .
T ( ω ) = 0 τ ( υ ) cos ω υ d ω .
τ ( υ ) = 2 υ x t ( x ) d x ( x 2 υ 2 ) 1 2 .
τ ( υ ) = 0 T ( ω ) cos υ ω d ω ,
T ( ω ) = n = 1 τ ( n π 2 ) cos n π 2 ω + 1 2 τ ( 0 ) .
τ ( υ ) = 2 n = 1 t ( α n / 2 ) J 1 2 ( α n ) $ 1 ( α n , υ ) ,
$ 1 ( α n , υ ) = 0 1 J 0 ( α n x ) cos 2 υ xdx .
$ 1 ( α n , 0 ) = ( π / 2 ) J 1 ( α n ) H 0 ( α n ) ,
0 1 J 0 ( α n x ) x 2 d x = Λ 1 ( α n ) 1 α n 2 [ π 2 J 1 ( α n ) H 0 ( α n ) ] ,
0 1 J 0 ( α n x ) x 4 d x = [ 1 9 α n 2 ] Λ 1 ( α n ) + 9 α n 4 [ π 2 J 1 ( α n ) H 0 ( α n ) ] ,
0 1 J 0 ( α n x ) x 6 d x = [ 1 25 α n 2 + 225 α n 4 ] Λ 1 ( α n ) 225 α n 6 [ π 2 J 1 ( α n ) H 0 ( α n ) ] ,
$ 1 ( α n , υ ) = π 2 J 1 ( α n ) H 0 ( α n ) × [ 1 ( 2 υ ) 2 2 ! α n 2 + 9 ( 2 υ ) 4 4 ! α n 4 + 225 ( 2 υ ) 6 6 ! α n 6 + ] + Λ 1 ( α n ) [ ( 2 υ ) 2 2 ! + ( 1 9 α n 2 ) ( 2 υ ) 4 4 ! ( 1 25 α n 2 + 225 α n 4 ) ( 2 υ ) 6 6 ! + ] .
τ ( 0 ) = π n = 1 t ( α n / 2 ) J 1 ( α n ) H 0 ( α n ) .
τ ( υ ) = H 1 ( 2 υ ) / υ 2 , τ ( 0 ) = 8 / 3 π ,
1 = 12 π 2 8 n = 1 J 1 2 ( α n / 2 ) α n 2 J 1 ( α n ) H 0 ( α n ) .
L i ( υ 0 ) = 0 υ 0 τ ( υ ) d υ .
L i ( υ 0 ) = 2 π 0 2 T ( ω ) sin υ 0 ω ω d ω ,
L i ( υ 0 ) = 2 π n = 1 t ( α n / 2 ) J 1 2 ( α n ) $ 2 ( α n , υ 0 ) ,
$ 2 ( α n , υ 0 ) = 0 1 J 0 ( α n x ) sin 2 υ 0 x x d x .
$ 2 ( α n , υ 0 ) 2 υ 0 0 1 J 0 ( α n x ) d x + 0 ( υ 0 2 ) , 2 υ 0 $ 1 ( α n , 0 ) .
L i ( υ 0 ) 4 π υ 0 n = 1 t ( α n / 2 ) J 1 2 ( α n ) $ 1 ( α n , 0 ) 2 π υ 0 τ ( 0 ) ,
$ 2 ( α n , υ 0 ) = π 2 J 1 ( α n ) H 0 ( α n ) × [ ( 2 υ 0 ) + ( 2 υ 0 ) 3 3 ! α n 2 + 9 ( 2 υ 0 ) 5 5 ! α n 4 + 225 ( 2 υ 0 ) 7 7 ! α n 6 + ] + Λ 1 ( α n ) [ ( 2 υ 0 ) 3 3 ! + ( 1 9 α n 2 ) ( 2 υ 0 ) 5 5 ! ( 1 25 α n 2 + 225 α n 4 ) ( 2 υ 0 ) 7 7 ! + ] .
$ 1 ( α n , υ 0 ) π / 2 , ( large υ 0 ) .
L i ( υ 0 ) = 1 π τ ( 0 ) S i ( 2 υ 0 ) + 2 π n = 1 τ ( n π 2 ) × [ S i ( υ 0 + n π 2 ) + S i ( υ 0 n π 2 ) ] ,
0 2 cos ( n π ω 2 ) sin υ 0 ω ω d ω = 1 2 S i ( υ 0 + n π 2 ) + 1 2 S i ( υ 0 n π 2 ) .