Abstract

The form of discs considered has λ/4-deep pits along the track, the spacings and lengths of which are modulated. A laser beam is focused on the track by means of a high-aperture objective, and the light which is scattered back through the objective is detected. The diffraction theory of this system is treated, both for simple periodic and for more complicated forms of disc. Formulas are obtained for the fundamental and harmonics of the read-out signal. The influences of focus error and aberrations are studied, for certain cases of which numerical results are given. Also treated are cross-talk between adjacent tracks and intermodulation between the pit-spacing frequencies and those of the length modulation of the pits.

© 1979 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Compaan and P. Kramer, “The Philips VLP System,” Philips Tech. Rev.,  33, 178–180 (1973).
  2. G. Bouwhuis and P. Burgstede, “The Optical Scanning System of the Philips VLP Record Player,” Philips Tech. Rev.,  33, 186–189 (1973).
  3. H. H. Hopkins and Maria J. Izuel, Opt. Acta,  17, 157–182 (1970).
    [Crossref]

1973 (2)

K. Compaan and P. Kramer, “The Philips VLP System,” Philips Tech. Rev.,  33, 178–180 (1973).

G. Bouwhuis and P. Burgstede, “The Optical Scanning System of the Philips VLP Record Player,” Philips Tech. Rev.,  33, 186–189 (1973).

1970 (1)

H. H. Hopkins and Maria J. Izuel, Opt. Acta,  17, 157–182 (1970).
[Crossref]

Bouwhuis, G.

G. Bouwhuis and P. Burgstede, “The Optical Scanning System of the Philips VLP Record Player,” Philips Tech. Rev.,  33, 186–189 (1973).

Burgstede, P.

G. Bouwhuis and P. Burgstede, “The Optical Scanning System of the Philips VLP Record Player,” Philips Tech. Rev.,  33, 186–189 (1973).

Compaan, K.

K. Compaan and P. Kramer, “The Philips VLP System,” Philips Tech. Rev.,  33, 178–180 (1973).

Hopkins, H. H.

H. H. Hopkins and Maria J. Izuel, Opt. Acta,  17, 157–182 (1970).
[Crossref]

Izuel, Maria J.

H. H. Hopkins and Maria J. Izuel, Opt. Acta,  17, 157–182 (1970).
[Crossref]

Kramer, P.

K. Compaan and P. Kramer, “The Philips VLP System,” Philips Tech. Rev.,  33, 178–180 (1973).

Opt. Acta (1)

H. H. Hopkins and Maria J. Izuel, Opt. Acta,  17, 157–182 (1970).
[Crossref]

Philips Tech. Rev. (2)

K. Compaan and P. Kramer, “The Philips VLP System,” Philips Tech. Rev.,  33, 178–180 (1973).

G. Bouwhuis and P. Burgstede, “The Optical Scanning System of the Philips VLP Record Player,” Philips Tech. Rev.,  33, 186–189 (1973).

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.


Figures (18)

FIG. 1
FIG. 1

Optical scanning system.

FIG. 2
FIG. 2

Read-out and tracking spots.

FIG. 3
FIG. 3

Object and image-plane coordinates.

FIG. 4
FIG. 4

Pupil sphere coordinates.

FIG. 5
FIG. 5

Region of integration for D(m,n;m + μ,n + ν).

FIG. 6
FIG. 6

Special case of the circles C1, C2, and C3.

FIG. 7
FIG. 7

Form of each pit.

FIG. 8
FIG. 8

Regions of integration for (a) D(m,n;m + μ,n + ν) and (b) D(m,n;m + μ,nν).

FIG. 9
FIG. 9

Mesh points for the base tabulation.

FIG. 10
FIG. 10

Ray paths for the general case.

FIG. 11
FIG. 11

Disc with staggered tracks and length-modulated pits.

FIG. 12
FIG. 12

Disc with adjacent tracks of different period.

FIG. 13
FIG. 13

Influence of nonuniformity of amplitude on the read-out signal.

FIG. 14
FIG. 14

Intermodulation between length-modulation and fundamental read-out frequencies. Curve A: full aperture, ϕ = π; curve B: push-pull operation, ϕ = π/2.

FIG. 15
FIG. 15

Influence of focus error on the read-out signal.

FIG. 16
FIG. 16

Influence of coma on the read-out signal.

FIG. 17
FIG. 17

Influence of nonuniformity of amplitude on the read-out signal in the presence of coma.

FIG. 18
FIG. 18

Influence of focus error, coma, spherical aberration, and astigmatism on the read-out signal.

Equations (207)

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

u = ( n sin α / λ ) ξ υ = ( n sin α / λ ) η
u = ( n sin α ) / λ ) ξ υ = ( n sin α / λ ) η ,
M = η sin α / η sin α
x = X / h , y = Y / h , x = X h , y = Y / h ,
sin θ / sin α = sin θ / sin α
X / h = X / h , Y / h = Y / h
U D = exp [ i k ( ξ 2 + η 2 2 R 0 ) ] F ( u , υ ) ,
F ( u , υ ) = + + f ( x , y ) × exp [ i 2 π ( u x + υ y ) ] d x d y
U D = F ( u , υ )
τ ( x , y ) = exp [ ( 1 / 2 ) σ ( x 2 + y 2 ) ]
I ( x , y ) = exp [ σ ( x 2 + y 2 ) ]
I m / I 0 = exp ( σ )
f ( x , y ) = τ ( x , y ) exp [ i 2 π W ( x , y ) ] ( x 2 + y 2 ) 1 , = 0 ( x 2 + y 2 ) > 1 ,
F ( u , υ ) = + + f ( x , y ) exp [ i 2 π ( u x + υ y ) ] d x d y ;
a ( x , y ) = + + F ( u , υ ) R ( u , υ ) × exp [ i 2 π ( u x + υ y ) ] d x d y
a ( x , y ) = + + f ( x x , y y ) r ( x , y ) d x d y
u = ( n sin α / λ ) ω a t
R ( u , υ ) = R 0 ( u u , υ υ ) ,
R 0 ( u , υ ) = m = + n = + R ( m , n ) × exp { i 2 π [ ( m p ) u + ( n q ) υ ] } ,
R ( m , n ) = 1 p q p / 2 + p / 2 q / 2 + q / 2 R 0 ( u , υ ) × exp { i 2 π [ ( m p ) u + ( n q ) υ ] } d u d υ
R ( u , υ ) = m = + n = + exp { i 2 π [ ( m p ) u + ( n q ) υ ] } × R ( m , n ) exp { i 2 π { ( m p ) u + ( n q ) υ ] }
r ( x , y ) = m = + n = + exp { i 2 π [ ( m p ) u + ( n q ) υ ] } × R ( m , n ) δ ( x m p , y n q ) ,
a ( x , y ) = m = + n = + exp { i 2 π [ ( m p ) u + ( n q ) υ ] } × R ( m , n ) f ( x m p , y n q ) ,
I ( x , y ) = m n m n exp { i 2 π [ ( m m p ) u + ( n n q ) υ ] } × R ( m , n ) R * ( m , n ) f ( x m p , y n q ) f * ( x m p , y n q ) ,
I = m n m n exp { i 2 π [ ( m m p ) u + ( n n q ) υ ] } × R ( m , n ) R * ( m , n ) D ( m , n ; m , n ) ,
D ( m , n ; m , n ) = ( x 2 + y 2 ) 1 f ( x m p , y n q ) × f * ( x m p , y n q ) d x d y ,
( m m ) = μ , ( n n ) = ν
I ( t ) = m n μ ν exp { i 2 π { ( μ p ) u + ( ν q ) υ ] } R ( m , n ) × R * ( m + μ , n + ν ) D ( m , n ; m + μ , n + ν ) ,
ω μ t = 2 π ( μ / p ) u = μ { 2 π ( n sin α / λ ) ω a / p } t
p = ( n sin α / λ ) ξ
ω μ = μ { 2 π ( ω a / ξ ) } = μ ω 1 , ω 1 = 2 π ( ω a / ξ ) .
I ( t ) = μ { ν m n exp [ i 2 π ( ν q ) υ ] R ( m , n ) R * ( m + μ , n + ν ) × D ( m , n ; m + μ , n + ν ) } exp ( i ω μ t )
I ( t ) = μ = + I μ exp { i μ ω 1 t }
I μ = ν m n exp [ i 2 π ( ν q ) υ ] R ( m , n ) R * ( m + μ , n + ν ) × D ( m , n ; m + μ , n + ν )
I ( t ) = I 0 + μ = 1 [ I μ exp ( i μ ω , t ) + I μ * exp [ i μ ω , t ) ]
I ( t ) = A 0 + μ = 1 [ A μ cos ( ω μ t ) + B μ sin ( ω μ t ) ]
A 0 = I 0 , A μ = 2 Re ( I μ ) , B μ = 2 Im ( I μ )
I μ = ν exp [ i 2 π ( ν q ) υ ] R ( 0 , 0 ) R * ( μ , ν ) D ( 0 , 0 ; μ , ν )
I 0 = | R ( 0 , 0 ) | 2 D ( 0 , 0 ; 0 , 0 ) .
I 0 = ( x 2 + y 2 ) 1 | f ( x , y ) | 2 d x d y ,
A = ( x 2 + y 2 ) 1 exp [ σ ( x 2 + y 2 ) ] d x d y ,
A = ( π / σ ) [ 1 exp ( σ ) ]
D ( m , n ; m + μ , n + ν ) = x 2 + y 2 1 f ( x m p , y n q ) × f * ( x m + μ p , y n + ν q ) d x d y ,
( x m / p ) 2 + ( y n / q ) 2 = 1
[ x ( m + μ ) / p ] 2 + [ y ( n + ν ) / q ] 2 = 1
( μ / p ) 2 + ( ν / q ) 2 < 4
0 μ μ 0 ,
μ 0 < 2 p
f = 2 ( n sin α / λ ) ω a
q [ 4 ( μ / p ) 2 ] 1 / 2 < ν < q [ 4 ( μ / p ) 2 ] 1 / 2
ν 0 < q [ 4 ( μ / p ) 2 ] 1 / 2
ν = 0 , ± 1 , ± 2 , , ± ν 0 ,
I ( t ) = A 0 + μ = 1 μ 0 [ A μ cos ( ω μ t ) + B μ sin ( ω μ t ) ]
A 0 = I 0 , A μ = 2 Re ( I μ ) , B μ = 2 Im ( I μ ) ,
I μ = m n R ( m , n ) R * ( m + μ , n ) D ( m , n ; m + μ , n ) + ν = 1 ν 0 exp [ i 2 π ( ν q ) υ ] m n R ( m , n ) R * ( m + μ , n + ν ) × D ( m , n ; m + μ , n + ν ) + ν = 1 ν 0 exp [ i 2 π ( ν q ) υ ] m n R ( m , n ) R * ( m + μ , n ν ) × D ( m , n ; m + μ , n ν ) ,
[ ( m + μ ) / p ] 2 + [ ( n ± ν ) / q ] < 4
( m / p ) 2 + ( n / q ) 2 < 4 .
( 2 p ) < ( m + μ ) < ( + 2 p )
( 2 p μ ) < m < ( 2 p μ )
2 p < m < + 2 p .
m 1 > ( 2 p ) m 2 < ( 2 p μ )
m 1 m m 2
[ 4 ( m + μ p ) 2 ] 1 / 2 < ( n + ν q ) < + [ 4 ( m + μ p ) 2 ] 1 / 2 ;
{ q [ 4 ( m + μ p ) 2 ] 1 / 2 ν } < n < { + q [ 4 ( m + μ p ) 2 ] 1 / 2 ν }
{ q [ 4 ( m p ) 2 ] 1 / 2 } < n < { + q [ 4 ( m p ) 2 ] 1 / 2 }
n 2 < q { 4 [ ( m + μ ) / p ] 2 } 1 / 2 ν ,
n 1 > q { 4 [ ( m + μ ) / p ] 2 } 1 / 2 ν , n 1 > q [ 4 ( m / p ) 2 ] 1 / 2 ,
n 1 n n 2
n 1 > q [ 4 ( m / p ) 2 ] 1 / 2 , ( μ = 0 ) n 2 < + q [ 4 ( m / p ) 2 ] 1 / 2 ν ,
{ q [ 4 ( m + μ p ) 2 ] 1 / 2 + ν } < n < { + q [ 4 ( m + μ p ) 2 ] 1 / 2 + ν }
{ q [ 4 ( m / p ) 2 ] 1 / 2 } < n < { + q [ 4 ( m / p ) 2 ] 1 / 2 }
n 3 > q { 4 [ ( m + μ ) ] 2 / p } 1 / 2 + ν
n 4 < + q { 4 [ ( m + μ ) / p ] 2 } 1 / 2 + ν , n 4 < + q [ 4 ( m / p ) 2 ] 1 / 2 ,
n 3 n n 4
n 3 > q [ 4 ( m / p ) 2 ] 1 / 2 + ν n 4 < + q [ 4 ( m / p ) 2 ] 1 / 2 ( μ = 0 )
I μ = m = m 1 m 2 n = n 1 n 2 R ( m , n ) R * ( m + μ , n ) D ( m , n ; m + μ , n ) + ν = 1 ν 0 exp { i 2 π ( ν q ) υ } m = m 1 m 2 n = n 1 n 2 R ( m , n ) × R * ( m + μ , n + ν ) D ( m , n ; m + μ , n + ν ) + ν = 1 ν 0 exp { i 2 π ( ν q ) υ } m = m 1 m 2 n = n 1 n 4 R ( m , n ) × R * ( m + μ , n ν ) D ( m , n ; m + μ , n ν )
( n 2 n 1 + 2 ) 4 q + 2 ( n 4 n 3 + 2 ) 4 q + 2 ( m 2 m 1 + 2 ) 4 p + 2
A 0 = m 1 m 2 n 1 n 2 R ( m , n ) R * ( m , n ) D ( m , n ; m , n ) + ν = 1 ν 0 exp { i 2 π ( ν q ) υ } m 1 m 2 n 1 υ 2 R ( m , n ) × R * ( m , n + ν ) D ( m , n ; m , n + ν ) + ν = 1 ν 0 exp { i 2 π ( ν q ) υ } m 1 m 2 n 3 n 4 R ( m , n ) × R * ( m , n ν ) D ( m , n ; m , n ν )
D ( m , n ; m , n ) = ( x 2 + y 2 ) 1 f ( x m p , y n q ) × f * ( x m p , y n q ) d x d y ,
D ( m , n ; m , n ) = ( x 2 + y 2 ) 1 | f ( x m p , y n q ) | 2 d x d y
n 1 > q [ 4 ( m / p ) 2 ] 1 / 2 n 2 < + q [ 4 ( m / p ) 2 ] 1 / 2 ν n 3 > q [ 4 ( m / p ) 2 ] 1 / 2 + ν n 4 < + q [ 4 ( m / p ) 2 ] 1 / 2
R ( m , ñ ) R * ( m , ñ + ν ) D ( m , ñ ; m , ñ + ν )
R ( m , ñ + ν ) R * ( m , ñ ) D ( m , ñ + ν ; m , ñ )
D ( m , ñ + ν ; m , ñ ) = D * ( m , ñ ; m , ñ + ν )
A 0 = m 1 m 2 n 1 n 2 R ( m , n ) R * ( m , n ) D ( m , n ; m , n ) + 2 Re { ν = 1 ν 0 exp [ i 2 π ( ν q ) υ ] m 1 m 2 n 1 n 2 R ( m , n ) × R * ( m , n + ν ) D ( m , n ; m , n + ν ) }
R ( m , n ) = G ( m , n ) + i K ( m , n ) ,
R ( m , n ) R * ( m , n ) = M ( m , n ; m , n ) + i N ( m , n ; m , n ) ,
M ( m , n ; m , n ) = G ( m , n ) G ( m , n ) + K ( m , n ) K ( m , n ) ,
N ( m , n ; m , n ) = K ( m , n ) G ( m , n ) G ( m , n ) K ( m , n ) ,
D ( m , n ; m , n ) = C ( m , n ; m , n ) + i S ( m , n ; m , n ) ,
A 0 = m 1 m 2 n 1 n 2 M ( m , n ; m , n ) C ( m , n ; m , n ) + 2 ν = 1 ν 0 cos [ 2 π ( ν q ) υ ] m 1 m 2 n 1 n 2 { M ( m , n ; m , n + ν ) × C ( m , n ; m , n + ν ) N ( m , n ; m , n + ν ) S ( m , n ; m , n + ν ) } + 2 ν = 1 ν 0 sin [ 2 π ( ν q ) υ ] m 1 m 2 n 1 n 2 { M ( m , n ; m , n + ν ) × S ( m , n ; m , n + ν ) N ( m , n ; m , n + ν ) C ( m , n ; m , n + ν ) }
A μ = 2 m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n ) C ( m , n ; m + μ , n ) N ( m , n ; m + μ , n ) S ( m , n ; m + μ , n ) ] + 2 ν = 1 ν 0 cos [ 2 π ( ν q ) υ ] m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n + ν ) × C ( m , n ; m + μ , n + ν ) N ( m , n ; m + μ , n + ν ) × S ( m , n ; m + μ , n + ν ) ] + 2 ν = 1 ν 0 sin [ 2 π ( ν q ) υ ] m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n + ν ) × S ( m , n ; m + μ , n + ν ) N ( m , n ; m + μ , n + ν ) × C ( m , n ; m + μ , n + ν ) ] + 2 ν = 1 ν 0 cos [ 2 π ( ν q ) υ ] m 1 m 2 n 3 n 4 [ M ( m , n ; m + μ , n ν ) × C ( m , n ; m + μ , n ν ) N ( m , n ; m + μ , n ν ) × S ( m , n ; m + μ , n ν ) ] + 2 ν = 1 ν 0 sin [ 2 π ( ν q ) υ ] m 1 m 2 n 3 n 4 [ M ( m , n ; m + μ , n ν ) × S ( m , n ; m + μ , n ν ) + N ( m , n ; m + μ , n ν ) × C ( m , n ; m + ν ) ]
B μ = 2 m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n ) S ( m , n ; m + μ , n ) N ( m , n ; m + μ , n ) C ( m , n ; m + μ , n ) ] + 2 ν = 1 ν 0 cos [ 2 π ( ν q ) υ ] m 1 m 2 n 1 υ 2 [ M ( m , n ; m + μ , n + ν ) × S ( m , n ; m + μ , n + ν ) N ( m , n ; m + μ , n + ν ) × C ( m , n ; m + μ , n + ν ) ] + 2 ν = 1 ν 0 sin [ 2 π ( ν q ) υ ] m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n + ν ) × C ( m , n ; m + μ , n + ν ) + N ( m , n ; m + μ , n + ν ) × S ( m , n ; m + μ , n + ν ) ] + 2 ν = 1 ν 0 cos [ 2 π ( ν q ) υ ] m 1 m 2 n 3 n 4 [ M ( m , n ; m + μ , n ν ) × S ( m , n ; m + μ , n ν ) N ( m , n ; m + μ , n ν ) × C ( m , n ; m + μ , n ν ) ] + 2 ν = 1 ν 0 sin [ 2 π ( ν q ) υ ] m 1 m 2 n 3 n 4 [ M ( m , n ; m + μ , n ν ) × C ( m , n ; m + μ , n ν ) N ( m , n ; m + μ , n ν ) × S ( m , n ; m + μ , n ν ) ]
P 1 ( μ , ν ) = m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n + ν ) C ( m , n ; m + μ , n + ν ) N ( m , n ; m + μ , n + ν ) S ( m , n ; m + μ , n + ν ) ] ,
P 2 ( μ , ν ) = m 1 m 2 n 1 n 2 [ M ( m , n ; m + μ , n + ν ) S ( m , n ; m + μ , n + ν ) + N ( m , n ; m + μ , n + ν ) C ( m , n ; m + μ , n + ν ) ] ,
Q 1 ( μ , ν ) = m 1 m 2 n 0 n 4 [ M ( m , n ; m + μ , n ν ) C ( m , n ; m + μ , n ν ) N ( m , n ; m + μ , n ν ) S ( m , n ; m + μ , n ν ) ] ,
Q 2 ( μ , ν ) = m 1 m 2 n 3 n 4 [ M ( m , n ; m + μ , n ν ) S ( m , n ; m + μ , n ν ) + N ( m , n ; m + μ , n ν ) C ( m , n ; m + μ , n ν ) ] ,
A 0 = P 1 ( 0 , 0 ) + 2 ν = 1 ν 0 { P 1 ( 0 , ν ) cos [ 2 π ( ν q ) υ ] P 2 ( 0 , ν ) sin [ 2 π ( ν q ) υ ] }
A ν = 2 P 1 ( μ , 0 ) + 2 ν = 1 ν 0 { [ P 1 ( μ , ν ) + Q 1 ( μ , ν ) cos [ 2 π ( ν q ) υ ] + [ P 2 ( μ , ν ) + Q 2 ( μ , ν ) ] sin [ 2 π ( ν q ) υ ] } ,
B μ = 2 P 2 ( μ , 0 ) + 2 ν = 1 ν 0 { P 2 ( μ , ν ) Q 2 ( μ , ν ) cos [ 2 π ( ν q ) υ ] + [ P 1 ( μ , ν ) + Q 1 ( μ , ν ) ] sin [ 2 π ( ν q ) υ ] } ,
R ( m , n ) = 1 p q 1 / 2 p + 1 / 2 p 1 / 2 q + 1 / 2 q R 0 ( u , υ ) × exp { i 2 π [ ( m p ) u + ( n q ) υ ] } d u d υ
R ( m , n ) = R ( m , n ) , R ( m , n ) = R ( m , n ) , R ( m , n ) = R ( m , n ) ,
r 00 ( x , y ) = 1 / 2 p + 1 / 2 p 1 / 2 q + 1 / 2 q R 00 ( u , υ ) × exp { i 2 π ( x u + y υ ) } d u d υ
R ( m , n ) = ( 1 / p q ) r 00 ( m / p , n / q )
R 00 ( u , υ ) = R 1 ( u , υ ) + R 2 ( u , υ ) + R 3 ( u , υ ) + R 4 ( u , υ )
R 00 ( u , υ ) = R 1 ( u , υ ) + R 2 ( u , υ ) + R 5 ( u + β / 2 , υ ) + R 5 ( u β / 2 , υ ) ,
G 1 ( u ; p ) = 1 , | u | ( 1 / 2 ) p 0 , | u | > ( 1 / 2 ) p
G 2 ( u , υ ; γ ) = 1 , u 0 ; ( u 2 + υ 2 ) ( γ / 2 ) 2 = 0 , u < 0 ; ( u 2 + υ 2 ) > ( γ / 2 ) 2
R 1 ( u , υ ) = G 1 ( u ; p ) G 1 ( υ ; q ) G 1 ( u ; β ) G 1 ( υ ; γ ) G 2 ( u β / 2 , υ ; γ ) G 2 ( u β / 2 , υ ; γ ) ,
R 2 ( u , υ ) = G 1 ( u ; β ) G 1 ( υ ; γ ) exp [ i H ( υ ) ] ,
R 5 ( u , υ ) = G 2 ( u , υ ; γ ) exp { i H [ ( u 2 + υ 2 ) 1 / 2 ] }
R 00 ( u , υ ) = G 1 ( u ; p ) G 1 ( υ ; q ) + G 1 ( u ; β ) G 1 ( υ ; γ ) { exp [ i H ( υ ) ] 1 } + G 2 ( u β / 2 , υ ; γ ) [ exp { i H [ ( u β / 2 ) 2 + υ 2 ) } 1 ) + G 2 ( u β / 2 , υ ; γ ) [ exp { i H ( V ( u β / 2 ) 2 + υ 2 ) } 1 )
r 00 ( x , y ) = [ sin ( π p x ) / π x ] [ sin ( π q y ) / π y ] + [ sin ( π β x ) / π x ] I 1 ( y ; γ ) + exp ( i π β x ) I 2 ( x , y ; γ ) + exp ( + i π β x ) I 2 ( x , y ; γ ) ,
I 1 ( y , γ ) = γ / 2 + γ / 2 exp [ i H ( υ ) ] exp { i 2 π y υ } d υ
I 2 ( x , y ; γ ) = R 5 [ exp { i H [ u 2 + υ 2 ) 1 / 2 } 1 ] × exp { i 2 π ( u x + υ y ) } d u d υ
I 2 ( x , y , ; γ ) = 0 γ / 2 π / 2 + π / 2 { exp [ i H ( ρ ) ] 1 } × exp [ i 2 π ( x ρ cos θ + y ρ sin θ ) ] ρ d ρ d θ ,
r 00 ( x , y ) = [ sin ( π p x ) / π x ] [ sin ( π q y ) / π y ] + [ i ( 2 π x ) I 1 ( y ; γ ) + I 2 ( x , y , γ ) ] exp ( i π β x ) + [ i ( 2 π x ) I 1 ( y ; γ ) + I 2 ( x , y , γ ) ] exp ( + i π β x ) ,
R ( m , n ) = sin ( π m ) π m sin ( π n ) π n + [ ( i ( 2 π m q ) ) I 1 ( n q ; γ ) + ( 1 p q ) I 2 ( m p , n q γ ) ] + [ exp [ i π ( m p ) β ] + [ ( i 2 π m q ) I 1 ( n q ; γ ) + ( 1 p q ) I 2 ( m p , n q ; γ ) ] exp [ + i π ( m p ) β ] ,
D ( m , n ; m , n ) = ( x 2 + y 2 ) 1 × f ( x m p , y n q ) f * ( x m p , y n q ) d x d y
f ( x , y ) = τ ( x , y ) exp [ i 2 π W ( x , y ) ] , ( x 2 + y 2 ) 1 = 0 ( x 2 + y 2 ) > 1
τ ( x , y ) = exp [ ( σ / 2 ) ( x 2 + y 2 ) ]
D ( m , n ; m + μ , n ± ν ) = ( x m / p ) 2 + ( y n / q ) 2 1 × f ( x , y ) f * ( x μ p , y ± ν q ) d x d y
f ( x , y ) f * ( x μ / p , y ν / q )
D ( m , n ; m + μ , n ± ν ) = D f ( r , φ ) f * ( r , φ ) r d r d ϕ ,
r = { [ ( r sin ϕ μ / p ) 2 + ( r cos α ν / q ) 2 ] } 1 / 2 , cos ϕ = r cos ϕ ( ν / q ) r
f ( r , ϕ ) = exp [ σ / 2 ) r 2 ] exp [ i 2 π W ( r , φ ) ] ,
W ( r , ϕ ) = ( W 20 r 2 + W 22 r 2 cos 2 φ ) + ( W 40 r 6 + W 60 r 6 + W 80 r 8 + W 100 r 10 ) + ( W 31 r 3 cos φ + W 51 r 5 cos φ + W 71 r 7 cos φ + W 91 n 9 cos φ ) ,
D ( m , n ; m + μ , n ± ν ) = 1 2 D f ( t , φ ) f * ( t , φ ) d t d ϕ ,
r = [ ( t sin φ μ / p ) 2 + ( t cos ϕ ν / q ) 2 ] 1 / 2 , cos ϕ = t cos ϕ ( ν / q ) r ,
f ( t , ϕ ) = exp [ ( 1 / 2 ) σ t ] exp [ i 2 π W ( t , ϕ ) ] ,
W ( t , φ ) = U ( t , φ ) + V ( t , φ ) ,
U ( t , φ ) = W 20 t + W 22 t cos 2 ϕ + W 40 t 2 + W 60 t 3 + W 80 t 4 + W 100 t 5
V ( t , φ ) = ( W 31 t + W 51 t 2 + W 71 t 3 + W 91 t 4 ) t cos ϕ
T ( t ) = exp [ ( σ / 2 ) t ] Φ ( t , φ ) = 2 π W ( t , φ ) = 2 π U ( t , φ ) + 2 π V ( t , φ )
t j = 1 ( 2 j 1 ) / 2 J , φ k = [ ( 2 k 1 ) / 4 K ] π ,
Φ ( t j , ϕ 2 K k ) = 2 π U ( t j , ϕ k ) 2 π V ( t j , ϕ k ) , Φ ( t j , ϕ 2 K + k ) = 2 π U ( t j , ϕ k ) 2 π V ( t j , φ k ) , Φ ( t j , ϕ 4 K k ) = 2 π U ( t j , ϕ k ) + 2 π V ( t j , φ k ) ,
δ D = π / ( 2 J K )
D ( m , n ; m + μ , n ± ν ) = D [ 1 D j k T ( t j ) T ( t j ) × exp { i [ Φ ( t j , ϕ k ) Φ ( t j , ϕ k ) ] } π 2 J K ] ,
D ( m , n ; m + μ , n ± ν ) = D [ 1 N j k T ( t j ) T ( t j ) × exp { i [ Φ ( t j , ϕ k ) Φ ( t j , ϕ k ) ] } ]
x ¯ = X ¯ / h , y ¯ = / h
D ( m , n ; m + μ , n ± ν ) = ( x x ) 2 + ( y y ¯ ) 2 1 f ( x m p , y n q ) × f * ( x m + μ p , y n ± ν q ) d x d y ,
D ( m , n ; m + μ , n ± ν ) = f ( x , y ) f * ( x μ p , y ± ν q ) d x d y , [ x { m p x ¯ } ] 2 + [ y { n q y ¯ } ] 2 1 ,
x 0 = X 0 / h y 0 = Y 0 / h
τ ( x , y ) = exp { ( 1 / 2 ) σ [ ( x x 0 ) 2 + ( y y 0 ) 2 ] }
β m = β [ 1 + M cos ( 2 π m p / P ) ] ,
m = + R ̂ 00 ( u m p , υ )
m = + R ̂ 00 ( u m p n s , υ n q )
R ̂ 0 ( u , υ ) = m = + n = + R ̂ 00 ( u m p n s , υ n q )
r ̂ 0 ( x , y ) = + m = + n = + R ̂ 00 ( u m p n s , υ n q ) × exp { i 2 π ( u x + υ y ) } d u d υ
r ̂ 0 ( x , y ) = m = + n = + + R ̂ 00 ( α , β ) × exp [ i 2 π ( α x + β y ) ] d α d β × exp { i 2 π [ ( m p + n s ) x + n q y ] } ;
r ̂ 0 ( x , y ) = m = + n = + r ̂ 00 ( x , y ) × exp { i 2 π [ m ( p x ) + n ( q y + s x ) ] } ,
r ̂ 00 ( x , y ) = E 1 ( x , y ) + E 2 ( x , y ) × exp { i π β [ 1 + M cos ( 2 π m p / p ) ] x } + E 3 ( x , y ) exp { + i π β [ 1 + M cos ( 2 π m p / p ) ] x } ,
E 1 ( x , y ) = [ sin ( π p x ) / ( π x ) ] [ sin ( π q x ) / ( π y ) ] , E 2 ( x , y ) = ( i / 2 π x ) I 1 ( y ; γ ) + I 2 ( x , y ; γ ) , E 3 ( x , y ) = ( i / 2 π x ) I 1 ( y ; γ ) + I 2 ( x , y ; γ ) ,
exp ( ± i r cos ϕ ) = r = + ( ± i ) r J r ( r ) exp ( ± i r ϕ ) ,
r ̂ 00 ( x , y ) = E 1 ( x , y ) + E 2 ( x , y ) exp ( i π β x ) × r = + ( i ) r J r ( π β M x ) exp [ i r ( 2 π m p p ) ] + E 3 ( x , y ) exp ( + i π β x ) × r = + ( + i ) r J r ( π β M x ) exp [ + i r ( 2 π m p p ) ] ,
r ̂ 0 ( x , y ) = E 1 ( x , y ) m = + exp [ i 2 π m ( p x ) ] × n = + exp [ i 2 π n ( q y + s x ) ] + E 2 ( x , y ) exp ( i π β x ) r = + ( i ) r J r ( π β M x ) m = + × exp [ i 2 π m ( p x + r p P ) ] n = + exp [ i 2 π ( q y + s x ) ] + E 3 ( x , y ) exp ( + i π β x ) r = + ( + i ) r J r ( π β M x ) m = + × exp [ i 2 π m ( p x r p P ) ] n = + exp [ i 2 π n ( q y + s x ) ] ,
r ̂ 0 ( x , y ) = E 1 ( x , y ) m = + exp [ i 2 π m ( p x ) ] × n = + exp [ i 2 π n ( q y + s x ) ] + r = + [ E 2 ( x , y ) exp ( i π β x ) ( i ) r + E 3 ( x , y ) exp ( + i π β x ) ( i ) r ] × J r ( π β M x ) m = + exp [ i 2 π m ( p x + r p P ) ] × n = + exp [ i 2 π n ( q y + s x ) ] ,
n = + exp [ ± i n ( 2 π Δ ) ] = lim N n = N + N exp [ ± i n ( 2 π Δ ) ] = lim n { 1 + 2 n = 1 N cos [ n ( 2 π Δ ) ] }
n = + exp { ± i n ( 2 π Δ ) } = lim N sin ( N + 1 ) π Δ sin ( π Δ ) ,
n = + exp { ± i n ( 2 π Δ ) } = n = + δ ( Δ n ) ,
r ̂ 00 ( x , y ) = E 1 ( x , y ) m = + δ ( p x m ) × m = + δ ( q y + s x n ) × r = + { E 2 ( x , y ) exp { i π β x } ( i ) r + E 3 ( x , y ) exp { + i π β x } ( i ) r } . J r ( π β M x ) m = + δ ( p x + r p P m ) m = + δ ( q y + s x n )
r ̂ 0 ( x , y ) = E 1 ( x , y ) ( 1 p q ) m = + δ ( x m p ) n = + × δ ( y + s x q n p ) + r = + [ E 2 ( x , y ) × exp ( i π β x ) ( i ) r + E 3 ( x , y ) exp ( + i π β x ) ( i ) r ] × J 1 ( π β M x ) ( 1 p q ) m = + δ ( x + r P m p ) n = + × δ ( y + s x q n q ) ,
R ̂ ( u , υ ) = R ̂ 0 ( u u , υ υ ) ;
r ̂ ( x , y ) = exp [ i 2 π ( u x + υ y ) ] r ̂ 0 ( x , y )
a ( x , y ) = + f ( x x , y y ) exp [ i 2 π ( u x + υ y ) × r ̂ 0 ( x + y ) d x d y
x = m / p r / P , y = n / q s x / q ,
x = m ̂ / p , y = n ̂ / q ,
m ̂ = m p ( r / P ) , n ̂ = n s ( m / p r / P )
a ( x , y ) = m n exp { i 2 π [ ( m ̂ p ) u + ( n ̂ q ) υ ] } × ( 1 p q ) E 1 ( m ̂ p , n ̂ q ) f ( x m ̂ p , y n ̂ q ) + r m n exp { i 2 π [ ( m ̂ p ) u + ( n ̂ q ) υ ] } ( 1 p q ) × [ E 2 ( m ̂ p , n ̂ q ) exp ( i π β m ̂ p ) ( i ) r + E 3 ( m ̂ p , n ̂ q ) exp ( + i π β m ̂ p ) ( i ) r ] × J r ( π β M m ̂ p ) f ( x m ̂ p , y n ̂ q ) ,
a ( x , y ) = r m n exp { i 2 π [ ( m ̂ p ) u + ( n ̂ q ) υ } ( 1 p q ) × { δ m n E 1 ( m ̂ p , n ̂ q ) + [ E 2 ( m ̂ p , n ̂ q ) exp ( i π β m ̂ p ) ( i ) r + E 3 ( m ̂ p , n ̂ q ) exp ( + i π β m ̂ p ) ( i ) r ] J r ( ( π β M m ̂ p ) × f ( x m ̂ p , y n ̂ q ) } ,
I ( t ) = ( x 2 + y 2 ) 1 | a ( x , y ) | 2 d x d y
exp [ i 2 π ( m ̂ m ̂ p ) u ]
ω μ ̂ t = 2 π ( μ ̂ / p ) u ,
μ ̂ = m ̂ m ̂ = ( m m ) p / P ( r r )
μ = m m , ρ = r r , μ ̂ = ( μ ( p / P ) ρ ) ,
ω μ ̂ t = 2 π ( μ / p ) u 2 π ( ρ / P ) u ,
ω μ ̂ t = μ [ 2 π ( n sin α λ ) ω a p ] t + ρ [ 2 π ( n sin α λ ) ω a P ] t
P = ( n sin α λ ) ξ
ω μ ̂ = μ ω 1 + ρ ω 1
ω 1 = 2 π ( ω a / ξ )
ω 1 = 2 π ( ω a / ξ )
D ( m ̂ , n ̂ ; m ̂ , n ̂ ) = ( x 2 + y 2 ) 1 f ( x m ̂ p , y n ̂ q ) × f * ( x m ̂ p , y n ̂ q ) d x d y
μ ̂ = ( m ̂ m ̂ ) = μ ( p / P ) ρ
ν ̂ = ( n ̂ n ̂ ) = ( n n ) ( s / p ) [ ( m m ) ( p / P ) ( r r ) ]
ν ̂ = ν ( s / p ) [ μ ( p / P ) ρ ]
D ( m ̂ , n ̂ ; m ̂ + μ , n ̂ + ν ̂ ) = ( x m ̂ / p ) 2 + ( y n ̂ / q ) 2 1 f ( x , y ) f * ( x μ ̂ p , y ν ̂ q ) d x d y
tracks ( + 1 + 3 k ) p + , tracks ( 0 + 3 k ) p 0 , tracks ( 1 + 3 k ) p ,
R 0 0 ( u , υ ) = m 0 n 0 R m 0 n 0 0 exp { i 2 π [ ( m 0 p 0 ) u + ( n 0 3 q ) υ ] }
R m 0 n 0 0 = ( 1 3 q p 0 ) ( p 0 / 2 ) + ( p 0 / 2 ) ( q / 2 + ( q / 2 ) R 0 0 ( u , υ ) × exp { i 2 π [ ( m 0 p 0 ) u + ( n 0 3 q ) υ ] } d u d υ ,
R 0 ( u , υ ) = m + n + R m + , n + + exp { i 2 π [ ( m + p + ) u + ( n + 3 q ) υ ] }
R m + n + + = ( 1 3 q p + ) ( p + / 2 ) + ( p + / 2 ) ( q / 2 ) + ( q / 2 ) R 0 + ( u , υ ) × exp { i 2 π [ ( m + p + ) u + ( n + 3 q ) υ ] } d u d υ ,
R 0 ( u , υ ) = m n R m n exp { i 2 π [ ( m p ) u + ( n 3 q ) υ ] }
R m n = ( 1 3 q p ) ( p / 2 ) + ( p / 2 ) ( q / 2 ) + ( q / 2 ) R 0 ( u , υ ) × exp { i 2 π [ ( m p ) u + ( n 3 q ) υ ] } d u d υ
R 0 ( u , υ ) = R 0 ( u , υ + q ) + R 0 0 ( u , υ ) + R 0 + ( u , υ q )
R 0 ( u , υ ) = m n ( R m n exp ( i 2 π n 3 ) × exp { i 2 π [ ( m p ) u + ( n 3 q ) υ ] } + R m n 0 exp { 2 π [ ( m p 0 ) u + ( n 3 q ) υ ] } + R m n + exp { i 2 π n 3 } exp { i 2 π [ ( m p + ) u + ( n 3 q ) υ ] } )
r ( x , y ) = m n exp { i 2 π [ ( m p ) u + ( n 3 q ) υ ] } × R m n exp ( i 2 π n 3 ) δ ( x m p , y n 3 q ) + m n exp { i 2 π [ ( m p 0 ) u + ( n 3 q ) υ ] } × R m n 0 δ ( x m p 0 , y n 3 q ) + m n exp { i 2 π [ ( m p + ) u + ( n 3 q ) υ ] } × R m n + exp ( i 2 π n 3 ) δ ( x m p + , y n 3 q )
a ( x , y ) = m n exp { i 2 π [ ( m p ) u + ( n 3 q ) υ ] } × R m n exp ( i 2 π n 3 ) f ( x m p , y n 3 q ) + m n exp { i 2 π [ ( m p 0 ) u + ( n 3 q ) υ ] } × R m n 0 f ( x m p 0 , y n 3 q ) + m n exp { i 2 π [ ( m p ) u + ( n 3 q ) υ ] } × R m n + exp ( i 2 π n 3 ) f ( x m p + , y n 3 q ) ,
I ( t ) = ( x 2 + y 2 ) 1 | a ( x , y ) | 2 d x d y ,
I ( t ) = A 0 + μ = 1 { A μ cos ( ω μ t ) + B μ sin ( ω μ t ) }
ω μ = μ [ 2 π ( ω a / ξ ) ]
I μ = ( A μ 2 + B μ 2 ) 1 / 2
β m = β [ 1 + M cos ( 2 π m p / P ) ] ,
ω μ ̂ = μ [ 2 π ( ω a / ξ ) ] + ρ [ 2 π ( ω a / ξ ) ]
W ( r , ϕ ) = W 20 r 2 ,
W ( r , ϕ ) = W 31 r 3 cos ϕ + W 11 r cos ϕ
W ( r , ϕ ) = W 40 r 4 + W 20 r 2
W ( r , ϕ ) = W 22 r 2 cos 2 ϕ ,