Abstract

The suitability of various binary encoding methods for electron-beam recording of computer generated holograms is systematically evaluated. Subjected to the limitations of computing resources, a set of criteria is established according to which these encoding schemes are evaluated and compared. This comparison can be used to determine the optimum encoding method for desired wavefront properties.

© 1987 Optical Society of America

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References

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  1. S. M. Arnold, “Electron Beam Fabrication of Computer Generated Holograms,” Opt. Eng. 24, 803 (1985).
    [CrossRef]
  2. S. M. Arnold, “E-Beam Written Computer Generated Holograms,” Final Technical Report E47069, Honeywell Corporation Technology Center, Bloomington, MN (1983).
  3. K. M. Leung, J. C. Lindquist, L. T. Shepherd, “E-Beam Computer-Generated Holograms for Aspheric Testing,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).
  4. J. L. Freyer, R. J. Perlmutter, J. W. Goodman, “Digital Holography: Algorithms, E-Beam Lithography, and 3-D Display,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).
  5. R. A. Athale, C. L. Giles, J. A. Blodgett, “Use of E-beam Written CGH in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 48 (1983).
  6. R. A. Gabel, “Reconstruction Errors in Computer Generated Binary Holograms: A Comparative Study,” Appl. Opt. 14, 2252 (1975).
    [CrossRef] [PubMed]
  7. R. A. Gabel, B. Liu, “Minimization of Reconstruction Errors in Computer Generated Binary Holograms,” Appl. Opt. 9, 1180 (1970).
    [CrossRef] [PubMed]
  8. J. Bucklew, N. C. Gallagher, “Comprehensive Error Models and a Comparative Study of Some Detour-Phase Holograms,” Appl. Opt. 18, 2861 (1979).
    [CrossRef] [PubMed]
  9. D. R. Herriott, G. R. Brewer, “Electron-Beam Lithography Machines,” in Electron-Beam Technology in Microelectronic Fabrication, G. R. Brewer, Ed. (Academic, New York, 1980), Chap. 3.
  10. E. E. Erny, J. Webster, M. Salsali, “State-of-the-Art Electron-beam Maskmaking Production Tools,” Report 60, Varian Associates, Inc., Lithography Products Division (1984).
  11. J. P. Allebach, “Representation-Related Errors in Binary Digital Holograms: A Unified Analysis,” Appl. Opt. 20, 290 (1981).
    [CrossRef] [PubMed]
  12. B. R. Brown, A. W. Lohmann, “Complex Spatial Filtering with Binary Masks,” Appl. Opt. 5, 967 (1967).
    [CrossRef]
  13. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer Holograms Generated by Computers,” Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  14. W. H. Lee, “Sampled Fourier Transform Hologram Generated by Computer,” Appl. Opt. 9, 639 (1970).
    [CrossRef] [PubMed]
  15. J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” Proc. IEEE 55, 599 (1967).
    [CrossRef]
  16. B. R. Brown, A. W. Lohmann, “Computer Generated Binary Holograms,” IBM J.Res. Dev. 13, 160 (1969).
    [CrossRef]
  17. W. H. Lee, “Binary Computer-Generated Holograms,” Appl. Opt. 18, 3661 (1979).
    [CrossRef] [PubMed]
  18. W. H. Lee, “Binary Synthetic Holograms,” Appl. Opt. 13, 1677 (1974).
    [CrossRef] [PubMed]
  19. W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
    [CrossRef]
  20. N. S. Jayant, P. Noll, Digital Coding of Waveforms, Principles and Applications to Speech and Video (Prentice-Hall, Englewood Cliffs, NJ, 1984), pp. 31–35.
  21. J. W. Goodman, A. M. Silvestri, “Some Effects of Fourier-Domain Phase Quantization,” IBM J. Res. Dev. 14, 478 (1970).
    [CrossRef]
  22. See, for example, S. Haykin, Communication Systems (Wiley, New York, 1978), pp. 492–496.
  23. P. B. Fellgett, E. H. Linfoot, “On the Assessment of Optical Images,” Philos. Trans. R. Soc. London Ser. A 247, 369 (1954).
  24. I. J. Cox, C. J. R. Sheppard, “Information Capacity and Resolution in an Optical System,” J. Opt. Soc. Am. A 3, 1152 (1986).
    [CrossRef]
  25. P. Chavel, J. P. Hugonin, “High Quality Holograms: The Problem of Phase Representation,” J. Opt. Soc. Am. 66, 989 (1976).
    [CrossRef]

1986

1985

S. M. Arnold, “Electron Beam Fabrication of Computer Generated Holograms,” Opt. Eng. 24, 803 (1985).
[CrossRef]

1983

J. L. Freyer, R. J. Perlmutter, J. W. Goodman, “Digital Holography: Algorithms, E-Beam Lithography, and 3-D Display,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).

R. A. Athale, C. L. Giles, J. A. Blodgett, “Use of E-beam Written CGH in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 48 (1983).

1981

1980

K. M. Leung, J. C. Lindquist, L. T. Shepherd, “E-Beam Computer-Generated Holograms for Aspheric Testing,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).

1979

1978

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
[CrossRef]

1976

1975

1974

1970

1969

B. R. Brown, A. W. Lohmann, “Computer Generated Binary Holograms,” IBM J.Res. Dev. 13, 160 (1969).
[CrossRef]

1967

1954

P. B. Fellgett, E. H. Linfoot, “On the Assessment of Optical Images,” Philos. Trans. R. Soc. London Ser. A 247, 369 (1954).

Allebach, J. P.

Arnold, S. M.

S. M. Arnold, “Electron Beam Fabrication of Computer Generated Holograms,” Opt. Eng. 24, 803 (1985).
[CrossRef]

S. M. Arnold, “E-Beam Written Computer Generated Holograms,” Final Technical Report E47069, Honeywell Corporation Technology Center, Bloomington, MN (1983).

Athale, R. A.

R. A. Athale, C. L. Giles, J. A. Blodgett, “Use of E-beam Written CGH in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 48 (1983).

Blodgett, J. A.

R. A. Athale, C. L. Giles, J. A. Blodgett, “Use of E-beam Written CGH in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 48 (1983).

Brewer, G. R.

D. R. Herriott, G. R. Brewer, “Electron-Beam Lithography Machines,” in Electron-Beam Technology in Microelectronic Fabrication, G. R. Brewer, Ed. (Academic, New York, 1980), Chap. 3.

Brown, B. R.

B. R. Brown, A. W. Lohmann, “Computer Generated Binary Holograms,” IBM J.Res. Dev. 13, 160 (1969).
[CrossRef]

B. R. Brown, A. W. Lohmann, “Complex Spatial Filtering with Binary Masks,” Appl. Opt. 5, 967 (1967).
[CrossRef]

Bucklew, J.

Burch, J. J.

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” Proc. IEEE 55, 599 (1967).
[CrossRef]

Chavel, P.

Cox, I. J.

Erny, E. E.

E. E. Erny, J. Webster, M. Salsali, “State-of-the-Art Electron-beam Maskmaking Production Tools,” Report 60, Varian Associates, Inc., Lithography Products Division (1984).

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, “On the Assessment of Optical Images,” Philos. Trans. R. Soc. London Ser. A 247, 369 (1954).

Freyer, J. L.

J. L. Freyer, R. J. Perlmutter, J. W. Goodman, “Digital Holography: Algorithms, E-Beam Lithography, and 3-D Display,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).

Gabel, R. A.

Gallagher, N. C.

Giles, C. L.

R. A. Athale, C. L. Giles, J. A. Blodgett, “Use of E-beam Written CGH in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 48 (1983).

Goodman, J. W.

J. L. Freyer, R. J. Perlmutter, J. W. Goodman, “Digital Holography: Algorithms, E-Beam Lithography, and 3-D Display,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).

J. W. Goodman, A. M. Silvestri, “Some Effects of Fourier-Domain Phase Quantization,” IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

Haykin, S.

See, for example, S. Haykin, Communication Systems (Wiley, New York, 1978), pp. 492–496.

Herriott, D. R.

D. R. Herriott, G. R. Brewer, “Electron-Beam Lithography Machines,” in Electron-Beam Technology in Microelectronic Fabrication, G. R. Brewer, Ed. (Academic, New York, 1980), Chap. 3.

Hugonin, J. P.

Jayant, N. S.

N. S. Jayant, P. Noll, Digital Coding of Waveforms, Principles and Applications to Speech and Video (Prentice-Hall, Englewood Cliffs, NJ, 1984), pp. 31–35.

Lee, W. H.

Leung, K. M.

K. M. Leung, J. C. Lindquist, L. T. Shepherd, “E-Beam Computer-Generated Holograms for Aspheric Testing,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).

Lindquist, J. C.

K. M. Leung, J. C. Lindquist, L. T. Shepherd, “E-Beam Computer-Generated Holograms for Aspheric Testing,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).

Linfoot, E. H.

P. B. Fellgett, E. H. Linfoot, “On the Assessment of Optical Images,” Philos. Trans. R. Soc. London Ser. A 247, 369 (1954).

Liu, B.

Lohmann, A. W.

Noll, P.

N. S. Jayant, P. Noll, Digital Coding of Waveforms, Principles and Applications to Speech and Video (Prentice-Hall, Englewood Cliffs, NJ, 1984), pp. 31–35.

Paris, D. P.

Perlmutter, R. J.

J. L. Freyer, R. J. Perlmutter, J. W. Goodman, “Digital Holography: Algorithms, E-Beam Lithography, and 3-D Display,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).

Salsali, M.

E. E. Erny, J. Webster, M. Salsali, “State-of-the-Art Electron-beam Maskmaking Production Tools,” Report 60, Varian Associates, Inc., Lithography Products Division (1984).

Shepherd, L. T.

K. M. Leung, J. C. Lindquist, L. T. Shepherd, “E-Beam Computer-Generated Holograms for Aspheric Testing,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).

Sheppard, C. J. R.

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some Effects of Fourier-Domain Phase Quantization,” IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

Webster, J.

E. E. Erny, J. Webster, M. Salsali, “State-of-the-Art Electron-beam Maskmaking Production Tools,” Report 60, Varian Associates, Inc., Lithography Products Division (1984).

Appl. Opt.

IBM J. Res. Dev.

J. W. Goodman, A. M. Silvestri, “Some Effects of Fourier-Domain Phase Quantization,” IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

IBM J.Res. Dev.

B. R. Brown, A. W. Lohmann, “Computer Generated Binary Holograms,” IBM J.Res. Dev. 13, 160 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

S. M. Arnold, “Electron Beam Fabrication of Computer Generated Holograms,” Opt. Eng. 24, 803 (1985).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A

P. B. Fellgett, E. H. Linfoot, “On the Assessment of Optical Images,” Philos. Trans. R. Soc. London Ser. A 247, 369 (1954).

Proc. IEEE

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” Proc. IEEE 55, 599 (1967).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

K. M. Leung, J. C. Lindquist, L. T. Shepherd, “E-Beam Computer-Generated Holograms for Aspheric Testing,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).

J. L. Freyer, R. J. Perlmutter, J. W. Goodman, “Digital Holography: Algorithms, E-Beam Lithography, and 3-D Display,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).

R. A. Athale, C. L. Giles, J. A. Blodgett, “Use of E-beam Written CGH in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 48 (1983).

Prog. Opt.

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
[CrossRef]

Other

N. S. Jayant, P. Noll, Digital Coding of Waveforms, Principles and Applications to Speech and Video (Prentice-Hall, Englewood Cliffs, NJ, 1984), pp. 31–35.

S. M. Arnold, “E-Beam Written Computer Generated Holograms,” Final Technical Report E47069, Honeywell Corporation Technology Center, Bloomington, MN (1983).

D. R. Herriott, G. R. Brewer, “Electron-Beam Lithography Machines,” in Electron-Beam Technology in Microelectronic Fabrication, G. R. Brewer, Ed. (Academic, New York, 1980), Chap. 3.

E. E. Erny, J. Webster, M. Salsali, “State-of-the-Art Electron-beam Maskmaking Production Tools,” Report 60, Varian Associates, Inc., Lithography Products Division (1984).

See, for example, S. Haykin, Communication Systems (Wiley, New York, 1978), pp. 492–496.

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Figures (2)

Fig. 1
Fig. 1

(a) Coordinate system and notation in the image plane. (b) Coordinate system in the CGH plane. (c) The CGH cell structure.

Fig. 2
Fig. 2

Sampling periods and representation of different encoding schemes. (a) Lohmann method in which the height and location of the apertures are determined by the amplitude and phase, respectively, of the wavefront at the center of the cell. (b) Revised Lohmann method, where the wavefront is evaluated at each sampling point, and a single aperture is placed at the sampling point which most nearly satisfies Eq. (1). The aperture height is determined by the amplitude of the wavefront at the sampling point. (c) In the Lee(1) method the wavefront is sampled at the center of each cell, and the height of each aperture is determined by the component of the sample along the appropriate basis vectors [0,π/2,π,(3π)/2]. (d) Lee(2) method is similar to Lee(1) method except that the wavefront is sampled at the center of each of the four apertures. (e) Burch's method in which the hologram transmittance function given in Eq. (2) is sampled at the center of each cell and a square aperture of area proportional to the sampled value is placed at the center of the cell. (f) In Arnold's method the hologram is drawn fringe by fringe. For each fringe, the location of the next sampling point is determined by beginning at a center line point and moving in the y direction by an amount equal to the local step size that is updated at each sampling point.

Tables (6)

Tables Icon

Table I Bandwidths, SNR, Diffraction Efficiency, NPS, and the Interpolation Formulas for Various Encoding Methods

Tables Icon

Table II Comparison of Encoding Methods for Amplitude and Phase Varying Wavefronts with Minimum Interpolation, Constant NPS = 223 ≈ 8.4 × 106, na = 5, np = 12, df = 0.5μm, dr = 0.125 μm, and δdr = 1/32 μm

Tables Icon

Table III Comparison of Encoding Methods for Amplitude and Phase Varying Wavefronts with Constant MxMy = 24, Constant NPS = 223 ≈ 8.4 × 106, na = 5, np = 12, df = 0.5μm, dr = 0.125 μm, and δdr = 1/32 μm

Tables Icon

Table IV Comparison of Encoding Methods for Amplitude and Phase Varying Wavefronts with Constant MxMy = 24, Constant N 0 2 = 10 6, na = 5, np = 12, df = 0.5 μm, dr = 0.125 μm, and δdr = 1/32 μm

Tables Icon

Table V Comparison of Encoding Methods for Phase Only Varying Wavefronts with Constant M x M y N 0 2 = 24 × 10 6, Variable Number of Quantization Levels, df = 0.5 μm, dr = 0.125 μm, and δdr = 1/32 μm

Tables Icon

Table VI Comparison of Information Capacity of Encoding Methods for Amplitude and Phase Varying Wavefronts. The NPS is held Fixed and the Interpolation Factors are kept at their Minimum: na = 5, np = 12, df = 0.5 μm, dr = 0.125 μm, and δdr = 1/32 μm

Equations (79)

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2 π x t c + ϕ ( x , y ) = 2 π n ,
T ( x , y ) = 0.5 [ 1 + | G ( x , y ) | cos ( 2 π x t c ϕ ( x , y ) ) ] ,
N x N y = M x M y N 0 2 ,
N 0 2 = B ox B oy X Y = B x B y X Y .
B x = B ox X X = B hx M x , B y = B oy Y Y = B hy M y .
SNR = average intensity in the desired image mean square error = 1 X Y | g ( x , y ) | 2 dxdy 1 X Y | h ( x , y ) g ( x , y ) | 2 dxdy ,
g ( x , y ) = m = 0 N x 1 n = 0 N y 1 A mn exp ( j ϕ mn ) exp [ j 2 π λ F ( xm δ x + yn δ y ) ] .
( SNR D ) min 12 π 2 ( δ d r ) 2 [ 4 t c 2 + 1 ( δ y M y ) 2 ] ,
SNR 1 = [ i SNR i ] 1 i = Q , R , D .
η t = P ideal P inc ,
η = P s P inc ,
η = r η t ,
NPS = ( number of hologram cells ) × N c = X Y δ x δ y N c = M x M y N 0 2 N c ,
NPS = P r q M y N 0 2 = P r 1 n p M x M y N 0 2 ,
C = B log 2 ( 1 + SNR ) .
C = 2 X Y B hx B hy log 2 ( 1 + SNR ) .
C = B x B y log 2 ( 1 + SNR ) .
B x 1
B y 1
[ 1 4 n a 2 + π 2 3 n p 2 ] 1
[ 1 1344 ( π M y ) 4 + 1 3 ( γ M x ) 2 ] 1
M x M y N 0 2
[ 1 1344 ( π M y ) 4 + 5 64 ( γ M x ) 2 ] 1
2 M x M y N 0 2
1 2
( n a 1 ) d f M y
( n a 1 ) d f M y
4 n a 2
960 π 4 ( M x 2 + M y 2 ) 2
M s ( n a 1 ) d f
( n a 1 ) d f
M x M y N 0 2
[ 1 4 n a 2 + π 2 3 n p 2 ] 1
1344 ( M y π ) 4
1 M s M x M y N 0 2
M x 4
2 n a 2
1344 ( M y π ) 4
1 2 M x M y N 0 2
3 n p 2 π 2
12 ( δ ϕ ) 2 12 M y 2 π 2
P r M s M x M y N 0 2
N 0 2 × 1 0 6
N 0 2 × 1 0 5
N 0 2 × 10 5
B x
B y
MSE = N x N y 12 n a 2 + 2 [ 1 sinc ( 1 n p ) ] M = 0 N x 1 n = 0 N y 1 | A mn | 2 ,
SNR Q = 1 N x N y M = 0 N x 1 n = 0 N y 1 | A mn | 2 1 12 n a 2 + 2 N x N y [ 1 sinc ( 1 n p ) ] M = 0 N x 1 n = 0 N y 1 | A mn | 2 .
1 N x N y m = 0 N x 1 n = 0 N y 1 | A mn | 2 = .
1 N x N y m = 0 N x 1 n = 0 N y 1 | ϕ mn | 2 = 4 3 π 2 .
SNR Q = 1 1 4 n a 2 + 2 [ 1 sinc ( 1 n p ) ] 1 1 4 n a 2 + π 2 3 n p 2 .
E mn = 1 sinc ( y δ y A mn λ F ) exp [ j ( x X 0 ) δ x ϕ mn λ F M ] ( π y λ F δ y A mn ) 2 6 j ( x X 0 ) δ x ϕ mn λ F ,
( SNR R ) min 1 1 1344 ( π M y ) 4 + 1 3 ( π M x ) 2 1 60 ( π M x ) 4 1 120 ( π 2 M x M y ) 2 .
h ( x , y ) = δ x 8 sinc ( 1 4 + x δ x 4 λ F ) exp ( j 3 π 4 ) m = 0 N x 1 n = 0 N y 1 d y × exp [ j 2 π λ F ( n x δ x + my δ y ) ] × { ( 1 E mn r 1 ) A 1 mn ( 1 E mn r 3 ) A 3 mn + j [ ( 1 E mn r 2 ) A 2 mn + ( 1 E mn r 4 ) A 4 mn ] } ,
E mn ( r , i ; k ) = 1 sinc ( y δ y A kmn λ F ) exp [ j P π ( x x 0 ) δ x 4 λ F ] ( π y λ F δ y A kmn ) 2 6 j P π ( x x 0 ) δ x 4 λ F .
( SNR R ) min 1 1 1344 ( π M y ) 4 + 5 16 ( π M x ) 2 1 1200 ( π M x ) 4 1 512 ( π 2 M x M y ) 2 .
( E mn ) R , aperture = 1 sinc ( y δ y A mn λ F ) sinc [ ( x x 0 ) δ x A mn λ F ] .
( SNR R ) min 960 π 4 [ 1 1 M x 2 + 1 M y 2 ] 2 .
SNR R = 12 ( δ ϕ ) 2 = 12 M y 2 π 2
SNR D = 1 2 [ 1 sinc ( Φ 2 π ) ] 12 Φ 2 ,
Φ = ( Φ x 2 + Φ y 2 ) 1 / 2 , Φ x = 2 π δ d r t c , Φ y = 2 π δ d r ( t y ) av ,
T ( x , y ) = 1 δ x 1 δ y rect ( x δ x ) rect ( y δ y ) * [ G ( x , y ) × comb ( x δ x ) comb ( y δ y ) ] ,
t ( x , y ) = sinc ( δ x x λ F ) sinc ( δ y y λ F ) n m × g ( x n λ F δ x , y m λ F δ y ) .
g 00 ( x , y ) = sinc ( δ x x λ F ) sinc ( δ y y λ F ) g ( x , y ) ,
r = x 0 X 2 x 0 + X 2 Y 2 + Y 2 sinc 2 ( δ x x λ F ) sinc 2 ( δ y y λ F ) | g 00 ( x , y ) | 2 dxdy | g 00 ( x , y ) | 2 dxdy .
r = 1 X Y x 0 X 2 x 0 + X 2 Y 2 + Y 2 sinc 2 ( δ x x λ F ) sinc 2 ( δ y y λ F ) dxdy = q 0.5 q + 0.5 sinc 2 ( u M x ) d u 0.5 0.5 sinc 2 ( u M y ) d υ .
X ( a , b ) = a 0.5 a + 0.5 sinc 2 u b du
r = X ( q , M x ) X ( 0 , M y ) ,
lim r M x = sinc 2 ( 1 M s ) = k M s = constant ,
r = X ( q , M x ) X ( 0 , M y ) sinc 2 ( 1 M s ) .
1 2 π δ ϕ | ϕ y | | ϕ x | ,
NPS = 1 2 π δ ϕ x y | ϕ y | | ϕ x | d x d y .
1 π | ϕ x | = constant = 2 t c = 2 q B x and y | ϕ y | d y = Y ( | ϕ y | ) av ,
NPS = P r q π δ ϕ N 0 2 , P r = ( | ϕ y | ) av ( | ϕ y | ) max .
M y = 1 B y 1 minimum sampling period ,
NPS = P r q M y N 0 2 = P r 1 n p M x M y N 0 2 .
2 P r q π δ ϕ N 0 2 ,
( M x M Y N 0 2 ) Arnold's method = n p 2 P r ( M x M Y N 0 2 ) conventional methods .

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