Abstract

Incoherent optical spatial filtering systems have a number of advantages over their coherent counterparts; however, they are limited in their conventional form to operating with nonnegative–real input, output, and pointspread distributions. Most serious is the limitation to nonnegative–real impulse responses. A broad class of hybrid methods is investigated that employs two pupils in the synthesis of bipolar–real impulse responses. The mathematical structure of these syntheses is presented along with limitations in the presence of various constraints. Minimization of image plane bias is considered. Methods for implementation are described and categorized.

© 1978 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.
  2. See, e.g., Proc. IEEE, 65, No. 1 (Jan.1977), special issue on optical computing.
  3. J. D. Armitage, A. W. Lohmann, Appl. Opt. 4, 461 (1965);S. Lowenthal, A. Werts, C. R. Acad. Sci. Ser. B 266, 542 (1968);W. T. Maloney, Appl. Opt. 10, 2127 (1971);W. T. Rhodes, W. R. Limburg, in Proceedings of the 1972 Electro-Optical Systems Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1972), pp. 314–320.
    [Crossref] [PubMed]
  4. G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
    [Crossref]
  5. J. W. Goodman, in Optical Information Processing, Y. E. Nesterikhin, G. W. Stroke, W. E. Kock, Eds. (Plenum, New York, 1976), pp. 85–103.
    [Crossref]
  6. S. Lowenthal, P. Chevel, in Proceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).
  7. A. Lohmann, Opt. Acta 6, 319 (1959);J. D. Armitage, A. W. Lohmann, R. B. Herrick, Appl. Opt. 4, 445 (1965);D. H. Kelly, J. Opt. Soc. Am. 51, 1095 (1961);E. A. Trabka, P. G. Roetling, J. Opt. Soc. Am. 54, 1242 (1964);R. V. Shack, Pattern Recognition 2, 123 (1970);W. Swindell, Appl. Opt. 9, 2459 (1970).
    [Crossref] [PubMed]
  8. P. Chavel, S. Lowenthal, J. Opt. Soc. Am. 66, 14 (1976).
    [Crossref]
  9. G. Hausler, A. Lohmann, in Preceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).
  10. B. Braunecker, R. Hauck, Opt. Commun. 20, 234 (1977).
    [Crossref]
  11. W. T. Rhodes, in Proceedings of the 1976 International Optical Computing Conference, Capri, Italy, S. Horwitz, Ed. (IEEE, New York, 1977).
  12. W. T. Rhodes, Appl. Opt. 16, 265 (1977).
    [Crossref] [PubMed]
  13. A. W. Lohmann, Appl. Opt. 16, 261 (1977).
    [Crossref] [PubMed]
  14. D. Görlitz, F. Lanzl, Opt. Commun. 20, 68 (1977).
    [Crossref]
  15. In order for the object to be considered quasi-monochromatic, the condition λ/Δλ > SW must be satisfied, where SW is the 1-D space–bandwidth product of the optical system.See, e.g., A. W. Lohmann, Appl. Opt. 7, 561 (1968).
    [Crossref]
  16. W. Lukosz, J. Opt. Soc. Am. 52, 827 (1962);Opt. Acta 9, 361 (1962).
    [Crossref]
  17. The independence of FR(x) and FI(x) also allows the synthesis of complex-valued impulse responses. See Ref. 13 for a discussion of suitable techniques employing modified two-pupil incoherent spatial filtering operations.
  18. D. C. Chu, J. R. Fienup, J. W. Goodman, Appl. Opt. 12, 1386 (1973).
    [Crossref] [PubMed]
  19. J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, pp. 145–149.
    [Crossref]
  20. The existence of an antisymmetric synthesis component FI(x) with real pupil transparencies is perhaps somewhat surprising. Note, however, that the effective pupil function of the imaging system, Eq. (6), is not itself restricted to real values. For example, with ϕ=−π/2,p∼(u)=ip∼1(u)+p∼2(u), which is, of course, fully complex for real p∼1, p∼2. The corresponding PSF is F(x) = |p(x)|2 = |p1(x)|2 + |p2(x)|2 + Im[p1(x)p2*(x))], where for real pupils the first two terms are even, the third odd. It is easily shown that the relative magnitude of the odd part is maximized when p∼2(u)=p∼1(−u). The PSF can then be written as F(x) = e(x) + o(x), the sum of even and odd parts, where e(x) = 2|p2(x)| and (x) = Im[p2(x)]. [The PSF is of course subject to the constraint e(x) + o(x) ≥ 0.]
  21. A. F. Metherell, in Acoustical Holography, A. F. Metherell, H. M. A. El-Sum, L. Larmore, Eds. (Plenum, New York, 1969), Vol. 1, Chap. 14.
  22. M. Severcan, Ph.D. thesis, Stanford University, University Microfilm, Ann Arbor, Order 74-13687 (1973).
  23. D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
    [Crossref] [PubMed]
  24. The functions p1(x) and p2(x) can be adequately represented by samples taken at the sampling rate for Fs(x) if the resulting pupil functions, obtained by the discrete Fourier transform, are replicated a number of times in the pupil plane. This point is discussed in “Pupil Function Replication in Optical Transfer Function Synthesis,” by B. Braunecker, R. Hauck, and W. Rhodes, in preparation. The resultant PSFs will not generally be the minimum-bias PSFs, however.
  25. S. F. Dashiell, A. W. Lohmann, J. D. Michaelson, Opt. Commun. 8, 105 (1973).
    [Crossref]

1977 (5)

See, e.g., Proc. IEEE, 65, No. 1 (Jan.1977), special issue on optical computing.

B. Braunecker, R. Hauck, Opt. Commun. 20, 234 (1977).
[Crossref]

W. T. Rhodes, Appl. Opt. 16, 265 (1977).
[Crossref] [PubMed]

A. W. Lohmann, Appl. Opt. 16, 261 (1977).
[Crossref] [PubMed]

D. Görlitz, F. Lanzl, Opt. Commun. 20, 68 (1977).
[Crossref]

1976 (1)

1975 (1)

G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
[Crossref]

1973 (2)

S. F. Dashiell, A. W. Lohmann, J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

D. C. Chu, J. R. Fienup, J. W. Goodman, Appl. Opt. 12, 1386 (1973).
[Crossref] [PubMed]

1972 (1)

1968 (1)

1965 (1)

1962 (1)

1959 (1)

A. Lohmann, Opt. Acta 6, 319 (1959);J. D. Armitage, A. W. Lohmann, R. B. Herrick, Appl. Opt. 4, 445 (1965);D. H. Kelly, J. Opt. Soc. Am. 51, 1095 (1961);E. A. Trabka, P. G. Roetling, J. Opt. Soc. Am. 54, 1242 (1964);R. V. Shack, Pattern Recognition 2, 123 (1970);W. Swindell, Appl. Opt. 9, 2459 (1970).
[Crossref] [PubMed]

Armitage, J. D.

Braunecker, B.

B. Braunecker, R. Hauck, Opt. Commun. 20, 234 (1977).
[Crossref]

Chavel, P.

Chevel, P.

S. Lowenthal, P. Chevel, in Proceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

Chu, D. C.

Dashiell, S. F.

S. F. Dashiell, A. W. Lohmann, J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

Fienup, J. R.

Goodman, J. W.

D. C. Chu, J. R. Fienup, J. W. Goodman, Appl. Opt. 12, 1386 (1973).
[Crossref] [PubMed]

D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
[Crossref] [PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

J. W. Goodman, in Optical Information Processing, Y. E. Nesterikhin, G. W. Stroke, W. E. Kock, Eds. (Plenum, New York, 1976), pp. 85–103.
[Crossref]

Görlitz, D.

D. Görlitz, F. Lanzl, Opt. Commun. 20, 68 (1977).
[Crossref]

Hauck, R.

B. Braunecker, R. Hauck, Opt. Commun. 20, 234 (1977).
[Crossref]

Hausler, G.

G. Hausler, A. Lohmann, in Preceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

Lanzl, F.

D. Görlitz, F. Lanzl, Opt. Commun. 20, 68 (1977).
[Crossref]

Lohmann, A.

A. Lohmann, Opt. Acta 6, 319 (1959);J. D. Armitage, A. W. Lohmann, R. B. Herrick, Appl. Opt. 4, 445 (1965);D. H. Kelly, J. Opt. Soc. Am. 51, 1095 (1961);E. A. Trabka, P. G. Roetling, J. Opt. Soc. Am. 54, 1242 (1964);R. V. Shack, Pattern Recognition 2, 123 (1970);W. Swindell, Appl. Opt. 9, 2459 (1970).
[Crossref] [PubMed]

G. Hausler, A. Lohmann, in Preceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

Lohmann, A. W.

Lowenthal, S.

P. Chavel, S. Lowenthal, J. Opt. Soc. Am. 66, 14 (1976).
[Crossref]

S. Lowenthal, P. Chevel, in Proceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

Lukosz, W.

Metherell, A. F.

A. F. Metherell, in Acoustical Holography, A. F. Metherell, H. M. A. El-Sum, L. Larmore, Eds. (Plenum, New York, 1969), Vol. 1, Chap. 14.

Michaelson, J. D.

S. F. Dashiell, A. W. Lohmann, J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

Rhodes, W. T.

W. T. Rhodes, Appl. Opt. 16, 265 (1977).
[Crossref] [PubMed]

W. T. Rhodes, in Proceedings of the 1976 International Optical Computing Conference, Capri, Italy, S. Horwitz, Ed. (IEEE, New York, 1977).

Rogers, G. L.

G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
[Crossref]

Severcan, M.

M. Severcan, Ph.D. thesis, Stanford University, University Microfilm, Ann Arbor, Order 74-13687 (1973).

Tsujiuchi, J.

J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, pp. 145–149.
[Crossref]

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

A. Lohmann, Opt. Acta 6, 319 (1959);J. D. Armitage, A. W. Lohmann, R. B. Herrick, Appl. Opt. 4, 445 (1965);D. H. Kelly, J. Opt. Soc. Am. 51, 1095 (1961);E. A. Trabka, P. G. Roetling, J. Opt. Soc. Am. 54, 1242 (1964);R. V. Shack, Pattern Recognition 2, 123 (1970);W. Swindell, Appl. Opt. 9, 2459 (1970).
[Crossref] [PubMed]

Opt. Commun. (3)

S. F. Dashiell, A. W. Lohmann, J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

D. Görlitz, F. Lanzl, Opt. Commun. 20, 68 (1977).
[Crossref]

B. Braunecker, R. Hauck, Opt. Commun. 20, 234 (1977).
[Crossref]

Opt. Laser Technol. (1)

G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
[Crossref]

Proc. IEEE (1)

See, e.g., Proc. IEEE, 65, No. 1 (Jan.1977), special issue on optical computing.

Other (11)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

G. Hausler, A. Lohmann, in Preceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

J. W. Goodman, in Optical Information Processing, Y. E. Nesterikhin, G. W. Stroke, W. E. Kock, Eds. (Plenum, New York, 1976), pp. 85–103.
[Crossref]

S. Lowenthal, P. Chevel, in Proceedings of the ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

W. T. Rhodes, in Proceedings of the 1976 International Optical Computing Conference, Capri, Italy, S. Horwitz, Ed. (IEEE, New York, 1977).

J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, pp. 145–149.
[Crossref]

The existence of an antisymmetric synthesis component FI(x) with real pupil transparencies is perhaps somewhat surprising. Note, however, that the effective pupil function of the imaging system, Eq. (6), is not itself restricted to real values. For example, with ϕ=−π/2,p∼(u)=ip∼1(u)+p∼2(u), which is, of course, fully complex for real p∼1, p∼2. The corresponding PSF is F(x) = |p(x)|2 = |p1(x)|2 + |p2(x)|2 + Im[p1(x)p2*(x))], where for real pupils the first two terms are even, the third odd. It is easily shown that the relative magnitude of the odd part is maximized when p∼2(u)=p∼1(−u). The PSF can then be written as F(x) = e(x) + o(x), the sum of even and odd parts, where e(x) = 2|p2(x)| and (x) = Im[p2(x)]. [The PSF is of course subject to the constraint e(x) + o(x) ≥ 0.]

A. F. Metherell, in Acoustical Holography, A. F. Metherell, H. M. A. El-Sum, L. Larmore, Eds. (Plenum, New York, 1969), Vol. 1, Chap. 14.

M. Severcan, Ph.D. thesis, Stanford University, University Microfilm, Ann Arbor, Order 74-13687 (1973).

The independence of FR(x) and FI(x) also allows the synthesis of complex-valued impulse responses. See Ref. 13 for a discussion of suitable techniques employing modified two-pupil incoherent spatial filtering operations.

The functions p1(x) and p2(x) can be adequately represented by samples taken at the sampling rate for Fs(x) if the resulting pupil functions, obtained by the discrete Fourier transform, are replicated a number of times in the pupil plane. This point is discussed in “Pupil Function Replication in Optical Transfer Function Synthesis,” by B. Braunecker, R. Hauck, and W. Rhodes, in preparation. The resultant PSFs will not generally be the minimum-bias PSFs, however.

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Figures (5)

Fig. 1
Fig. 1

Single-pupil incoherent spatial filtering system.

Fig. 2
Fig. 2

Example of two-pupil spatial filtering system (Mach-Zehnder configuration). Pupils p 1 and p 2 are effectively superposed. Attenuators A1 and A2 and phase shifter Φ are adjustable. Second lens (dotted) at output used for simultaneous subtraction processing (Sec. VI).

Fig. 3
Fig. 3

Synthesis of impusle responses Fs(x) = 2 sinc2x cos2π2x and Fs(x) = 2 sinc2x sin2πx using, respectively, noninteractive and interactive syntheses: (a) two pupil functions; (b) corresponding auto correlations and cross-correlations; (c) synthesized OTF and impulse response in noninteraction regime; (d) same for interaction regime.

Fig. 4
Fig. 4

Minimum-bias synthesis in interaction regime: (a) desired synthesized impulse response; (b), (c), and (d) presynthesis impulse response for various values of adjustable phase constants ϕ.

Fig. 5
Fig. 5

Presynthesis OTFs obtained with carrier methods of synthesis: (a) spatial carrier method; (b) temporal carrier method.

Tables (2)

Tables Icon

Table I Pupil Specification Flexibility

Tables Icon

Table II Two-Pupil Synthesis Implementation

Equations (60)

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I ( x ) = O ( x ) * F ( x ) ,
F ( x ) = | p ( x ) | 2 ,
Ĩ ( u ) = Õ ( u ) F ( u ) ,
F ( u ) = p ( ξ + u ) p * ( ξ ) d ξ p ( u ) p ( u ) ,
| p ( ξ ) | 2 d ξ = F ( O ) = 1 ,
p ( u ) = A 1 p 1 ( u ) exp ( i ϕ ) + A 2 p 2 ( u ) ,
F ( u ) = A 1 2 [ p 1 ( u ) p 1 ( u ) ] + a 2 2 [ p 2 ( u ) p 2 ( u ) ] + A 1 A 2 { p 1 ( u ) p 2 ( u ) exp ( i ϕ ) + [ p 2 ( u ) p 1 ( u ) ] exp ( i ϕ ) } .
F ( x ) = A 1 2 | p 1 ( x ) | 2 + A 2 2 | p 2 ( x ) | 2 + A 1 A 2 [ p 1 ( x ) p 2 * ( x ) exp ( i ϕ ) + p 1 * ( x ) p 2 ( x ) exp ( i ϕ ) ] .
F 1 ( x ) = F ( x ; 1 , 0 , 0 ) = | p 1 ( x ) | 2 ,
F 2 ( x ) = F ( x ; 0 , 1 , 0 ) = | p 2 ( x ) | 2 ,
F R ( x ) = 1 4 [ F ( x ; 1 , 1 , 0 ) F ( x ; 1 , 1 , π ) ] , = 1 2 [ p 1 ( x ) p 2 * ( x ) + p 1 * ( x ) p 2 ( x ) ] , = Re [ p 1 ( x ) p 2 * ( x ) ] = | p 1 ( x ) p 2 ( x ) | cos [ θ 1 ( x ) θ 2 ( x ) ] ,
F I ( x ) = 1 4 { F [ x ; 1 , 1 , ( π / 2 ) ] F [ x ; 1 , 1 , ( π / 2 ) ] } , = ( i / 2 ) [ p 1 ( x ) p 2 * ( x ) p 1 * ( x ) p 2 ( x ) ] , = Im [ p 1 ( x ) p 2 * ( x ) ] = | p 1 ( x ) p 2 ( x ) | sin [ θ 1 ( x ) θ 2 ( x ) ] ,
F 1 ( u ) = p 1 ( u ) p 1 ( u ) ,
F 2 ( u ) = p 2 ( u ) p 2 ( u ) ,
F R ( u ) = 1 2 [ p 1 ( u ) p 2 ( u ) + p 2 ( u ) p 1 ( u ) ] ,
F I ( u ) = ( i / 2 ) [ p 1 ( u ) p 2 ( u ) p 2 ( u ) p 1 ( u ) ] .
F s ( x ) = α F R ( x ) + β F I ( x ) ;
F s ( x ) = γ 1 F 1 ( x ) + γ 2 F 2 ( x ) ;
F s ( x ) = α F R ( x ) + β F I ( x ) + γ 1 F 1 ( x ) + γ 2 F 2 ( x ) .
F 1 ( x ) = | p 1 ( x ) | 2 is symmetric , real , nonnegative ,
F 2 ( x ) = | p 2 ( x ) | 2 is symmetric , real , nonnegative ,
F R ( x ) = Re [ p 1 ( x ) p 2 * ( x ) ] is symmetric , real ,
F I ( x ) = Im [ p 1 ( x ) p 2 * ( x ) ] is antisymmetric , 20 real .
F s ( x ) = f ( x Δ , y ) + f ( x Δ , y ) ,
p 1 ( u ) = p 0 ( u ) ,
p 2 ( u ) = p 0 * ( u ) .
p 1 ( x ) = p 0 ( x ) ,
p 2 ( x ) = p 0 * ( x ) ,
F R ( x ) = Re [ p 0 2 ( x ) ] ,
F I ( x ) = Im [ p 0 2 ( x ) ] 0 .
p 1 ( u ) = rect ( u Δ u 1 ) exp [ i ϕ 1 ( x ) ] ,
p 2 ( u ) = rect ( u Δ u 2 ) exp [ i ϕ 2 ( x ) ] .
I s ( x ) = I + ( x ) I ( x ) ,
I + ( x ) = O ( x ) * F 1 ( x ) ,
I ( x ) = O ( x ) * F 2 ( x ) .
E n I + ( x ) dx + I ( x ) dx .
E n Õ ( 0 ) [ F 1 ( 0 ) + F 2 ( 0 ) ] .
( a ) F 1 ( x ) F 2 ( x ) = F s ( x ) , ( b ) F 1 ( x ) 0 , F 2 ( x ) 0 , ( c ) [ F 1 ( 0 ) + F 2 ( 0 ) ] is a minimum , consistent with ( a ) and ( b ) .
F 1 ( x ) = 1 2 [ F s ( x ) + | F s ( x ) | ] = F s ( x ) H [ F s ( x ) ] ,
F 2 ( x ) = 1 2 [ F s ( x ) + | F s ( x ) | ] = F s ( x ) H [ F s ( x ) ] ,
m ( x ) = | F 1 ( x ) F 2 ( x ) | F 1 ( x ) + F 2 ( x ) .
F ( x ) = | p 0 ( x ) | 2 + | p 0 ( x ) | 2 + 2 | p 0 ( x ) | 2 cos [ ϕ + 2 θ 0 ( x ) ] .
F 1 ( x ) = 2 | p 0 ( x ) | 2 [ 1 + cos 2 θ 0 ( x ) ] ,
F 2 ( x ) = 2 | p 0 ( x ) | 2 [ 1 cos 2 θ 0 ( x ) ] ,
F s ( x ) = F 1 ( x ) F 2 ( x ) = 4 | p 0 ( x ) | 2 cos 2 θ 0 ( x ) = 4 F R ( x ) .
m ( x ) = 4 | p 0 ( x ) | 2 | cos 2 θ 0 ( x ) | 4 | p 0 ( x ) | 2 = | cos 2 θ 0 ( x ) | .
p a ( u ) = p 1 ( u ) exp ( i ϕ 0 ) + p 2 ( u ) ,
p b ( u ) = p 1 ( u ) exp [ i ( ϕ 0 + π ) ] + p 2 ( u ) ,
F s ( x ) = 4 cos ϕ 0 | p 1 ( x ) p 2 ( x ) | cos [ θ 1 ( x ) θ 2 ( x ) ] 4 sin ϕ 0 | p 1 ( x ) p 2 ( x ) | sin [ θ 1 ( x ) θ 2 ( x ) ] = α F R ( x ) + β F I ( x ) ,
F sc ( x ) = | p 1 ( x ) | 2 + | p 2 ( x ) | 2 + 2 | p 1 ( x ) p 2 ( x ) | cos [ ω x x θ 1 ( x ) θ 2 ( x ) ] ,
F tc ( x , t ) = | p 1 ( x ) | 2 + | p 2 ( x ) | 2 + 2 | p 1 ( x ) p 2 ( x ) | cos [ ω t t + θ 1 ( x ) θ 2 ( x ) ] ,
p k ( u ) = s k ( u ) + ã k ( u ) ,
s k ( u ) | a k ( u ) | 0 .
p k ( x ) = s k ( x ) + ia k ( x ) ,
s k ( u ) s k ( x ) ,
ã k ( u ) ia k ( x ) .
F 1 ( x ) = s 1 2 ( x ) + a 1 2 ( x ) ,
F 2 ( x ) = s 2 2 ( x ) + a 2 2 ( x ) ,
F R ( x ) = s 1 ( x ) s 2 ( x ) + a 1 ( x ) a 2 ( x ) ,
F I ( x ) = a 1 ( x ) s 2 ( x ) a 2 ( x ) s 1 ( x ) .

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