Abstract

The class of second-order curvilinear gratings consists of all the curvilinear gratings that are obtained by second-order spatial transformations of periodic gratings. It includes, for example, circular, elliptic, and hyperbolic gratings as well as circular, elliptic, and hyperbolic zone plates. Such structures occur quite frequently in optics, and their Fourier transforms may arise, for instance, in connection with the Fraunhofer diffraction patterns generated by these structures. I present the two-dimensional Fourier spectra of the most important second-order curvilinear gratings for gratings having any desired intensity profile (cosinusoidal, sawtooth wave, square wave, etc.). These analytic results are also illustrated by figures showing the various gratings and their spectra as they are obtained on a computer by two-dimensional fast Fourier transform.

© 1998 Optical Society of America

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  1. I. Amidror, “Fourier spectrum of radially periodic images,” J. Opt. Soc. Am. A 14, 816–826 (1997).
    [CrossRef]
  2. A. V. Baez, “Fresnel zone plate for optical image formation using extreme ultraviolet and soft x radiation,” J. Opt. Soc. Am. 51, 405–412 (1961).
    [CrossRef]
  3. T. R. Welberry, R. P. Williams, “On certain non-circular zone plates,” Opt. Acta 23, 237–244 (1976).
    [CrossRef]
  4. N. Abramson, “The holo-diagram. IV: a practical device for simulating fringe patterns in hologram interferometry,” Appl. Opt. 10, 2155–2161 (1971).
    [CrossRef] [PubMed]
  5. J. Feldman, “Diffraction gratings,” U.S. patent3,628,849 (1971); S. H. Macomber, “Curved grating surface emitting distributed feedback semiconductor laser,” U.S. patent5,345,466 (1994); T. Shiono, “Optical disk head including a light path having a thickness and width greater than the light beam wavelength by a predetermined, amount,” U.S. patent5,317,551 (1994).
  6. R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y.1986), p. 241. On alternative conventions used in literature and the relationships between them, see also pp. 7 and 17.
  7. Note that the term “curvature” is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.
  8. A. B. Ivanov, “Surface of the second order,” in Encyclopaedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1993), Vol. 9, p. 82.
  9. I. N. Bronshtein, K. A. Semendyayev, Handbook of Mathematics (Springer-Verlag, Berlin, 1997), pp. 209–214.
  10. O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
    [CrossRef]
  11. H. H. M. Chau, “Zone plates produced optically,” Appl. Opt. 8, 1209–1211 (1969).
    [CrossRef] [PubMed]
  12. A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 24.
  13. R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), p. 166.
  14. A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 23.
  15. R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y., 1986), p. 244.
  16. The hyperbolic zone grating is also known as Girard grille; see, for example, S. Ananda Rao, “On the holographic simulation of a Girard grille,” reprinted in Selected Papers on Zone Plates, SPIE Milestone Series, Vol. MS 128 (SPIE; Bellingham, Wash., 1996), pp. 386–387.
  17. F. Oberhettinger, Tables of Bessel Transforms (Springer-Verlag, Berlin, 1972), p. 12. Note that in Eq. 2.57 y1/2 is erroneously missing in the right-hand side of the transform.
  18. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), pp. 136–139.
  19. A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 27.
  20. R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 130–131.
  21. A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, p. 36.
  22. A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, p. 39.
  23. F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions (Springer-Verlag, Berlin, 1990), p. 27.
  24. See, for example, in J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 308–309; or in R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 159–161.
  25. A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]

1997 (1)

1976 (1)

T. R. Welberry, R. P. Williams, “On certain non-circular zone plates,” Opt. Acta 23, 237–244 (1976).
[CrossRef]

1971 (1)

1969 (1)

1967 (1)

1961 (1)

1951 (1)

O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
[CrossRef]

Abramson, N.

Amidror, I.

Ananda Rao, S.

The hyperbolic zone grating is also known as Girard grille; see, for example, S. Ananda Rao, “On the holographic simulation of a Girard grille,” reprinted in Selected Papers on Zone Plates, SPIE Milestone Series, Vol. MS 128 (SPIE; Bellingham, Wash., 1996), pp. 386–387.

Baez, A. V.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y.1986), p. 241. On alternative conventions used in literature and the relationships between them, see also pp. 7 and 17.

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), p. 166.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y., 1986), p. 244.

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 130–131.

Bronshtein, I. N.

I. N. Bronshtein, K. A. Semendyayev, Handbook of Mathematics (Springer-Verlag, Berlin, 1997), pp. 209–214.

Chau, H. H. M.

Courant, R.

Note that the term “curvature” is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.

Feldman, J.

J. Feldman, “Diffraction gratings,” U.S. patent3,628,849 (1971); S. H. Macomber, “Curved grating surface emitting distributed feedback semiconductor laser,” U.S. patent5,345,466 (1994); T. Shiono, “Optical disk head including a light path having a thickness and width greater than the light beam wavelength by a predetermined, amount,” U.S. patent5,317,551 (1994).

Gaskill, J. D.

See, for example, in J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 308–309; or in R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 159–161.

Ivanov, A. B.

A. B. Ivanov, “Surface of the second order,” in Encyclopaedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1993), Vol. 9, p. 82.

Lohmann, A. W.

Myers, O. E.

O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
[CrossRef]

Oberhettinger, F.

F. Oberhettinger, Tables of Bessel Transforms (Springer-Verlag, Berlin, 1972), p. 12. Note that in Eq. 2.57 y1/2 is erroneously missing in the right-hand side of the transform.

F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions (Springer-Verlag, Berlin, 1990), p. 27.

Paris, D. P.

Semendyayev, K. A.

I. N. Bronshtein, K. A. Semendyayev, Handbook of Mathematics (Springer-Verlag, Berlin, 1997), pp. 209–214.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), pp. 136–139.

Welberry, T. R.

T. R. Welberry, R. P. Williams, “On certain non-circular zone plates,” Opt. Acta 23, 237–244 (1976).
[CrossRef]

Williams, R. P.

T. R. Welberry, R. P. Williams, “On certain non-circular zone plates,” Opt. Acta 23, 237–244 (1976).
[CrossRef]

Am. J. Phys. (1)

O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

T. R. Welberry, R. P. Williams, “On certain non-circular zone plates,” Opt. Acta 23, 237–244 (1976).
[CrossRef]

Other (18)

J. Feldman, “Diffraction gratings,” U.S. patent3,628,849 (1971); S. H. Macomber, “Curved grating surface emitting distributed feedback semiconductor laser,” U.S. patent5,345,466 (1994); T. Shiono, “Optical disk head including a light path having a thickness and width greater than the light beam wavelength by a predetermined, amount,” U.S. patent5,317,551 (1994).

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y.1986), p. 241. On alternative conventions used in literature and the relationships between them, see also pp. 7 and 17.

Note that the term “curvature” is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.

A. B. Ivanov, “Surface of the second order,” in Encyclopaedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1993), Vol. 9, p. 82.

I. N. Bronshtein, K. A. Semendyayev, Handbook of Mathematics (Springer-Verlag, Berlin, 1997), pp. 209–214.

A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 24.

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), p. 166.

A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 23.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y., 1986), p. 244.

The hyperbolic zone grating is also known as Girard grille; see, for example, S. Ananda Rao, “On the holographic simulation of a Girard grille,” reprinted in Selected Papers on Zone Plates, SPIE Milestone Series, Vol. MS 128 (SPIE; Bellingham, Wash., 1996), pp. 386–387.

F. Oberhettinger, Tables of Bessel Transforms (Springer-Verlag, Berlin, 1972), p. 12. Note that in Eq. 2.57 y1/2 is erroneously missing in the right-hand side of the transform.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), pp. 136–139.

A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 27.

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 130–131.

A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, p. 36.

A. Erdélyi, (ed.), Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, p. 39.

F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions (Springer-Verlag, Berlin, 1990), p. 27.

See, for example, in J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 308–309; or in R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 159–161.

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Figures (4)

Fig. 1
Fig. 1

Various curvilinear gratings r(x, y) with a periodic-profile waveform of cos(2πfx) (with f=1), and their spectra R(u, v) as obtained on computer by 2D discrete Fourier transform (DFT): (a) straight grating, cos(2πfx); (b) circular zone grating, cos[2πf(x2+y2)/8]; (c) elliptic zone grating, cos[2πf(14x2+y2)/8]; (d) hyperbolic zone grating, cos[2πf(x2-y2)/8]; (e) linear zone grating, cos(2πfx2/8); (f) sphere projection grating, cos(2πf64-x2-y2); (g) ellipsoid projection grating, cos(2πf64-x2-2y2); (h) cylinder projection grating, cos(2πf36-x2); (i) circular grating, cos(2πfx2+y2); (j) elliptic grating, cos(2πf14x2+y2); (k) hyperbolic grating, cos(2πfx2-y2); (l) one-sheet hyperboloid projection grating, cos(2πfx2+y2-4); (m) two-sheet hyperboloid projection grating, cos(2πfx2+y2+16); (n) laterally opened hyperbolic cylinder projection grating, cos(2πfx2-1); (o) top-opened hyperbolic cylinder projection grating, cos(2πfx2+16). Since all these cases are centrosymmetric, their spectra have no imaginary parts. The right-hand figure in each row shows a cross section through the horizontal u axis of the spectrum. Note that the oscillations in the spectra of the zone gratings (b)–(e) fade out short of the spectrum border since the DFT cannot find higher frequencies in the corresponding finite-sized, sampled functions in the image domain; in reality these spectra oscillate ad infinitum without fading out. Note also the DFT rippling artifacts in some of the spectra.

Fig. 2
Fig. 2

Rc(u) and Rs(u), the 1D Fourier transforms of the functions rc(x)=cos(2πfx2) and rs(x)=sin(2πfx2) (with f=1).

Fig. 3
Fig. 3

Some examples of curvilinear gratings r(x, y) with a square-wave periodic profile (with opening ratio τ/T=0.6), and their respective spectra R(u, v): (a) parabolic grating, g1(x, y)=y-0.15x2; (b) circular grating, g1(x, y)=x2+y2; (c) circular zone grating, g1(x, y)=(x2+y2)/8. The amplitudes of the different harmonics in the spectra are weighted by the Fourier series coefficients an of the square wave [see Eq. (12)]: a1=0.303, a2=-0.094, a3=-0.062, etc; the sign inversions in the second and third harmonics are clearly visible in the spectra of cases (a) and (b). Notice the various DFT artifacts in the spectra (folding over due to aliasing; rippling).

Fig. 4
Fig. 4

Some further curvilinear gratings r(x, y) with a periodic-profile waveform of cos(2πfx) (with f=1), and their spectra R(u, v) as obtained on computer by 2D DFT: (a) equispaced parabolic grating, cos[2πf(0.15x2-y)]; (b) equispaced circle-arc grating, cos[2πf(36-x2+y)]; (c) equispaced hyperbola-arc grating, cos[2πf(x2+16-y)]; (d) eccentric equispaced parabolic grating, cos[2πf(x2+y2-y)]; (e) eccentric equispaced elliptic grating, cos[2πf(x2+y2-y/2)]; (f) eccentric equispaced hyperbolic grating, cos[2πf(x2+y2-1.5y)]; (g) eccentric ellipsoid projection grating, cos[2πf(64-x2-2y2+x)]. In each case the central column shows the real part of the spectrum, and the right-hand column shows the imaginary part of the spectrum. Note the DFT rippling artifacts in some of the spectra.

Equations (97)

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F(u, ν)=--f(x, y)exp[-i2π(ux+νy)]dxdy.
a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+a34z+a44=0.
r1(x, y)=cos[2πf(x2+y2)].
r1(x, y)=cos(2πfx2)cos(2πfy2)-sin(2πfx2)sin(2πfy2).
Rc(u)=12fcosπ2fu2+sinπ2fu2=12 fcosπ2fu2-π4
cos(2πfax2)cos(2πfby2)12fcosπ2fu2+sinπ2fu2×12fcosπ2fv2+sinπ2fv2,
=14fcos π2f(u2-v2)+sin π2f(u2+v2).
sin(2πfax2)sin(2πfby2)14fcos π2f(u2-v2)-sin π2f(u2+v2);
R1(u, v)=12fsinπ2f(u2+v2).
Rs(u)=12fcosπ2fu2-sinπ2fu2=12 fsinπ2fu2+π4
r1(x, y)=cos{2πf[(bx)2+(cy)2]}.
R1(u, v)=12f|bc|sinπ2fub2+vc2.
r1(x, y)=cos{2πf[(bx)2-(cy)2]}
R1(u, v)=12f|bc|cosπ2fub2-vc2.
r1(x, y)=sin(2πfa2-r2)0r<a0ar<,
r2(x, y)=cos(2πfa2-r2)0r<a0ar<,
R1(u, v)=πfa3/2 1(f2+q2)3/4J3/2(2πaf2+q2),
R2(u, v)=-πfa3/2 1(f2+q2)3/4Y3/2(2πaf2+q2),
R3(u, v)=1|bc|R1ub, vc=1|bc|πfa3/21(f2+q2)3/4J3/2(2πaf2+q2),
R4(u, v)=1|bc|R2ub, vc=-1|bc|πfa3/21(f2+q2)3/4Y3/2(2πaf2+q2),
r1(x, y)=sin(2πf a2-x2)0|x|<a0a|x|<,
r2(x, y)=cos(2πf a2-x2)0|x|<a0a|x|<,
R1(u, v)=πfa 1f2+u2J1(2πaf2+u2)δ(v).
R2(u, v)=-πfa 1f2+u2Y1(2πaf2+u2)δ(v).
R1(u, v)=-f2π1(f2-q2)3/20q<f0f<q<,
R1(u, v)=fπ1(f+q)3/2δ(1/2)(f-q).
R2(u, v)=00q<f-f2π1(q2-f2)3/2f<q<,
R2(u, v)=fπ1(f+q)3/2δ(1/2)(q-f).
r3(x, y)=r1(bx, cy)=cos[2πf(bx)2+(cy)2],
r4(x, y)=r2(bx, cy)=sin[2πf(bx)2+(cy)2].
R3(u, v)=1|bc|R1ub, vc=-1|bc|f2π1(f2-q2)3/20q<f0f<q<,
R4(u, v)=1|bc|R2ub, vc=00q<f-1|bc|f2π1(q2-f2)3/2f<q<,
R3(u, v)=fπ1(f+q)3/2δ(1/2)(f-q),
R4(u, v)=fπ1(f+q)3/2δ(1/2)(q-f).
r3(x, y)=cos[2πf(bx)2-(cy)2](bx)2-(cy)2>00(bx)2-(cy)20,
r4(x, y)=sin[2πf(bx)2-(cy)2](bx)2-(cy)2>00(bx)2-(cy)20.
R3(u, v)=1|bc|R1ub, vc=-1|bc|f2π1(f2-q2)3/20q<f0f<q<,
R4(u, v)=1|bc|R2ub, vc=00q<f-1|bc|f2π1(q2-f2)3/2f<q<,
R3(u, v)=fπ1(f+q)3/2δ(1/2)(f-q),
R4(u, v)=fπ1(f+q)3/2δ(1/2)(q-f).
r1(x, y)=00r<asin(2πfr2-a2)ar<,
r2(x, y)=00r<acos(2πfr2-a2)ar<,
R1(u, v)=0πfa3/2 1(q2-f2)3/4J-3/2(2πaq2-f2),
0q<f,f<q<,
R2(u, v)=-2 fa3/2 1(f2-q2)3/4K-3/2(2πaf2-q2)πfa3/2 1(q2-f2)3/4Y-3/2(2πaq2-f2),
0q<f,f<q<,
R1(u, v)=-πfa3/2 1(f2-q2)3/4J3/2(2πaf2-q2)-2 fa3/2 1(q2-f2)3/4K3/2(2πaq2-f2),
0f,f<q<,
R2(u, v)=πfa3/2 1(f2-q2)3/4Y3/2(2πaf2-q2)0,
0q<f,f<q<.
r1(x, y)=00|x|<asin(2πfx2-a2)a|x|<,
r2(x, y)=00|x|<acos(2πfx2-a2)a|x|<
R1(u, v)=2fa 1f2-u2K-1(2πaf2-u2)δ(v)-πfa 1u2-f2Y-1(2πau2-f2)δ(v),
0u<f,f<u<.
R2(u, v)=0πfa 1u2-f2J-1(2πau2-f2)δ(v),
0u<f,f<u<.
R1(u, v)=-πfa 1f2-u2Y1(2πaf2-u2)δ(v)-2fa 1u2-f2K1(2πau2-f2)δ(v),
0u<f,f<u<.
R2(u,v)=-πfa1f2-u2J1(2πaf2-u2)δ(v)0,
0u<f,f<u<.
p(x)=n=-cn exp(i2πnfx),
r(x, y)=p[g1(x, y)]=n=-cn exp[i2πnfg1(x, y)].
p(x)=a0+2n=1an cos(2πnx/T)+2n=1bn sin(2πnx/T),
p(x)=a0+2n=1an cos(2πnx/T),
r(x, y)=p[g1(x, y)]=a0+2n=1an cos(2πng1(x, y)/T)
cos(2πfx)12δ(u-f)+12δ(u+f)
p(x)=n=-an cos(2πnx/T)P(u)=n=-anδ(u-n/T).
r(x, y)=n=-an cos(2πng1(x, y)/T)R(u, v)=n=-anRn(u, v).
r(x, y)=n=-cn exp[i2πnfg1(x, y)]R(u, v)=n=-cnRn(u, v),
p(x)=n=-an cos(2πnx/T),
an=(τ/T)sinc(nτ/T).
r(x, y)=p(y-ax2)=n=-an cos(2πn(y-ax2)/T).
R(u, v)=n=-anRn(u, v),
Rn(u, v)=12[Rc(u)+iRs(u)]δ(v-nf)+12[Rc(u)-iRs(u)]δ(v+nf),
Rc(u)=12nfacosπ2nfau2+sinπ2nfau2,
Rs(u)=12nfacosπ2nfau2-sinπ2nfau2,
r(x, y)=p(x2+y2)=a0+2n=1an cos(2πnx2+y2/T)
R(u, v)=a0R0(u, v)+2n=1anRn(u, v),
r(x, y)=a0+2n=1an cos[2πn(x2+y2)/T],
R(u, v)=a0R0(u, v)+2n=1anRn(u, v),
r3(x, y)=cos{2πf[g1(x, y)+(bx+cy)]}=cos[2πfg1(x, y)]cos[2πf(bx+cy)]-sin[2πfg1(x, y)]sin[2πf(bx+cy)].
r1(x, y)cos[2πf(bx+cy)]12R1(u+bf, v+cf)+12R1(u-bf, v-cf),
r2(x, y)sin[2πf(bx+cy)]12iR2(u+bf, v+cf)-12iR2(u-bf, v-cf),
R3(u, v)
=12[R1(u+bf, v+cf)+R1(u-bf, v-cf)]
+12i[R2(u+bf, v+cf)-R2(u-bf, v-cf)]
R4(u, v)
=12[R2(u+bf, v+cf)+R2(u-bf, v-cf)]
+12i[R1(u+bf, v+cf)+R1(u-bf, v-cf)].
R5(u, v)=12[R1(u, v-f)+R1(u, v+f)]+12i[R2(u, v-f)-R2(u, v+f)],
=12[Rc(u)+iRs(u)]δ(v-f)+12[Rc(u)-iRs(u)]δ(v+f),
R6(u, v)=12[Rs(u)+iRc(u)]δ(v-f)+12[Rs(u)+iRc(u)]δ(v+f).
r5(x, y)=sin[2πf(a2-x2+y)]0|x|<a0a|x|<,
r6(x, y)=cos[2πf(a2-x2+y)]0|x|<a0a|x|<.
r3(x, y)=cos(2πf[g1(x, y)+c])=r1(x, y)cos(2πfc)-r2(x, y)sin(2πfc).
R3(u, v)=cos(2πfc)R1(u, v)-sin(2πfc)R2(u, v),
R4(u, v)=sin(2πfc)R1(u, v)+cos(2πfc)R2(u, v).

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