Abstract

In this study the general algorithm for the fractionalization of the linear cyclic integral transforms is established. It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform. The main properties of the fractional transforms used in optics are considered. As an example, two different types of fractional Hartley transform are introduced, and the experimental setups for their optical implementation are proposed.

© 2000 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  2. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.
  3. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
    [CrossRef]
  4. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  5. A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997).
    [CrossRef]
  6. L. Yu, Y. Lu, X. Zeng, M. Huang, M. Chen, W. Huang, Z. Zhu, “Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform,” Opt. Lett. 23, 1158–1160 (1998).
    [CrossRef]
  7. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
    [CrossRef]
  8. C. C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
    [CrossRef]
  9. S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
    [CrossRef]
  10. T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
    [CrossRef]
  11. D. A. Linden, “A discussion of sampling theorems,” Proc. IRE 47, 1219–1226 (1959).
    [CrossRef]
  12. R. P. Kanwal, Linear Integral Equations: Theory and Techniques (Academic, New York, 1971), Chaps. 8 and 9.
  13. T. Alieva, A. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
    [CrossRef]
  14. R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Streibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
    [CrossRef] [PubMed]

1999 (2)

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
[CrossRef]

1998 (1)

1997 (3)

S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
[CrossRef]

A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997).
[CrossRef]

T. Alieva, A. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

1996 (1)

1995 (1)

C. C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
[CrossRef]

1993 (1)

1985 (1)

1982 (1)

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
[CrossRef]

1959 (1)

D. A. Linden, “A discussion of sampling theorems,” Proc. IRE 47, 1219–1226 (1959).
[CrossRef]

Alieva, T.

T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
[CrossRef]

T. Alieva, A. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Barbe, A.

T. Alieva, A. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Bartelt, H.

Bastiaans, M. J.

Bracewell, R. N.

Chen, M.

Dickinson, B. W.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
[CrossRef]

Huang, M.

Huang, W.

Jiang, J.

S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
[CrossRef]

Kanwal, R. P.

R. P. Kanwal, Linear Integral Equations: Theory and Techniques (Academic, New York, 1971), Chaps. 8 and 9.

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

Linden, D. A.

D. A. Linden, “A discussion of sampling theorems,” Proc. IRE 47, 1219–1226 (1959).
[CrossRef]

Liu, S.

S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
[CrossRef]

Lohmann, A. W.

Lu, Y.

Mendlovic, D.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

Ozaktas, H. M.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

Ramirez, J. G.

Shih, C. C.

C. C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
[CrossRef]

Steiglitz, K.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
[CrossRef]

Streibl, N.

Tepichin, E.

Yu, L.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

Zeng, X.

Zhang, J.

S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
[CrossRef]

Zhang, Y.

S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
[CrossRef]

Zhu, Z.

Adv. Imaging Electron Phys. (1)

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Acoust. Speech Signal Process. (1)

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. A (2)

S. Liu, J. Jiang, Y. Zhang, J. Zhang, “Generalized fractional Fourier transforms,” J. Phys. A 30, 973–981 (1997).
[CrossRef]

T. Alieva, A. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Opt. Commun. (1)

C. C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
[CrossRef]

Opt. Lett. (2)

Proc. IRE (1)

D. A. Linden, “A discussion of sampling theorems,” Proc. IRE 47, 1219–1226 (1959).
[CrossRef]

Other (2)

R. P. Kanwal, Linear Integral Equations: Theory and Techniques (Academic, New York, 1971), Chaps. 8 and 9.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

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Figures (2)

Fig. 1
Fig. 1

Harmonic content for the different types of fractional Fourier transform, which correspond to the kernels defined by the following equations: (a) the optical fractional Fourier transform [Eq. (35)]; (b) Eq. (22) and Eq. (44) for N=4; (c) Eq. (21), where N=4 and φ0=0, φ1=4, φ2=0, φ3=-4; (d) Eq. (45) for N=4; (e) Eq. (47).

Fig. 2
Fig. 2

Setup for the optical realization of the fractional Hartley transform defined by Eq. (56) (see text for details). PP, phase plate; BS, beam splitter; AP, absorbing plate; M’s, mirrors.

Equations (72)

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R[f(x)](u)=K(x, u)f(x)dx.
RN=I.
K(x, u)=1iBexp[iπ(Ax2+Du2-2xu)/B],
Ψ(x)=1Nn=0N-1 exp(-i2πnL/N)Rn[g(u)](x).
R[Ψ(x)](u)=1Nn=0N-1 exp(-i2πnL/N)Rn+1[g(x)](u)=exp(i2πL/N)1Nn=0N-1 exp[-i2π(n+1)L/N]Rn+1[g(x)](u)=exp(i2πL/N)Ψ(u).
K(α, x, u)=n=-kn(x, u)expi2παnN,
K(0, x, u)=n=-kn(x, u),K(1, x, u)=n=-kn(x, u)expi2πnN,K(N-1, x, u)=n=-kn(x, u)expi2π(N-1)nN.
kn(x, u)km(u, y)du=δn,mkn(x, y),
K(α, x, u)K(β, u, y)du=K(α+β, x, y),
n=-m=-
×expi2π(αn+βm)Nkn(x, u)km(u, y)du
=l=-kl(x, y)expi2π(α+β)lN,
K(0, x, u)=n=-kn(x, u)=n=0N-1m=-kn+mN(x, u),K(1, x, u)=n=0N-1 expi2πnNm=-kn+mN(x, u),K(N-1, x, u)=n=0N-1 expi2π(N-1)nN×m=-kn+mN(x, u).
Cn(x, u)=m=-kn+mN(x, u),
B×C=K,
B=11111expi2πNexpi4πNexpi2π(N-1)N1expi2π(N-1)Nexpi4π(N-1)Nexpi2π(N-1)2N,
C=C0(x, u)C1(k, u)CN-1(x, u),K=K(0, x, u)K(1, x, u)K(N-1, x, u).
B-1=1N111111expi2π(N-1)Nexpi2π(N-2)Nexpi2πN1expi2π(N-1)2Nexpi2π(N-1)(N-2)Nexpi2π(N-1)N,
Cn(x, u)=1Nl=0N-1 exp-i2πlnNK(l, x, u).
Cn(x, u)Cm(u, y)du=δn,mCn(x, y).
C0(x, u)=12[K(0, x, u)+K(1, x, u)],
C1(x, u)=12[K(0, x, u)-K(1, x, u)],
C0(x, u)=14[K(0, x, u)+K(1, x, u)+K(2, x, u)+K(3, x, u)],
C1(x, u)=14[K(0, x, u)-iK(1, x, u)-K(2, x, u)+iK(3, x, u)],
C2(x, u)=14[K(0, x, u)-K(1, x, u)+K(2, x, u)-K(3, x, u)],
C3(x, u)=14[K(0, x, u)+iK(1, x, u)-K(2, x, u)-iK(3, x, u)].
K(α, x, u)=n=0N-1kn+φn(x, u)expi2πα(n+φn)N=1Nl=0N-1K(l, x, u)n=0N-1 exp-i2πlnN×expi2πα(n+ϕn)N.
K(α, x, u)=1Nl=0N-1 expiπ(N-1)(α-l)N×sin[π(α-l)]sin[π(α-l)/N]K(l, x, u),
K(α, x, u)=1Nl=0N-1sin[π(α-l)]sin[π(α-l)/N]K(l, x, u).
K(α, x, u)=14l=03K(l, x, u)n=03 exp-iπln2×expiπα(n+φn)2=14l=03K(l, x, u)S(l),
S(0)=expiπαφ02+expiπα(1+φ1)2+expiπα(2+φ2)2+expiπα(3+φ3)2,
S(1)=expiπαφ02-i expiπα(1+φ1)2-expiπα(2+φ2)2+i expiπα(3+φ3)2,
S(2)=expiπαφ02-expiπα(1+φ1)2+expiπα(2+φ2)2-expiπα(3+φ3)2,
S(3)=expiπαφ02+i expiπα(1+φ1)2-expiπα(2+φ2)2-i expiπα(3+φ3)2.
K(α, x, u)=12K(0, x, u)expiπα(3+φ3)2+expiπα(1+φ1)2-i2K(1, x, u)×expiπα(3+φ3)2-expiπα(1+φ1)2=exp[iπα(1+m1+m3)]×K(0, x, u)cosπα12+m3-m1-K(1, x, u)sinπα12+m3-m1,
K(α, x, u)=exp(iπα)K(0, x, u)cosπα2-K(1, x, u)sinπα2,
K(α, x, u)=K(0, x, u)cosπα2+K(1, x, u)sinπα2.
Φn(x)Φm*(x)dx=δn,m.
Φn(x)=(π2nn!)-1/2 exp(-12x2)Hn(x),
- f(x)H[f(u)](x)dx=0,
- Ψ2(x)dx=0.
K(p, x, u)=n=0Φn(x)(An)pΦn*(u)=n=0Φn(x)expi2πpLnNΦn*(u).
K(α, x, u)=n=0Φn(x)expi2π(Ln+lnN)NαΦn*(u),
KF(α, x, u)=n=0Φn(x)exp-iπnα2Φn*(u)=exp(iπα/4)[i2π sin(πα/2)]-1/2×expi(x2+u2)cos(πα/2)-2xu2 sin(πα/2).
K(α, x, u)=KF(α, x, u)exp(i2πlα).
K(α, x, u)=KF(α, x, u)+[exp(i2πα)-1].
K(α, x, u)=n=-zn(x, u)expi2πnNα.
K(α, x, u)=n=0 exp-iπn2αΦn(x)Φn*(u)=n=-0zn(x, u)expiπn2α,
K(α, x, u)=n=0 exp(-iπαn)z-n(x, u),
z-n(x, u)=Φ2n(x)Φ2n(u)+Φ2n+1(x)Φ2n+1(u).
K(α, x, u)=n=0M-1 expi2παnMm=-zn+Mm(x, u).
kn(M, x, u)=m=-zn+Mm(x, u),
K(α, x, u)=n=0M-1 exp-i2παn(1-M)M×m=0Φn+Mm(x)Φn+Mm*(u)=1Mn=0M-1 expiπ(M-1)(αl-n)M×sin[π(αl-n)]sin[π(αl-n)/M]KF(n/l, x, u),
K(α, x, u)=n=0N-1 exp-i2παn(1-N)N×m=0Φn+Nm(x)Φn+Nm*(u).
K(α, x, u)=m=0z-Nm(x, u)+n=1N-1m=0×exp-i2παnN+mz-n-Nm(x, u),
K(α, x, u)=n=04 exp-iπαn2k-n(x, u),
k0(x, u)=Φ0(x)Φ0*(u),
k-j(x, u)=m=0Φj+4m(x)Φj+4m*(u),j=1, 2, 3,
k-4(x, u)=m=1Φ4m(x)Φ4m*(u).
{RFα[f(x)](u)}*=RF-α[f(x)](u).
Ψ1/M(x)=1Mn=0M-1 exp(-i2πnL/M)Rn/M[g(u)](x).
ΨαL(x)=1N0N exp(-i2παL)Rα[g(u)](x)dα.
ΨαL(x)=aLm=0ΦL+mN(x),
Rα[ΨαL(u)](x)=exp(i2παL)ΨαL(x),
RH[f(x)](u)=f(x)cas(2πxu)dx,
RH[f(x)](u)=12{exp(iπ/4)RF[f(x)](u)+exp(-iπ/4)RF3[f(x)](u)}.
RH[Ψn(x)](u)
=Ψn(u)forn=4m,n=1+4m-Ψn(u)forn=2+4m,n=3+4m,
H(α, x, u)=n=0 exp(-iπαn)[Ψ2n(x)Ψ2n(u)+Ψ2n+1(x)Ψ2n+1(u)].
RHα=exp(iπα/4)[cos(πα/4)RFα-i sin(πα/4)RFα+2].
H(α, x, u)=expi2πα(n1+n2)×δ(x-u)cosπα2(n1-n2)-i cas(2πxu)sinπα2(n1-n2).
H(α, x, u)=expi2παδ(x-u)cosπα2-i cas(2πxu)sinπα2.

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