Abstract

The mode analysis of signals in a multimodal shallow harmonic waveguide whose eigenfrequencies are equally spaced and finite can be performed by an optoelectronic device, of which the optical part uses the guide to sample the wave field at a number of sensors along its axis and the electronic part computes their fast Fourier transform. We illustrate this process with the Kravchuk transform.

© 2000 Optical Society of America

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References

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  1. J. D. Secada, “Numerical evaluation of the Hankel transform,” Comput. Phys. Commun. 116, 278–294 (1999).
    [CrossRef]
  2. T. Alieva, M. J. Bastiaans, “Mode analysis through the fractional transforms in optics,” Opt. Lett. 24, 1206–1208 (1999).
    [CrossRef]
  3. B. Santhanam, J. H. McClellan, “The discrete rotational Fourier transform,” IEEE Trans. Signal Process. 44, 994–998 (1996).
    [CrossRef]
  4. S.-Ch. Pei, M.-H. Yeh, Ch.-Ch. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
    [CrossRef]
  5. H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Intro-duction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999);M. Alper Kutay, H. Özataş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999);Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
    [CrossRef]
  6. L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
    [CrossRef]
  7. N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
    [CrossRef]
  8. T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
    [CrossRef]
  9. M. Arik, N. M. Atakishlyev, K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371–L376 (1999).
    [CrossRef]
  10. N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Teor. Mat. Fiz. 85, 1055–1062 (1991).
    [CrossRef]
  11. A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).
  12. N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
    [CrossRef]
  13. N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).
  14. T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
    [CrossRef]
  15. L. C. Biedenharn, J. D. Louck, “Angular momentum in quantum physics,” Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, Mass., 1981), Vol. 8.
  16. P. Feinsilver, R. Schott, “Operator calculus approach to orthogonal polynomial expansions,” J. Comput. Appl. Math. 66, 185–199 (1996).
    [CrossRef]

2000 (2)

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

1999 (6)

J. D. Secada, “Numerical evaluation of the Hankel transform,” Comput. Phys. Commun. 116, 278–294 (1999).
[CrossRef]

T. Alieva, M. J. Bastiaans, “Mode analysis through the fractional transforms in optics,” Opt. Lett. 24, 1206–1208 (1999).
[CrossRef]

S.-Ch. Pei, M.-H. Yeh, Ch.-Ch. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Intro-duction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999);M. Alper Kutay, H. Özataş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999);Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
[CrossRef]

M. Arik, N. M. Atakishlyev, K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371–L376 (1999).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

1997 (2)

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

N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

1996 (2)

P. Feinsilver, R. Schott, “Operator calculus approach to orthogonal polynomial expansions,” J. Comput. Appl. Math. 66, 185–199 (1996).
[CrossRef]

B. Santhanam, J. H. McClellan, “The discrete rotational Fourier transform,” IEEE Trans. Signal Process. 44, 994–998 (1996).
[CrossRef]

1994 (1)

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).

1991 (1)

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Teor. Mat. Fiz. 85, 1055–1062 (1991).
[CrossRef]

Alieva, T.

T. Alieva, M. J. Bastiaans, “Mode analysis through the fractional transforms in optics,” Opt. Lett. 24, 1206–1208 (1999).
[CrossRef]

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

Alper Kutay, M.

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Intro-duction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999);M. Alper Kutay, H. Özataş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999);Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
[CrossRef]

Arik, M.

M. Arik, N. M. Atakishlyev, K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371–L376 (1999).
[CrossRef]

Atakishiyev, N. M.

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Teor. Mat. Fiz. 85, 1055–1062 (1991).
[CrossRef]

Atakishlyev, N. M.

M. Arik, N. M. Atakishlyev, K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371–L376 (1999).
[CrossRef]

Barbé, A.

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

Barker, L.

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Bastiaans, M. J.

Biedenharn, L. C.

L. C. Biedenharn, J. D. Louck, “Angular momentum in quantum physics,” Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, Mass., 1981), Vol. 8.

Candan, Ç.

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Feinsilver, P.

P. Feinsilver, R. Schott, “Operator calculus approach to orthogonal polynomial expansions,” J. Comput. Appl. Math. 66, 185–199 (1996).
[CrossRef]

Hakioglu, T.

T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Louck, J. D.

L. C. Biedenharn, J. D. Louck, “Angular momentum in quantum physics,” Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, Mass., 1981), Vol. 8.

McClellan, J. H.

B. Santhanam, J. H. McClellan, “The discrete rotational Fourier transform,” IEEE Trans. Signal Process. 44, 994–998 (1996).
[CrossRef]

Mendlovic, D.

H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Intro-duction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999);M. Alper Kutay, H. Özataş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999);Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
[CrossRef]

Nikiforov, A. F.

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

Ozaktas, H. M.

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Intro-duction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999);M. Alper Kutay, H. Özataş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999);Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
[CrossRef]

Pei, S.-Ch.

S.-Ch. Pei, M.-H. Yeh, Ch.-Ch. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Santhanam, B.

B. Santhanam, J. H. McClellan, “The discrete rotational Fourier transform,” IEEE Trans. Signal Process. 44, 994–998 (1996).
[CrossRef]

Schott, R.

P. Feinsilver, R. Schott, “Operator calculus approach to orthogonal polynomial expansions,” J. Comput. Appl. Math. 66, 185–199 (1996).
[CrossRef]

Secada, J. D.

J. D. Secada, “Numerical evaluation of the Hankel transform,” Comput. Phys. Commun. 116, 278–294 (1999).
[CrossRef]

Suslov, S. K.

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Teor. Mat. Fiz. 85, 1055–1062 (1991).
[CrossRef]

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

Tseng, Ch.-Ch.

S.-Ch. Pei, M.-H. Yeh, Ch.-Ch. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Uvarov, V. B.

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

Vicent, L. E.

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

Wolf, K. B.

T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

M. Arik, N. M. Atakishlyev, K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371–L376 (1999).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).

Yeh, M.-H.

S.-Ch. Pei, M.-H. Yeh, Ch.-Ch. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Adv. Imaging Electron Phys. (1)

H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Intro-duction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999);M. Alper Kutay, H. Özataş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999);Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
[CrossRef]

Comput. Phys. Commun. (1)

J. D. Secada, “Numerical evaluation of the Hankel transform,” Comput. Phys. Commun. 116, 278–294 (1999).
[CrossRef]

IEEE Trans. Signal Process. (2)

B. Santhanam, J. H. McClellan, “The discrete rotational Fourier transform,” IEEE Trans. Signal Process. 44, 994–998 (1996).
[CrossRef]

S.-Ch. Pei, M.-H. Yeh, Ch.-Ch. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

J. Comput. Appl. Math. (2)

P. Feinsilver, R. Schott, “Operator calculus approach to orthogonal polynomial expansions,” J. Comput. Appl. Math. 66, 185–199 (1996).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

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

J. Phys. A (4)

L. Barker, Ç. Candan, T. Hakioğlu, M. Alper Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

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

M. Arik, N. M. Atakishlyev, K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371–L376 (1999).
[CrossRef]

Opt. Lett. (1)

Rev. Mex. Fis. (1)

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).

Teor. Mat. Fiz. (1)

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Teor. Mat. Fiz. 85, 1055–1062 (1991).
[CrossRef]

Other (2)

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

L. C. Biedenharn, J. D. Louck, “Angular momentum in quantum physics,” Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, Mass., 1981), Vol. 8.

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Equations (14)

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

ϕmϕm=δm,m,n=-N/2N/2ϕm(n)*ϕm(n)=δm,m,
m=0Nϕmϕm=1,m=0Nϕm(n)*ϕm(n)=δn,n,
Hϕm=(m+κ)ϕmϕm(z)=exp(-izH)ϕm=exp(-izκ)exp(-izm)ϕm,
fw=Wf,fmw=n=N/2N/2ϕm(n)*f(n),
f=Wfw,f(n)=m=0Nϕm(n)fmw.
f(z)=Φzf,
f(n, z)=exp(-izκ)m=0Nexp(-imz)ϕm(n)fmw=exp(-izκ)n=-N/2N/2Φz(n, n)f(n),
Φz=Φz(n, n),
Φz(n, n)=m=0Nϕm(n) exp(-imz)ϕm(n)*.
zk=2πk/(N+1),k=1, 2 ,, N+1.
fmw=exp(izkκ)ϕm(n)k=0Nexp[2πimk/(N+1)]f(n, zk).
μ=-lldμ,μl(θ)*dμ,μl(θ)=dμ,μl(0)=δμ,μ.
ϕm(n)=(-1)l-mdn,l-ml12 π=12l-m2ll+n/2lm1/2km(l+n, 2l),
km(x, 2l)=(-1)m2m2lm2F1(-m, -x; -2l; 2),

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