Abstract

We apply the tools of the fractional calculus to introduce new fractional-order beam solutions to the paraxial wave equation that can be regarded as intermediate solutions between the known integral-order solutions. We restrict our attention to the fractionalization of the elegant and standard Hermite–Gaussian beams.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2004

2003

1998

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Prog. Opt. 38, 265 (1998).

1997

N. Engheta, IEEE Antennas Propag. Mag. 39, 35 (1997).
[CrossRef]

1996

N. Engheta, IEEE Trans. Antennas Propag. 44, 554 (1996).
[CrossRef]

1977

1973

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), Chap. 13.

Bandres, M. A.

Enderlein, J.

Engheta, N.

N. Engheta, IEEE Antennas Propag. Mag. 39, 35 (1997).
[CrossRef]

N. Engheta, IEEE Trans. Antennas Propag. 44, 554 (1996).
[CrossRef]

Felsen, L. B.

Gutiérrez-Vega, J. C.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Prog. Opt. 38, 265 (1998).

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Prog. Opt. 38, 265 (1998).

Oldham, K.

K. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Pampaloni, F.

Podlubny, I.

I. Podlubny, Fractional Differential Equations (Academic, 1999).

Seshadri, S. R.

Shin, S. Y.

Siegman, A. E.

A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986).

Spanier, J.

K. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), Chap. 13.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Prog. Opt. 38, 265 (1998).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

IEEE Antennas Propag. Mag.

N. Engheta, IEEE Antennas Propag. Mag. 39, 35 (1997).
[CrossRef]

IEEE Trans. Antennas Propag.

N. Engheta, IEEE Trans. Antennas Propag. 44, 554 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Prog. Opt.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Prog. Opt. 38, 265 (1998).

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), Chap. 13.

A. E. Siegman, Lasers (University Science, 1986).

I. Podlubny, Fractional Differential Equations (Academic, 1999).

K. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

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Figures (2)

Fig. 1
Fig. 1

Behavior of the fractional derivative of the Gaussian function on the ( x , α ) plane.

Fig. 2
Fig. 2

Amplitude and phase of the beam U α x , α y ( r ) at z = { 0 , 0.5 z R , z R } for α x = 2.5 , α y = 1.6 , and w 0 x = w 0 y . Square image size of 6 w 0 × 6 w 0 .

Equations (10)

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u 0 ( x , z ) = ( 2 π ) 1 4 μ w 0 exp ( x 2 μ w 0 2 ) ,
u m = m u 0 x m 1 μ m 2 H m ( x μ w 0 ) u 0 ( x , z ) ,
D x α a f ( x ) = 1 Γ ( n α ) d n d x n a x ( x ξ ) n α 1 f ( ξ ) d ξ ,
d m d x m D α = D α d m d x m = D α + m .
u α ( x , z ) = D α u 0 ( x , z ) = w 0 ( 2 π ) 3 4 × ( i k x ) α exp ( μ k x 2 w 0 2 4 + i k x x ) d k x .
u α ( x , z ) = N α μ α + 1 exp ( x 2 μ w 0 2 ) P α ( x μ w 0 ) ,
P α ( X ) cos ( π 2 α ) Γ ( 1 + α 2 ) Φ ( α 2 , 1 2 ; X 2 ) sin ( π 2 α ) α Γ ( α 2 ) X Φ ( 1 α 2 , 3 2 ; X 2 ) ,
u α ( ζ ) = A Φ ( σ , 1 2 ; ζ ) + B ζ 1 2 Φ ( σ + 1 2 , 3 2 ; ζ ) ,
T u α = [ d d x 2 + 2 γ x d d x + 2 γ ( 1 + α ) ] u α = 0 ,
u α S ( x , z ) = ( μ * μ ) α 2 u 0 P α ( 2 x w 0 μ ) .

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