Abstract

We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation.

© 2007 Optical Society of America

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References

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  1. H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE,  54, 1312-1329 (1966).
    [CrossRef]
  2. A. E. Siegman, Lasers (University Science, Mill Valley CA, 1986).
  3. E. G. Kalnins and W. MillerJr., "Lie theory and separation of variables. 5. The equations iUt +Uxx = 0 and iUt +Uxx-c/x2 U = 0," J. Math. Phys. 15, 1728-1737 (1974).
    [CrossRef]
  4. M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
    [CrossRef]
  5. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994).
    [CrossRef]
  6. K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy Packet in quantum mechanics, " Am. J. Phys. 64, 1034-1036 (1996).
    [CrossRef]
  7. G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
    [CrossRef] [PubMed]
  8. I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007).
    [CrossRef] [PubMed]
  9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  10. H. M. Osaktas, Z. Zalevski, and M. A. Kutay, The Fractional Fourier Transform with applications in Optics and Signal processing (Wiley, London, 2001).
  11. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1964).
  12. O. Vallée and M Soares, Airy functions and applications to physics (Imperial College Press, London, 2004).

2007 (3)

1996 (1)

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy Packet in quantum mechanics, " Am. J. Phys. 64, 1034-1036 (1996).
[CrossRef]

1994 (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

1974 (1)

E. G. Kalnins and W. MillerJr., "Lie theory and separation of variables. 5. The equations iUt +Uxx = 0 and iUt +Uxx-c/x2 U = 0," J. Math. Phys. 15, 1728-1737 (1974).
[CrossRef]

1966 (1)

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE,  54, 1312-1329 (1966).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Besieris, I. M.

I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Christodoulides, D. N.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
[CrossRef] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Kalnins, E. G.

E. G. Kalnins and W. MillerJr., "Lie theory and separation of variables. 5. The equations iUt +Uxx = 0 and iUt +Uxx-c/x2 U = 0," J. Math. Phys. 15, 1728-1737 (1974).
[CrossRef]

Kogelnik, H.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE,  54, 1312-1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE,  54, 1312-1329 (1966).
[CrossRef]

Miller, W.

E. G. Kalnins and W. MillerJr., "Lie theory and separation of variables. 5. The equations iUt +Uxx = 0 and iUt +Uxx-c/x2 U = 0," J. Math. Phys. 15, 1728-1737 (1974).
[CrossRef]

Rau, A. R. P.

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy Packet in quantum mechanics, " Am. J. Phys. 64, 1034-1036 (1996).
[CrossRef]

Shaarawi, A. M.

I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

Siviloglou, G. A.

G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Unnikrishnan, K.

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy Packet in quantum mechanics, " Am. J. Phys. 64, 1034-1036 (1996).
[CrossRef]

Ziolkowski, R. W.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

Am. J. Phys. (3)

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy Packet in quantum mechanics, " Am. J. Phys. 64, 1034-1036 (1996).
[CrossRef]

J. Math. Phys. (1)

E. G. Kalnins and W. MillerJr., "Lie theory and separation of variables. 5. The equations iUt +Uxx = 0 and iUt +Uxx-c/x2 U = 0," J. Math. Phys. 15, 1728-1737 (1974).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE,  54, 1312-1329 (1966).
[CrossRef]

Other (4)

A. E. Siegman, Lasers (University Science, Mill Valley CA, 1986).

H. M. Osaktas, Z. Zalevski, and M. A. Kutay, The Fractional Fourier Transform with applications in Optics and Signal processing (Wiley, London, 2001).

M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1964).

O. Vallée and M Soares, Airy functions and applications to physics (Imperial College Press, London, 2004).

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Figures (3)

Fig. 1.
Fig. 1.

Free space propagation of an Airy-Gaussian beam for several initial conditions.

Fig. 2.
Fig. 2.

Propagation of an AiG beam through a GRIN medium for two different initial conditions.

Fig. 3.
Fig. 3.

(a)–(c) Absolute value and phase of the Airy function Ai(w) on the complex plane w=u+iv. (d) The variation of x∈(-∞,∞) in the argument of the Airy function Ai((x+δ)/κ) corresponds to a straight line on the complex w=u+iv=(x+δ)/κ plane. φκ=arctan[Im(κ)/Re(κ)]. The figure shows the case Reκ>0 and Imκ>0.

Equations (26)

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( 2 x 2 + i 2 k z ) U ( x , z ) = 0 ,
U 1 ( x 1 ; κ 1 , δ 1 , S 1 , q 1 ) = Ai ( x 1 + δ 1 κ 1 ) exp [ i S 1 ( x 1 + δ 1 κ 1 ) + i 1 3 S 1 3 ] exp ( i k x 1 2 2 q 1 ) ,
U 2 ( x 2 ) = k i 2 π B U 1 ( x 1 ) exp [ ik 2 B ( A x 1 2 2 x 1 x 2 + D x 2 2 ) ] d x 1 ,
U 2 ( x 2 ; κ 2 , δ 2 , S 2 , q 2 ) = Ai ( x 2 + δ 2 κ 2 ) exp [ i S 2 ( x 2 + δ 2 κ 2 ) + i 1 3 S 2 3 ] GB ( x 2 , q 2 ) ,
GB ( x 2 , q 2 ) = 1 A + B q 1 exp ( i k x 2 2 2 q 2 ) ,
q 2 = A q 1 + B C q 1 + D , κ 2 = κ 1 ( A + B q 1 ) , S 2 = S 1 + B 2 k κ 1 κ 2 , δ 2 = δ 1 ( A + B q 1 ) B 2 k κ 1 ( S 1 + S 2 ) .
U 2 ( x 2 ) = Ai ( x 2 + δ 2 κ 1 A ) exp [ i S 2 ( x 2 + δ 2 κ 1 A ) + i 1 3 S 2 3 ] 1 A exp ( i C k x 2 2 2 A ) ,
S 2 = S 1 + B 2 k κ 1 2 A , δ 2 = B k κ 1 ( S 1 + B 4 k κ 1 2 A ) .
q 2 ( z ) = q 1 μ , κ 2 ( z ) = κ 1 μ , S 2 ( z ) = S 1 + ( 1 2 k κ 1 2 ) z μ , δ 2 ( z ) = δ 1 + ( δ 1 q 1 S 1 k κ 1 ) z ( 1 4 k 2 κ 1 3 ) z 2 μ ,
x c ( z ) = x ( z ) = x U ( x , z ) 2 d x U ( x , z ) 2 d x ,
U ( x , 0 ) Ai ( x + δ i κ 1 ) exp [ ( α + i β ) x ] exp ( x 2 w 1 2 ) exp ( i k x 2 2 R 1 ) ,
x c ( z ) = x c ( 0 ) + ( x c ( 0 ) R 1 + β k ) z z 0
[ A B C D ] = [ cos ( z a ) a sin ( z a ) sin ( z a ) a cos ( z a ) ] .
q 2 ( z ) = a q 1 cos ( z a ) + a sin ( z a ) q 1 sin ( z a ) + a cos ( z a ) ,
κ 2 ( z ) = κ 1 [ cos ( z a ) + ( a q 1 ) sin ( z a ) ] ,
S 2 ( z ) = S 1 + a 2 k κ 1 2 [ cot ( z a ) + a q 1 ] ,
δ 2 ( z ) = δ 1 [ cos ( z a ) + a sin ( z a ) q 1 ] a sin ( z a ) 2 k κ 1 ( S 1 + S 2 ) ,
q 2 ( L 4 ) = a 2 q 1 , κ 2 ( L 4 ) = κ 1 a q 1 , S 2 ( L 4 ) = S 1 + q 1 2 k κ 1 2 , δ 2 ( L 4 ) = δ 1 a q 1 a k κ 1 ( S 1 + q 1 4 k κ 1 2 ) .
U ( x ) Ai ( x + δ κ ) exp ( iS κ x ) exp ( i k x 2 2 q ) ,
w = x + δ κ = u ( x ) + iv ( x ) = [ x Re κ + ( κ · δ ) κ 2 ] + i [ x Im κ + ( κ × δ ) κ 2 ]
v s ( u ) = ( Im κ Re κ ) u + [ ( Im κ Re κ ) ( κ · δ ) + ( κ · δ ) κ 2 ] ,
Ai ( w ) exp ( 2 w 3 2 3 ) 2 π w 1 4 , arg w < π ,
Ai ( w ) sin ( 2 w 3 2 3 + π 4 ) π w 1 4 , arg w < 2 π 3 .
U x + 2 exp ( 2 α x ) x + Re δ 1 2 exp [ k Im ( 1 q ) x 2 ] exp ( 4 3 x + Re δ Re κ 3 2 ) ,
U x + 2 exp ( 2 α x ) x + Re δ 1 2 exp [ k Im ( 1 q ) x 2 ] sin 2 ( 2 3 x + Re δ Re κ 3 2 + π 4 ) ,
U x ± 2 exp ( 2 α x ) x + δ 1 2 exp [ k Im ( 1 q ) x 2 ] exp [ cos ( 3 2 φ ) 4 x + δ 3 2 3 κ 3 2 ] ,

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