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  • Editor: Alan E. Willner
  • Vol. 37, Iss. 22 — Nov. 15, 2012
  • pp: 4774–4776
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Virtual source for an Airy beam

Shaohui Yan, Baoli Yao, Ming Lei, Dan Dan, Yanlong Yang, and Peng Gao  »View Author Affiliations


Optics Letters, Vol. 37, Issue 22, pp. 4774-4776 (2012)
http://dx.doi.org/10.1364/OL.37.004774


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Abstract

We identify a virtual source for generating an Airy wave. A spectral integral expression is derived to describe the Airy wave, which, in the paraxial limit, yields the freely accelerating, nondiffracting, and finite energy Airy beam. From the spectral representation of the Airy wave, the first two orders of nonparaxial corrections to the paraxial Airy beam are determined. Also, a connection between the obtained Airy wave and the well-known complex source point spherical wave is given.

© 2012 Optical Society of America

The paraxial wave equation (PWE) admits some interesting solutions. One class of them is nondiffracting beams, which leave their transverse intensity distribution quasi-invariant during propagation. Among them, Bessel beams [1

1. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987). [CrossRef]

], Mathieu beams [2

2. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000). [CrossRef]

,3

3. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001). [CrossRef]

], and parabolic nondiffracting beams [4

4. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, Opt. Lett. 29, 44 (2004). [CrossRef]

] are typical examples obtained by solving the PWE in circular cylindrical coordinates, elliptic coordinates, and parabolic coordinates, respectively. Another family of solutions is accelerating beams, which, in addition to the nondiffracting property, have a propagation dynamics of freely accelerating in free space. Well-known examples of accelerating beams are Airy beams [5

5. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007). [CrossRef]

] and accelerating parabolic beams [6

6. M. A. Bandres, Opt. Lett. 33, 1678 (2008). [CrossRef]

]. Unlike the common Gaussian beam, whose maximum cross section intensity follows a straight line during propagation, the maximum intensity of the Airy beam forms a parabolic trajectory. In optical trapping, this property plus the nondiffracting property enables the Airy beam to transport particles along a parabolic trajectory [7

7. J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008). [CrossRef]

] instead of confining them to the center of the focal region as the Gaussian beam does. An important property of the Airy beam is that its spectral function contains a cubic phase factor relative to the spectrum of a fundamental Gaussian beam, which leads to the parabolic trajectory of the beam during the propagation. Therefore, one method for generating an Airy beam is to introduce a cubic phase factor in the spectral plane of a Gaussian beam by means of a spatial light modulator [8

8. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

]. It is natural to investigate the evolution of the Airy beams by analyzing their spectral representation. Based on the spectral representation, A. V. Novitsky and D. V. Novitsky [9

9. A. V. Novitsky and D. V. Novitsky, Opt. Lett. 34, 3430 (2009). [CrossRef]

] gave an expression for nonparaxial Airy beams and discussed the refraction of the Airy beam through an interface as well as the field of the transmitted Airy beam. Another method of investigating beams is the virtual source technique, which was pointed out by Deschamps [10

10. G. A. Deschamps, Electron. Lett. 7, 684 (1971). [CrossRef]

] and systematically developed by Felsen [11

11. L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976). [CrossRef]

] and Shin and Felsen [12

12. S. Y. Shin and L. B. Felsen, J. Opt. Soc. Am. 67, 699 (1977). [CrossRef]

]. The virtual source method has been widely applied to investigate characterization and propagation of different beams [13

13. S. R. Seshadri, Opt. Lett. 27, 998 (2002). [CrossRef]

16

16. Y. H. Zhang, Y. J. Song, Z. Chen, J. H. Ji, and Z. X. Shi, Opt. Lett. 32, 292 (2007). [CrossRef]

]. In this Letter, a corresponding treatment of the Airy wave is presented.

In the z=0 plane, the field distribution of a paraxial one-dimensional Airy beam is in general described by
ϕ(u,s=0)=Ai(u)exp(au),
(1)
where u=x/x0 and s=z/x0 represent dimensionless transverse and longitudinal coordinates with x0 an arbitrary transverse scale, Ai(u) is the Airy function, and the decay factor a>0 is a small positive quantity to ensure the finiteness of the beam’s energy. We now want to construct a scalar Airy wave function F(u,s), which travels along the positive z axis and satisfies the homogeneous Helmholtz equation for z>0. We require that in the paraxial approximation, the input field distribution of the beam described by F(u,s) reduces to Eq. (1). In the virtual source method, the beam is assumed to be generated by a source of strength Sext situated at z=zext exterior to the physical space z>0. The source strength Sext and the (complex-valued) source location zext are undetermined for the present. In the subsequent stage, the requirement of yielding the desired input field distribution will specify them. With the source included, the wave function F(u,s) satisfies the inhomogeneous Helmholtz equation
[2/u2+2/s2+(kx0)2]F(u,s)=Sextexp[D(u)]δ(u)δ(szext),
(2)
where k=2π/λ is the wave number and the operator D(u) is defined as
D(u)=13(3u33a2u).
(3)
By use of the Fourier transform pair
F(u,s)=12πF¯(p,s)exp(ipu)dp,
(4)
F¯(p,s)=F(u,s)exp(ipu)du,
(5)
F¯(p,s) is determined from Eq. (2). Inserting it into Eq. (4) gives
F(u,s)=12πdpexp(ipu)iSext2ζ×exp[i3(p33a2p)]exp[iζ(szext)]
(6)
for Re(szext)>0, where
ζ=[(kx0)2p2]1/2.
(7)

Now, we asymptotically evaluate the integral in Eq. (6) to recover the paraxial field distribution in the z=0 plane by suitably choosing Sext and zext. Applying the paraxial limit p2(kx0)2, we expand ζ in terms of p2 and retain the leading term for the amplitude factor and the first two terms for the phase factor. In this approximation, Eq. (6) becomes
Fp(u,s)=12πexp(ikz)[iSext2kx0exp(ikx0zext)]dpexp(ipu)exp[i3(p33a2p)]exp[ip22kx0s+ip22kx0zext],
(8)
where the suffix p denotes the paraxial approximation. When zext=i2akx0, the integral in Eq. (8) takes the form
Fp(u,s)=12πexp(ikz)[iSext2kx0exp(2ak2x02)]dpexp(ipu)×exp[i3(p33a2p)]exp[ip22kx0sap2].
(9)

Applying Airy’s integral
Ai(t)=12πdqexp(iqt)exp(iq3/3),
(10)
the integral in Eq. (9) is evaluated to be
Fp(u,s)=iSext2kx0e(ikz+2ak2x02a3/3)Ai[u(s1/2)2+ias1]×exp[auas12/2is13/12+ia2s1/2+ius1/2]
(11)
with s1=s/kx0. In the input plane z=0 as well as s=0, Eq. (11) becomes
Fp(u,0)=[iSext2kx0exp(2ak2x02a3/3)]Ai(u)exp(au).
(12)
For this field distribution to coincide with ϕ(u,0)=Ai(u)exp(au) in Eq. (1), the source strength Sext must be equal to i2kx0exp[a3/32a(kx0)2]. So, by setting the source strength Sext and the source location zext to be
zext=i2akx0ib,
(13)
Sext=i2kx0exp[a3/32a(kx0)2],
(14)
we obtain a beam whose field distribution in the input plane z=0 reduces to that of a one-dimensional finite energy Airy beam, provided the paraxial approximation is used. With the expressions for zext and Sext, we obtain the exact form of the beam from Eq. (6) as
F(u,s)=kx02πexp[a3/32a(kx0)2]dpexp(ipu)/ζ×exp[i3(p33a2p)]exp[iζ(sib)].
(15)

It is straightforward to show that the virtual source equation governing a two-dimensional Airy wave packet is
[t2+2/s2+(kx0)2]F(u,v,s)=Sextexp[D(u,v)]×δ(u)δ(v)δ(szext)
(16)
with
t2=2/u2+2/v2,
(17)
D(u,v)=13[(3u3+3a2u)(3v3+3a2v)].
(18)

Here, v=y/x0, zext is the same as in Eq. (13) and Sext=i2kx0exp[2a3/32a(kx0)2]. Equations (16)–(18) will yield a beam whose input field distribution in the plane z=0 is equal to
ϕ(u,v,s=0)=Ai(u)Ai(v)exp[a(u+v)],
(19)
in agreement with that of the two-dimensional Airy beam presented in [5

5. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007). [CrossRef]

].

The integral expression [Eq. (15)] for F(u,s) can be used to obtain a corrected series expression (the paraxial term plus the nonparaxial corrections) for the beam if we expand the amplitude factor 1/ζ and the phase factor exp[iζ(sib)] in the power series of some small quantity f. For Laguerre–Gauss and Bessel–Gauss beams, this small quantity is f=1/(kw0), where w0 is the waist size of the beam [13

13. S. R. Seshadri, Opt. Lett. 27, 998 (2002). [CrossRef]

,14

14. S. R. Seshadri, Opt. Lett. 27, 1872 (2002). [CrossRef]

]. For the Airy beam discussed here, this small quantity is f=1/(kx0). This can be seen from the input field distribution Fp(u,0)=exp(au)Ai(u) given by Eq. (1) or Eq. (12) of the paraxial beam. For a given decay factor a, the presence of the exponent factor exp(au) with u=x/x0 implies that the actual size of the beam is equal roughly to x0/a. From the spectral integral [Eq. (9)] of Fp(u,s), we note that the spectrum function contains an amplitude factor exp(ap2), indicating that the spectrum integral in Eq. (9) is mainly contributed by the components in the range p21/a. If x0 is large enough in comparison with a wavelength to ensure that f=1/kx01 and 1/a(kx0)2, the paraxial approximation p2(kx)2 will be satisfied. In general, the two conditions f1 and 1/a(kx0)2 hold. For example, when x0=100μm, a=0.1, and λ=0.5μm, typical parameters chosen in [5

5. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007). [CrossRef]

], it is straightforward to show the two conditions are safely satisfied. Having ascertained the expansion parameter f, we give one nonparaxial correction term to the paraxial beam. For this purpose, we retain the terms of the expansion of 1/ζ up to order of f2, and the same for the phase factor exp[iζ(sib)] expansion. In the product of the two series expansions, only terms up to the order f2=1/(kx0)2 are retained. Then, Eq. (15) can be approximated as
F(u,s)=12πexp[a3/3+ikz]dpexp(ipu)×exp[i3(p33a2p)ap2ip2s/(2kx0)]G(p,s).
(20)

Here the function G(p,s) is defined as
G(p,s)=1+f2[p22ap44q2(s)]
(21)
with q2(s)=(1+is/b)1. In evaluating the integral in Eq. (20), we can use integration by parts to simplify the integral. By repeated use of the differential equality p2exp(ip3/3)dp=id[exp(ip3/3)], we are finally led to an integral as follows:
F(u,s)=Fp(u,s)+f2F2(u,s),
(22)
where Fp(u,s) is the paraxial Airy beam given by Eq. (11), and the second-order correction Fp(u,s) is defined as
F2(u,s)=[A(u,s)+B(u,s)u]Fp(u,s).
(23)

Here, the functions A(u,s) and B(u,s) are defined as
A(u,s)=(ua2)22a24q4a(ua2)24q2a3(ua2)q6,
(24)
B(u,s)=3a2q2a2(ua2)q42a4q8.
(25)
From Eq. (11), it is seen that Fp(u,s) is the product of an exponential function and the Airy function. Note that the derivative of the Airy function can be written as dAi(t)/dt=t/(31/2π)K2/3(2/3t3/2) for Re(t)>0 and dAi(t)/dt=(t/3)[J2/3(2/3(t)3/2)J2/3(2/3(t)3/2)] for Re(t)<0, where Jν(t) and Kν(t) are the Bessel functions and the modified Bessel functions, respectively. The second-order correction term F2(u,s) can be expressed without any partial derivatives with respect to u.

Finally, we give the connection of the two-dimensional Airy wave function govern by Eq. (16) to the complex-source-point spherical wave (CSPSW) function. The CSPSW G(x,xext)) satisfies the inhomogeneous wave equation:
[t2+2/s2+(kx0)2]G(x,xext)=δ(u)δ(v)δ(sib).
(26)

Its explicit expression is given by
G(x,xext)=exp(ikx0R)/4πR,
(27)
where
R=[u2+v2+(sib)2]1/2.
(28)

Equation (29) includes the paraxial Airy beam as well as all other nonparaxial corrections to the paraxial beam.

This research is supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and the Natural Science Foundation of China (NSFC) under Grant No. 61205123.

References

1.

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987). [CrossRef]

2.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000). [CrossRef]

3.

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001). [CrossRef]

4.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, Opt. Lett. 29, 44 (2004). [CrossRef]

5.

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007). [CrossRef]

6.

M. A. Bandres, Opt. Lett. 33, 1678 (2008). [CrossRef]

7.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008). [CrossRef]

8.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

9.

A. V. Novitsky and D. V. Novitsky, Opt. Lett. 34, 3430 (2009). [CrossRef]

10.

G. A. Deschamps, Electron. Lett. 7, 684 (1971). [CrossRef]

11.

L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976). [CrossRef]

12.

S. Y. Shin and L. B. Felsen, J. Opt. Soc. Am. 67, 699 (1977). [CrossRef]

13.

S. R. Seshadri, Opt. Lett. 27, 998 (2002). [CrossRef]

14.

S. R. Seshadri, Opt. Lett. 27, 1872 (2002). [CrossRef]

15.

S. R. Seshadri, Opt. Lett. 28, 595 (2003). [CrossRef]

16.

Y. H. Zhang, Y. J. Song, Z. Chen, J. H. Ji, and Z. X. Shi, Opt. Lett. 32, 292 (2007). [CrossRef]

OCIS Codes
(260.1960) Physical optics : Diffraction theory
(260.2110) Physical optics : Electromagnetic optics
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 2, 2012
Manuscript Accepted: October 10, 2012
Published: November 14, 2012

Citation
Shaohui Yan, Baoli Yao, Ming Lei, Dan Dan, Yanlong Yang, and Peng Gao, "Virtual source for an Airy beam," Opt. Lett. 37, 4774-4776 (2012)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-22-4774


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