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J. Opt. Soc. Am. A / Vol. 31, No. 8 / August 2014

C. Huang and H. Lu

Accelerating propagation properties of misplaced Hermite–Gaussian beams Chaohong Huang* and Helin Lu Department of Electronic Engineering, Xiamen University, Xiamen 361005, China *Corresponding author: [email protected] Received May 1, 2014; accepted June 12, 2014; posted June 24, 2014 (Doc. ID 211183); published July 17, 2014 A new family of finite-energy accelerating beams was constructed through misplacing the Hermite polynomial and Gaussian window function. The closed-form solution of k-space spectra and paraxial propagation of these beams are derived from the Fourier transform and the scalar angle spectra integral. These beams have similar propagation properties to finite Airy beams and parabolic beams, but the accelerating trajectory is hyperbola rather than parabola. The beam family can be experimentally generated by exponentially truncating the high-order Hermite–Gaussian beams in the spatial domain. © 2014 Optical Society of America OCIS codes: (050.1960) Diffraction theory; (000.3860) Mathematical methods in physics. http://dx.doi.org/10.1364/JOSAA.31.001762

Since Siviloglou and co-workers theoretically and experimentally demonstrated the existence of Airy beams in 2007 [1,2], accelerating beams have attracted a great deal of attention for their unusual features: accelerating, shape-preserving, and self-healing. These novel features have found many applications in various fields, such as microparticle manipulation [3,4], plasma waveguiding [5], and generation of light bullets [6,7]. The concept “accelerating beams” originates from Berry and Balazs’ work on nondispersion Airy wavepacket solutions of the time-dependent potential-free Schrödinger equation [8]. In optics, the paraxial wave propagation equation has the same form as the time-dependent potential-free Schrödinger equation. Therefore, the Airy wavepacket function also describes a diffraction-free paraxial planar beam in a 1  1D wave propagation system. The amplitude distribution of the Airy beam is highly asymmetric transversely, and the lobes of the beam propagate along a parabolic trajectory instead of a straight line. For the 1  1D system, the Airy wavepacket is a unique nontrivial nondispersion solution of the paraxial wave equation in free space [9]. The 1D Airy planar beam can be easily extended to the 2  1D propagating system by multiplying two Airy functions in the x and y directions. However, the nondiffractive accelerating beams in 2D are not limited to the Airy beam, and other beams with similar behavior are also possible, e.g., the parabolic beam introduced by Bandres in [10]. Like conventional diffraction-free beams, such as the Bessel beam [11], the Mathieu beam [12], and the parabolic nondiffracting beam [13], perfect nondiffractive accelerating beams also carry infinite energy and spread in infinite space. Therefore, these nondiffractive beams exist only in theory. From the view of reality, only a finite-energy truncated version of these beams can be realized and employed. Although a mandatory amplitude truncation will degrade the performance of these beams to a certain extent, the unusual propagation properties of infinite-energy beams are still expected to 1084-7529/14/081762-04$15.00/0

be retainable along a propagation distance longer than the diffraction distance. The propagation properties and physical realization of several truncated accelerating beams have been investigated since 2007. The first example of such beams was introduced by Siviloglou and co-workers in [1,2] by multiplying an exponential window function to the ideal Airy beam. Another example is the parabolic beam introduced by Bandres in [10] and observated by Davis et al. in [14]. These truncated accelerating beams can maintain their topological structure and remain essentially diffraction free for a much longer distance than the Rayleigh length of Gaussian beams with the same beam width. Due to the truncation, the practical accelerating beams will spread out finally after propagating a long distance. When relaxing the constraint of strict nondiffraction, more possible finite-energy accelerating beams can be constructed and realized physically. In fact, finite-energy accelerating beams are not limited to the truncated version of infiniteenergy accelerating beams, and the accelerating trajectory is also not limited to parabola. For example, Greenfield et al. constructed finite-energy accelerating beams with arbitrary convex trajectory [15], Jiang et al. constructed Airy-related beams from flat-topped Gaussian beams [16], and Chremmos et al. constructed and Zhao et al. observed self-accelerating Bessel-like optical beams along arbitrary trajectories [17,18]. In this paper, we construct a family of finite-energy accelerating beams by truncating the high-order Hermite–Gaussian (HG) beams and derive these beams’ closed-form solutions of k-space spectra and the paraxial propagation formula. The main lobes of these beams are weakly diffractive and follow a hyperbolic curve trajectory. It is well known that the lobes of high-order HG beams propagate along hyperbolic trajectories. The broadening effect of each lobe with propagation is weaker than the whole beam. Figure 1(a) shows the beam broadening of a 1D 20order HG beam with propagation. As shown in this figure, the lobes of the beam show a much smaller broadening than © 2014 Optical Society of America

C. Huang and H. Lu

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Fig. 1. Propagation contour diagrams of 1D HG20 beam with λ  1.06 μm, ω0  0.5 mm, and its “hard” truncation version.

the whole beam. The result inspires us to construct an accelerating beam by truncating the beam using a proper window function. The simplest way for this purpose is to let field zero at x < 0. Figure 1(b) shows the propagation properties of the “hard” truncation version of the usual HG beam. It can be seen that the propagating trajectory of the main lobe of the half-HG beam is almost influenced by the hard truncation in a long propagation distance, but some small ripples appear. In the following section, we employ an exponential truncation window and derive the closed-form propagation solution for the beams, which satisfy the paraxial wave propagation equation. Consider a 1D m-order HG beam (HGm ) truncated exponentially at the z  0 plane: Ux  H m

p     2 x 2ax 2 ; x exp − 2 · exp ω0 ω20 ω0

(1)

where H m ⋅ is the m-order Hermite polynomial with physicist form. The parameter a denotes a truncation factor. ω0 is the waist radius of the original HGm beam. Equation (1) is equivalent to the following formula:  2 p    a 2 x − a2 Ux  exp 2 H m : (2) x exp − ω0 ω0 ω20 From Eq. (2), we can see the exponential truncation of the HGm beam only results in a coordinate translation a for the Gaussian function in the HG mode. The first constant factor on the right-hand side of Eq. (2) has no effect on the beam distribution. If the usual HGm modes are regarded as an m-order Hermite polynomial multiplied by a Gaussian window function with center at x  0, then Eq. (2) in fact represents the misplaced HGm beam with Gaussian windows with center at x  a. The parameter a is the misplacement factor between the Hermite polynomial and the Gaussian window. The k-space spectra of the misplaced HGm beam family can be obtained from the Fourier transform of Eq. (1) or Eq. (2):    p ω 2a Ukx   −im ω0 π H m p0 kx  i 2 ω0 2  2 2  ω 2a · exp − 0 kx  i 2 : 4 ω0

(3)

From Eq. (3), one can see the k-space spectra of the misplaced or exponentially truncated HGm beams have the same function form as that of usual HGm beams. The only difference is that the wave vector component kx is substituted by a complex argument for the misplaced HGm beams. Figure 2 shows the variation of the spectral intensity distribution with

Normalized spectral intensity

a/ω0=0.0 a/ω0=0.1 a/ω0=0.2 a/ω0=0.3 a/ω0=0.4 a/ω0=0.5 a/ω0=0.6 -0.010

-0.005

0.000

0.005

0.010

kxλ/2π

Fig. 2. Variation of k-space spectra of 1D exponentially truncated HG beams with truncation factor a (λ  1.06 μm, ω0  0.5 mm, and m  20).

truncation factor a. The results show that the spectral intensity varies from HG form to Gaussian form when increasing the truncation factor a. By the scalar angular spectrum theory, the beam propagation property in free space can be written as follows: Ux; z 

1 2π

Z

∞

−∞

 q Ukx  expikx x exp iz k2 − k2x dkx : (4)

For the paraxial approximation (kx ≪ k), Eq. (4) can be integrated analytically using the generation functions of the Hermite polynomial. Through a tedious process, we get  1∕2 p   2  ω0 a z 2 Ux; z  xi a exp 2 H m f0 ω ω ω   2 x − a · expikz · exp − ω2   x − a2 z · exp ik ; − im  1∕2arctg f0 2R

(5)

where πω2 f 0  0 ; ω  ω0 λ

s  2   2  z f : 1 ;R  z 1  0 f0 z

(6)

Obviously, Eqs. (5) and (6) are very similar with usual HGm beams. The misplaced HGm beams have only a real coordination translation a for the exponential part and a imaginary coordination translation iaz∕f 0 for the Hermite polynomial, compared with usual HGm beams. While the parameter a equals zero, the misplaced HGm beams become the usual HGm beams. The trajectory of the main lobe of the misplaced HGm beams along z can be represented approximately by the hyperbolic curve equation: s  2 z x  x0 1  : f0

(7)

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x0 is the position of the main lobe at the z  0 plane. The 1D misplaced HGm beams can easily extend to the 2D situation. Consider a 2D HGmn beam truncated exponentially at the z  0 plane: p  p  2 2 x Hn y ω0 ω0     2 x  y2 2ax  2by exp ; · exp − ω20 ω20

Ux; y  H m

(8)

where a and b are the truncation or misplacement factors along the x and y directions, respectively. Equation (8) can also be written as  2  p  p  a  b2 2 2 Ux; y  exp H x Hn y m ω0 ω0 ω20   x − a2  y − b2 : (9) · exp − ω20 The closed-form k-space spectra and paraxial propagation formula can be derived using the Fourier transform and the scalar angle spectra integral: Ukx ; ky   −imn ω20 π    2    ω0 ω0 2a 2a 2 · H m p kx  i 2 exp − kx  i 2 4 ω0 ω0 2    2    ω ω 2b 2b 2 · H n p0 ky  i 2 exp − 0 ky  i 2 ; 4 ω0 ω0 2 (10) p   p   2 za 2 zb Ux;y;z  H m Hn xi yi ω f0 ω f0     2 2 2 ω a b x−a y−b2 · 0 exp exp − expikz ω ω2 ω2   x−a2 y−b2 z ·exp ik −imn1arctg ; f0 2R (11)

Fig. 3. Propagation characteristics of 1D misplaced HG beams with different parameters (m  20, 40, and 60 for the first, second, and third rows, respectively; a∕ω0  0.15, 0.25, and 0.35 for the first, second, and third columns, respectively).

trajectories of these beams’ main lobes. For example, for the case of m  20 and a  0.15ω0 , the FWHM of the main lobe at the z  0 plane is about 175 μm and is invariant up to z  0.75 m. The Rayleigh length of the Gaussian beam with the same width is only 0.096 m. The shift distance transversely is about 0.8 mm while propagating to z  0.75 m. Similar results can be obtained for other situations. With increase of parameter a, the misplaced HGm beams show a clear transition from initial form at the z  0 plane to Gaussian-like form at the far field. For the 2D case, the field distributions are diverse with different values of m, n, a, and b. Figure 5 shows several typical 2D exponentially truncated HGmn beams. These beams follow a similar hyperbolic curve trajectory to that seen in 1D cases. In summary, we constructed the exponentially truncated or misplaced HG beams as finite-energy accelerating beams. The closed-form solutions of k-space spectra and paraxial propagation of these beams are derived from the Fourier transform and the scalar angular spectra integral. The results show that the

where f 0 , ω, and R are the same with Eq. (6). The trajectory of the main lobe of the 2D misplaced HGmn beams along z can be represented approximately by the hyperbolic curve equation: s  2 z r  r0 1  ; f0

(12)

where r is the transverse displacement of the beams. For clarity, we demonstrate several typical examples of the misplaced HGm or HGmn beams below. For all examples, the laser wavelength is 1.06 μm, and ω0 is 0.5 mm. Figure 3 shows the propagation contour diagrams of several 1D misplaced HGm beams. Figures 4(a)–4(c) show the intensity distributions of misplaced HG20 , HG40 , and HG60 beams with a∕ω0  0.15 at several propagation distances (z  0, 0.5, and 1 m). Clearly, the beams are accelerative transversally and shape-preservating while propagating a long distance. Figure 4(d) shows the hyperbolic propagating

Fig. 4. Propagation characteristics of 1D misplaced HG beams (HG20 , HG40 , and HG60 ) with a∕ω0  0.15. (a), (b), and (c) Show the intensity distributions at z  0, 0.5, and 1 m for beams with m  20, 40, and 60, respectively. (d) Shows the propagation trajectories of the main lobes of these beams.

C. Huang and H. Lu

Fig. 5. Propagation characteristics of 2D exponentially HG beams with different parameters (m  20, n  0, a∕ω0  0.15, and b∕ω0  0 for the first row; m  n  20 and a∕ω0  b∕ω0  0.15 for the second row; m  20, n  1, a∕ω0  0.15, and b∕ω0  0 for the third row; and z  0, 0.3, and 0.6 m for the first, second, and third columns).

misplaced HG beam family has similar self-acceleration and shape-preservation properties to other finite-energy accelerating beams, such as finite Airy beams and parabolic beams. The difference is that the misplaced HG beams demonstrate hyperbolic accelerating dynamics rather than parabolic. The misplaced HG beams can be realized by exponentially truncating the high-order HG laser modes in the spatial domain.

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3. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). 4. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011). 5. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009). 6. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). 7. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010). 8. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979). 9. K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034 (1996). 10. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008). 11. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987). 12. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). 13. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). 14. J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008). 15. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011). 16. Y. F. Jiang, K. K. Huang, and X. H. Lu, “Airy-related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29, 1412–1416 (2012). 17. I. D. Chremmos, Z. G. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37, 5003–5005 (2012). 18. J. Y. Zhao, P. Zhang, D. M. Deng, J. J. Liu, Y. M. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, “Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories,” Opt. Lett. 38, 498–500 (2013).

Accelerating propagation properties of misplaced Hermite-Gaussian beams.

A new family of finite-energy accelerating beams was constructed through misplacing the Hermite polynomial and Gaussian window function. The closed-fo...
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