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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3490–3500
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Accelerating light beams with arbitrarily transverse shapes

Adrian Ruelas, Jeffrey A. Davis, Ignacio Moreno, Don M. Cottrell, and Miguel A. Bandres  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 3490-3500 (2014)
http://dx.doi.org/10.1364/OE.22.003490


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Abstract

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. Their unique characteristics have opened the door to applications that range from optical micromanipulation and plasma-channel generation to laser micromachining. Here, we demonstrate, theoretically and experimentally, that accelerating beams can be generated with a variety of arbitrarily chosen transverse shapes. We present a general method to construct such beams in the paraxial and nonparaxial regime and demonstrate experimentally their propagation in the paraxial case. The key ingredient of our method is the use of the spectral representation of the accelerating beams, which offers a unique and compact description of these beams. The on-demand accelerating light patterns described here are likely to give rise to new applications and add versatility to the current ones.

© 2014 Optical Society of America

1. Introduction

Accelerating beams are wave packets of light that preserve their shape while propagating along curved trajectories. This exotic behavior was first encountered within the framework of quantum mechanics [1

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

]. This phenomenon requires no waveguiding structure or external potential, appearing even in free space as a result of pure intierference. The first optical accelerating beam, the paraxial Airy beam, was proposed and observed in 2007 [2

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

, 3

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

], and it represents the only exactly shape preserving self-bending solution in 2D paraxial optical systems. However, in three-dimensional (3D) paraxial systems, two separable families of solutions are possible: two-dimensional Airy beams [3

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

] and accelerating parabolic beams [4

4. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008). [CrossRef] [PubMed]

6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

]. Furthermore and more importantly for this work, researchers in [7

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]

] showed that any function on the real line can be mapped to a paraxial accelerating beam with a unique transverse shape, and vice versa.

In recent work [8

8. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012). [CrossRef]

10

10. P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012). [CrossRef] [PubMed]

] researchers overcame the paraxial limit by finding shape preserving accelerating solutions of the Maxwell equations. These beams propagate along semicircular trajectories and being nonparaxial they can reach bending of almost 180° and can display small features on the order of a few wavelengths or less. Subsequently, 2D nonparaxial accelerating wave packets following parabolic [11

11. M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New Journal of Physics 15, 013054 (2013). [CrossRef]

,12

12. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012). [CrossRef] [PubMed]

] and elliptical [12

12. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012). [CrossRef] [PubMed]

,13

13. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012). [CrossRef] [PubMed]

] trajectories and 3D nonparaxial accelerating beams based on spherical, parabolic, oblate and prolate spheroidal fields [13

13. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012). [CrossRef] [PubMed]

15

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]

] were found. And finally, in [15

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]

] it was demonstrated that any 3D nonparaxial accelerating beam can be encoded in a function over the semicircle, and vice versa.

Due to their unique properties, the concept of accelerating beams has led to many intriguing ideas and applications ranging from light-induced curved plasma channels [16

16. 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). [CrossRef] [PubMed]

], self-accelerating nonlinear beams [17

17. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012). [CrossRef] [PubMed]

], to self-bending electron beams [18

18. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013). [CrossRef] [PubMed]

]. In many of these applications, like particle and cell micromanipulation [19

19. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]

], and laser micromachining [20

20. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012). [CrossRef]

, 21

21. A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. Dudley, and F. Courvoisier, “Direct machining of curved trenches in silicon with femtosecond accelerating beams,” J. Euro. Opt. Soc. Rapid publications8(2013). [CrossRef]

], the transverse structure of the accelerating beams plays a crucial role. Therefore, a natural question is: is it possible to engineer the transverse structure of an accelerating beam? To what extent can we shape them?

In this work, we demonstrate that paraxial and nonparaxial accelerating beams with a variety of on-demand transverse shapes can be produced. In particular, we present a general method to construct these beams, and demonstrate, theoretically and experimentally, accelerating beams with a one-dimensional modulation of choice in their transverse intensity. To the best of our knowledge, this is the first method to shape the transverse distribution of accelerating beams on-demand.The key ingredient of our method is the use of the spectral representation of the accelerating beams [7

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]

, 15

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]

], which offers a unique and compact representation of these beams. By engineering the spectrum of the beam we are able to shape its transverse intensity. Our results broaden the applications of accelerating beams by providing with a means of tailoring their transverse shape to match the requirements of the particular application.

2. Shaped accelerating beams

Before starting our analysis it is useful to recall how the shaping of nondiffracting beams works [22

22. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). [CrossRef] [PubMed]

, 23

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010). [CrossRef] [PubMed]

]. Nondiffracting beams are shape preserving beams that move in a straight line, i.e., |ψ(x, y, 0)| = |ψ(x, y, z)|. For example, Bessel, Mathieu and parabolic (Weber) beams, represent three families of separable solutions [22

22. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). [CrossRef] [PubMed]

, 24

24. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

]. Nondiffracting beams are composed by plane waves with the same phase velocity in the propagation direction and such conical superpositions underscore their diffractionless property. In this way, the only degree of freedom of a nondiffracting beam is the angular modulation of their constituent plane waves over the cone. This angular modulation is called the angular spectrum of the nondiffracting beam and it completely characterizes the transverse structure of the beam. By choosing the correct angular spectrum one can generate nondiffracting beams with a variety of transverse intensity patterns of choice; this was demonstrated for the first time in [23

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010). [CrossRef] [PubMed]

]. Also, [25

25. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009). [CrossRef] [PubMed]

] presented a technique to generate Bessel beams with a tunable axial intensity.

Now, let us start our analysis of accelerating beam shaping. A paraxial accelerating beam is a wavepacket that satisfies the paraxial wave equation and the accelerating condition |ψ(x, y, z)| = |ψ(x + Az2, y, 0)|. Researchers in [7

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]

] demonstrated that any two-dimensional accelerating beam can be characterized by a one-dimensional function called the line spectrum, L(kv), and vice versa. For example, the 2D Airy beams have harmonic functions as line spectra, while the accelerating parabolic beams are characterized by the quartic oscillator functions and the Airy-plane-wave beams have a Dirac-delta function as line spectra [7

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]

]. This means that the only degree of freedom that characterizes the transverse shape of an accelerating beam is its line spectrum.

To begin the analysis we must consider the normalized paraxial wave equation ∇2ψ + i∂sψ = 0, where ∇2 = uu + vv, (u, v) = (x, y) /κ represents the dimensionless transverse coordinates, κ is an arbitrary transverse scale factor, s = z/22 is the normalized propagation distance, and k is the wave number. Here it is important to remember, that although a beam satisfies the paraxial equation it does not necessarily satisfy the paraxial condition that assumes that the angle between the optical axis and the wave vectors of the plane waves that constitute a beam is small enough to ensure that the wave does not deviate too much from its propagation direction. For example, the exact infinite energy Airy beams satisfy the paraxial wave equation. However, despite the fact that they carry infinite power they are not paraxial beams because their Fourier spectrum is not confined around the optical axis. In this case, the arbitrary transverse scale factor κ plays a crucial role. For example by increasing κ we can get a beam into the paraxial regime, since this will compress its Fourier spectrum by the scaling properties of the Fourier transform. Note also that s = z/22 depends on κ and therefore changing the transverse scale of the beam also affects its propagation behavior and in the case of accelerating beams their degree of bending. Finally, we can also make the accelerating beams finite and paraxial by introducing an exponential or a Gaussian apodization function. This can be readily done by using the results of [26

26. M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009). [CrossRef]

] where it is shown that one can get the analytic closed form propagation of solutions of the paraxial equation in an ABCD optical system (including misalignments) if one knows the closed form solution in free space.

For our analysis we will use, as an orthogonal basis, the complete family of Airy-plane-wave beams [7

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]

] because they are separable in the same Cartesian coordinate system as our target beam profile. The family of Airy-plane-wave beams is given by
AipwB(u,v,s;γ,kv)=exp(is(uγs2)+is3/3)Ai(u+kv2γs2)exp(ikvv),
(1)
where γ, kv ∈ ℜ are the family parameters. The orthogonally relation of Airy-plane-wave beams at the transverse plane s = 0 is given by
AipwB*(γ,kv)AipwB(γ,kv)dudv=2πδ(kvkv)δ(γγ).
(2)
All beams with the same γ belong to a subfamily of accelerating beams that maintain their relative phase upon propagation, i.e., the terms exp(is γ) is the same for the whole subfamily. Since different subfamilies at s = 0 just differ in a transverse translation in the x-axis we can set γ = 0 for the rest of this work without loss of generality.

Any accelerating beam can be decomposed as
ψ(u,v,0)=14π2L(kv)Ai(u+kv2)exp(ikvv)dkv=1[L(kv)exp(ikukv2+iku3/3)],
(3)
where −1 is the inverse Fourier transform operator. The last part of this equation shows that the k-spectrum of any accelerating beam is a one dimensional modulation of a cubic phase. The middle part of Eq. 3 shows that any accelerating beam is composed by a superposition of Airy-plane-waves with different frequency kv and modulation L(kv). It is important to note the behavior of the Airy-plane-wave basis. For example, Airy-plane-waves with different kv have different frequencies in the v-axis but also their Airy functions in the u-axis Ai(u+kv2) experience different translations. If this translation did not exist, we could construct our desired accelerating beams ψ(u, v, 0) ≈ X(u)Y (v) perfectly by just setting L(kv) = [Y (v)] where is the Fourier transform operator. However this is not possible because removing the translation implies changing the subfamily of the basis and breaking the accelerating properties of the beam.

Our solution to this problem consists of minimizing this translation by scaling the desired modulation function Y (v). Therefore, by using the scale parameter α, we can write Y (v) → Y (v/α) and by the Fourier scaling theorem we have L(kv) → L(αkv)/|α|. Consequently by increasing α we can compress L(αkv) and confine it to small values of kv. This implies a small translation of the Airy component of the Airy-plane-wave basis and therefore the accelerating beam will have the desired shape ψ(u, v, 0) ≈ X(u)Y (v).

Now, we will characterize the engineered accelerating beams as a function of the extent of its spectrum. For this characterization we used the value of kv that contains 95% of L (kv). We will call this parameter k95% and it is defined as
0.95=k95%k95%|L(kv)|dkv/|L(kv)|dkv.
(4)
We found that if k95% ≲ 0.25, the engineered accelerating beam satisfies ψX(u)Y (v/α) almost perfectly. As k95% increases to reach the range 0.25 ≲ k95% ≲ 0.6, the accelerating beam will start losing the desired modulation but at the x position of the first local maximum the modulation Y (v/α) will prevail qualitative to a good approximation at least until 0.6 ≲ k95%.

In Fig. 1 we depict three accelerating beams created using our method. The first row shows the desired modulation Y (v): (a) triangle function, (b) a cosine-Gauss function, and (c) a complex function with the triangle function in the real part (red line) and the cosine-Gauss in the imaginary part (blue line). The black lines depict the y-line profile of the engineered beam at the first 20 local x-maxima; these lines closely overlap with the desired modulation showing that our method truly encodes the desired function in the engineered accelerating beam. The spectrum of the modulation function is depicted in the second row of Fig. 1 and their extent is k95% ≈ 0.26. As mentioned above we need to have functions with a small spectral extent in order for our method to work. The third row shows the invariant intensity structure of the constructed accelerating beams; one can clearly appreciate that the beams have the desired form ψX(x)Y (y) where Y (y) is the arbitrary on-demand modulation depicted in the first row. The last row of Fig. 1 depicts the propagation of our shaped accelerating beams. For the triangular profile in (a), we did not use any apodization function. For the cosine-Gauss profile in (b), we used an exponential apodization function and we used a Gaussian apodization function for the complex profile in (c). As one can appreciate from the last row of Fig. 1, the accelerating characteristics of our on-demand accelerating beams are exactly the same as the Airy beam or any other two-dimensional paraxial accelerating beam.

Fig. 1 Accelerating beams with on-demand transverse shapes: (a) triangle, (b) cosine-Gauss and (c) triangle+i cosine-Gauss. First row, desired target modulation Y (v), blue/red line are the real/imaginary parts and the black line is the vertical y-line profile of the beam at the local maxima of the horizontal x-modulation. Second row, line spectrum of the desired beam. Third row, intensity of the engineered accelerating beams. Last row, propagation of the accelerating beams with no apodization, exponential apodization and Gaussian apodization.

Figure 2 shows the behavior of our method as a function of the extent of the spectrum of the desired modulation Y (y/α). The red lines show the desired target modulation while the blue lines are the engineered beam y-line profiles at different local x-maxima of the beam. The darkest tone of blue represents the first x-maximum cut while the lighter blue tones represent subsequent x-maxima cuts. From the first to the last row we change the extent of the desired modulation from k95% = 0.2 to k95% = 0.8. The first row with k95% = 0.2 has a spectrum confined between −0.2 < k < 0.2 and, in each subsequent row, we increased the extent of the spectral function by stretching it and consequently compressing the desired modulation Y (y/α), i.e., α decreases. Notice that for the first and second rows of Fig. 2, i.e., modulations with small spectral extent k95% ≲ 0.4, the y-line profiles of the beam (blue lines) perfectly overlap with the desired modulation (red lines) showing that our engineered accelerating beam accurately encodes the desired modulation. As we increase the extent of the spectrum to 0.4 ≲ k95%, as in the last two rows of Fig. 2, the beam y-line profile (blue lines) starts to deviate from the desired modulation (red line); the main x-maximum cut of the beam retains its y-modulation for the longest extent. For this reason the light blue lines that correspond to the y-line profile at subsequent x-maxima deviate more from the target modulation. It is important to have in mind that for most practical applications what is important is a qualitative agreement with a desired shape function. Hence, although for modulations with bigger spectral extent the quantitative agreement with the target modulation is not perfect, like the one in the last row of Fig. 2, the desired shape is qualitatively the same.

Fig. 2 The behavior of our method to shape accelerating beams as a function of the extent of the spectrum of the desired modulation Y (y/α), for (a) Mexican hat and (b) random function modulation. The red line is the desired target modulation; the blue lines are the beam y-line profile at different local x-maxima of the beam invariant structure. The darkest tone of blue stands for the first x-maximum. Each row represents a different extent of the desired modulation.

Finally, we just want to mention that our method is able to modulate the beam in the y-coordinate in amplitude and phase as shown in Fig. 1(c). If one wants to impose only an amplitude y-modulation, then it is possible to use the phase of the modulation as another degree of freedom to try to improve the beam shaping, for example using analogous iterative methods similar to the ones used in [23

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010). [CrossRef] [PubMed]

, 25

25. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009). [CrossRef] [PubMed]

] to shape nondiffracting beams. It is important to point out that the methods used in [23

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010). [CrossRef] [PubMed]

, 25

25. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009). [CrossRef] [PubMed]

] are specially tailored to shape nondiffractive beam that move in a straight line. In these cases the domain of the Fourier spectrum of the beam is constraint to a circular ring. This kind of “domain constrain” can be implemented readily in iterative methods like the GerchbergSaxton algorithm. However, in our case, the Fourier spectra of the accelerating beams have nontrivial amplitude and phase constraint and therefore different iterative schemes are needed. We developed several iterative schemes and we did not find significant improvement over our proposed and extremely simpler approach. However the possibility that an iterative method could broader the degree of shaping of accelerating beams that we present in this work is feasible.

3. Experimental realization

We proceed to demonstrate our method experimentally. We generate the shaped accelerating beams by combining both the amplitude and phase patterns onto a single LCD using a previously reported technique [6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

, 27

27. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999). [CrossRef]

]. In our experiments, linearly polarized light from an Argon laser is spatially filtered, expanded, and collimated. The optical elements are encoded onto a parallel-aligned nematic liquid crystal display (LCD) manufactured by Seiko Epson with 640 × 480 pixels with pixel spacing of Δ = 42 microns. Each pixel acts as an electrically controllable phase plate where the total phase shift exceeds 2π radians as a function of gray level at the Argon laser wavelength of 514.5 nm. Amplitude information is encoded by spatially varying the phase depth of a grating that multiplies the desired phase pattern [6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

, 27

27. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999). [CrossRef]

].

Because the width of the amplitude information is very narrow, we used a small period of 3.66 pixels for the linear phase grating. We have found that this approach for encoding amplitude information depends somewhat on the choice of the grating period and, in our experiments, we adjust this parameter for best results. The Fourier transform of the mask was formed in the focal plane of a 100 cm lens to create the accelerating beam. As in [6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

], the LCD is placed in the front focal plane of the Fourier transform lens and the Fourier transform is formed in the back focal plane. Our approach for encoding amplitude and phase information also generates a strong zero-order beam that was spatially filtered, by physically blocking it with an opaque tip. The output was then recorded with a WinCamD CCD camera at different distances from the focal plane of the Fourier transform system. This camera has 1024× 1024 pixels with pixel sizes of 4.65 × 4.65 microns. So the detector area is approximately 4.76 × 4.76 mm2. The direction of the transverse shift can be reversed using the complex conjugate of the phase mask. The deflection of the accelerating beam can be measured for instance by comparing the results obtained with a given pattern and with the complex conjugated pattern. The origin could then be found as the common center point [5

5. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008). [CrossRef] [PubMed]

, 6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

].

Figure 3 shows our experimentally generated accelerating optical beams with on-demand transverse shapes. Captured images here have been normalized to the maximum intensity value and represented in color for better visualization. The first row of Fig. 3 depicts the simulated optical beam intensities and the blue line shows the desired vertical modulation Y (y) for each case; notice the good agreement between the desired modulation and the beam structure. Also, the experimental (second row) and the simulated (first row) optical intensities at the initial plane are in good agreement with each other. The second to last row shows the experimentally measured optical beam at z = 0, 15, 30 and 45 cm measured from the back focal plane. Notice that the beams do not diffract during this 45 cm range of propagation and that they transversely shift (“accelerate”) under propagation. The last column of Fig. 3 shows the propagation of an 2D Airy beam for comparison; as one can see, our engineered accelerating beams have the same accelerating properties as an 2D Airy beam.

Fig. 3 Experimental realization of accelerating beams with on-demand transverse structures. The first row depicts the simulated optical intensity distributions of accelerating beams with different transverse patterns; the blue line shows the desired y-modulation. The second to fifth row show the experimental intensity distribution propagation of the generated beams at z =0, 15, 30 and 45 cm. Each rectangle depicts a 3.46 cm × 4.76 cm portion of the image. The last column shows an Airy beam for comparison.

In order to quantitatively visualize the parabolic evolution and to compare the transverse translation for all these different experimental beams we measured the location of the maximum of the beam for the different z-planes. We fitted this data with a parabola, ax2 + b, to calculate the transverse constant shift, “b” for each beam. The experimental transverse translation data (relative to the constant shift) are represented in Fig. 4. The results there show that all beams follow the same parabolic trajectory.

Fig. 4 Experimental transverse x-location of the maximum intensity as a function of the axial distance z (modulo transverse constant shift) for all six beams in Fig. 3.

4. Shaped nonparaxial accelerating beams

Analogous to the paraxial case, we found that it is possible to engineer nonparaxial accelerating beams with any on-demand vertical modulation. The advantage of nonparaxial accelerating beams is that they can reach bending angles of almost 180° and can have small features, on the order of a few wavelengths or less. In [15

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]

] it was demonstrated that any 3D nonparaxial accelerating beam can be encoded in a function over the semicircle g(θ), and vice versa. For a given spectral function g(θ) its corresponding nonparaxial accelerating beam is
ψ(r)=0ππ/2π/2g(θ)exp(imϕ)exp(ikru)sinθdϕdθ,
(5)
where m can be any positive real number and u = (sinθ sinϕ, cosθ, sinθ cosϕ) is a unit vector that runs over the unit sphere. By construction, all these waves share the same accelerating characteristics: their maxima propagate along a semicircular path of radius slightly larger than m/k, while approximately preserving their 2D transverse shape up to almost 90° bending angles.

Similar to the paraxial case, we found that we can encode the desired vertical modulation Y (y) in the angular spectral g(θ) of the nonparaxial accelerating beam by using the following relation,
g(θ)=L(ksinθ)cosθ,
(6)
where L(ky) is the spectrum of Y (y) as before. In this nonparaxial case we found that we can achieve a good shaping of the beam if g(θ) is confined to an angle of less than 10° around θ = π/2, for m ≲ 200. The encoding is better if the angular confinement is small and beams with larger m requires more confinement. In this way, the ky components of these beams are paraxial and control the vertical modulation of the beam; but the kx components are nonparaxial and control the steep bending of the beam. Several works [28

28. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011). [CrossRef] [PubMed]

30

30. A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. 38, 2218–2220 (2013). [CrossRef] [PubMed]

] have implemented caustics methods to create beams whose maximum follows different curves; however these methods cannot provide shape-preserving solutions. Therefore, it may be possible to bend our beams in other convex curve rather than a circle if we modify thephase factor exp(imϕ) in Eq. 5 by an engineered phase base on optical caustics, however, this is at the expense of losing the shape-preserving property of the beams.

In Fig. 5 we depict three nonparaxial accelerating beams created using our method. The first row shows the desired modulation Y (v): (a) Mexican hat shape, (b) transverse apodized Bessel shape, and (c) saw shape, for different values of m. The black lines depict the vertical profile of the engineered beam at the first 5 local horizontal maxima; these lines closely overlap with the desired modulation showing that our method truly encodes the desired function in the engineered nonparaxial accelerating beam. The angular spectrum g(θ) is depicted in the second row of Fig. 5 and their extent around θ = π/2 is θ95% ≲ 10°. The third row shows the invariant intensity structure of the constructed nonparaxial accelerating beams; one can clearly appreciate that the beams have the desired form ψX(x)Y (y) where Y (y) is the arbitrary on-demand modulation depicted in the first row. The last row of Fig. 5 depicts the quasi-invariant propagation of our shaped accelerating beams along a semicircular path or radius m/k.

Fig. 5 Nonparaxial accelerating beams with on-demand transverse shapes: (a) Mexican hat shape (m = 50), (b) transverse apodized Bessel shape (m = 100), and (c) saw shape (m = 200). First row, desired target modulation Y (y/k) (red line) and the vertical line profile of the beam at the first 5 local maxima of the horizontal modulation (black lines). Second row, angular spectrum of the desired beam. Third row, intensity of the engineered nonparaxial accelerating beams. Last row, propagation of the accelerating waves.

Acknowledgments

MAB thanks Manuel Guizar-Sicairos for helpful discussions. AR acknowledge financial support from the Consejo Nacional de Ciencia y Tecnología (Grant 182005), the Tecnológico de Monterrey (Grant CAT141). IM acknowledges financial support from Spanish Ministerio de Economía y Competitividad and FEDER funds through grant FIS2012-39158-C02-02.

References and links

1.

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

2.

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

3.

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

4.

M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008). [CrossRef] [PubMed]

5.

J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008). [CrossRef] [PubMed]

6.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

7.

M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]

8.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012). [CrossRef]

9.

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. 37, 1736–1738 (2012). [CrossRef] [PubMed]

10.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012). [CrossRef] [PubMed]

11.

M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New Journal of Physics 15, 013054 (2013). [CrossRef]

12.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012). [CrossRef] [PubMed]

13.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012). [CrossRef] [PubMed]

14.

M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37, 5175–5177 (2012). [CrossRef] [PubMed]

15.

M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]

16.

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). [CrossRef] [PubMed]

17.

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012). [CrossRef] [PubMed]

18.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013). [CrossRef] [PubMed]

19.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]

20.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012). [CrossRef]

21.

A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. Dudley, and F. Courvoisier, “Direct machining of curved trenches in silicon with femtosecond accelerating beams,” J. Euro. Opt. Soc. Rapid publications8(2013). [CrossRef]

22.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). [CrossRef] [PubMed]

23.

C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010). [CrossRef] [PubMed]

24.

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

25.

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009). [CrossRef] [PubMed]

26.

M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009). [CrossRef]

27.

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999). [CrossRef]

28.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011). [CrossRef] [PubMed]

29.

L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19, 16455–16465 (2011). [CrossRef] [PubMed]

30.

A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. 38, 2218–2220 (2013). [CrossRef] [PubMed]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(140.3300) Lasers and laser optics : Laser beam shaping
(230.6120) Optical devices : Spatial light modulators
(350.5500) Other areas of optics : Propagation
(070.3185) Fourier optics and signal processing : Invariant optical fields
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Physical Optics

History
Original Manuscript: October 22, 2013
Revised Manuscript: December 20, 2013
Manuscript Accepted: December 27, 2013
Published: February 6, 2014

Citation
Adrian Ruelas, Jeffrey A. Davis, Ignacio Moreno, Don M. Cottrell, and Miguel A. Bandres, "Accelerating light beams with arbitrarily transverse shapes," Opt. Express 22, 3490-3500 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3490


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References

  1. M. V. Berry, N.L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979). [CrossRef]
  2. G. A. Siviloglou, D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
  4. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008). [CrossRef] [PubMed]
  5. J. A. Davis, M. J. Mintry, M. A. Bandres, D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008). [CrossRef] [PubMed]
  6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]
  7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]
  8. I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012). [CrossRef]
  9. F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. 37, 1736–1738 (2012). [CrossRef] [PubMed]
  10. P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012). [CrossRef] [PubMed]
  11. M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New Journal of Physics 15, 013054 (2013). [CrossRef]
  12. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012). [CrossRef] [PubMed]
  13. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012). [CrossRef] [PubMed]
  14. M. A. Alonso, M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37, 5175–5177 (2012). [CrossRef] [PubMed]
  15. M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]
  16. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009). [CrossRef] [PubMed]
  17. I. Dolev, I. Kaminer, A. Shapira, M. Segev, A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012). [CrossRef] [PubMed]
  18. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013). [CrossRef] [PubMed]
  19. J. Baumgartl, M. Mazilu, K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]
  20. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012). [CrossRef]
  21. A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. Dudley, F. Courvoisier, “Direct machining of curved trenches in silicon with femtosecond accelerating beams,” J. Euro. Opt. Soc. Rapid publications8(2013). [CrossRef]
  22. M. A. Bandres, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). [CrossRef] [PubMed]
  23. C. López-Mariscal, K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010). [CrossRef] [PubMed]
  24. J. C. Gutiérrez-Vega, M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]
  25. T. Čižmár, K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009). [CrossRef] [PubMed]
  26. M. A. Bandres, M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009). [CrossRef]
  27. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999). [CrossRef]
  28. E. Greenfield, M. Segev, W. Walasik, O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011). [CrossRef] [PubMed]
  29. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19, 16455–16465 (2011). [CrossRef] [PubMed]
  30. A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. 38, 2218–2220 (2013). [CrossRef] [PubMed]

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