Open Access
Add to BibSonomyAbstract
We propose a non-paraxial hollow accelerating beam, which is formed by incoherently superposing two well-designed coherent accelerating beams. Very interestingly, this incoherent superposition does not hamper the acceleration dynamics pertaining to the coherent ones, but results in a hollow intensity pattern in the cross section transverse to the circular accelerating trajectory. By a simple optimization, this hollow cross section pattern can be effectively extended to an angle close to 90°. The magnitude and the phase of the angular spectrum of the beam are given followed by a suggested scheme to generate the beam in practice. Such highly self-bending hollow beams may find applications in some fields such as optical manipulation.
© 2016 Optical Society of America
Accelerating wave packets, also known as Airy wave packets, were first introduced by Berry and Balazs [1] in the context of quantum mechanics. Their optical variants, optical Airy beams, were demonstrated theoretically and experimentally in 2007 [2, 3]. Such beams develop some unique dynamic properties like non-diffraction and self-bending [2–5], making them become a rapidly growing field in optics. Since then, many applications based on accelerating dynamics have been demonstrated, including optical manipulation of particles [6–8], generation of curved plasma channels [9], and optical microscopy [10, 11]. It should be noted that Airy beams, as the solutions of the paraxial wave equation, can accelerate only over a finite distance; equivalently their trajectory is limited to small angles, thereafter their dynamic properties are getting worse and worse and finally break down. To remove this restriction, non-paraxial accelerating beams (NABs) were introduced theoretically and experimentally [12–14]. Unlike their paraxial counterparts, the NABs, as the exact solutions of the Helmholtz equation (HE), can bend themselves to a large angle close to 90°, while preserving shape during propagation. In addition, the NABs can travel along different curved trajectories such as circles [12], ellipse and parabola [13] and other fairly arbitrary trajectories [15]. Moreover, their transverse modes are not necessarily limited to Airy function. Instead, they can be described by some special functions. For example, circular accelerating beams may exhibit a transverse mode described by a combination of associated Legendre function and spherical Bessel function, or spheroidal wave functions [15, 16]. Recently, Bessel-like accelerating beams were suggested theoretically and realized experimentally [17, 18]. Of particular interest are their higher order modes, which have a hollow intensity profile, i.e., donut-like shape, during their self-accelerating propagation [19, 20] and were observed to cause spiraling of micro-particles in optical trapping [20]. Like Airy beams, hollow accelerating beams so far are subject to the paraxial limit. In many applications, non-paraxial optical fields are of practical importance. Then a question arises naturally: Can we generalize hollow accelerating beams to the non-paraxial domain as we do in the extension of Airy beams to NABs?
Here we report a non-paraxial (incoherent) hollow accelerating beam traveling along a circular path, following recent demonstration of incoherent Airy accelerating beams by Lumer et al. [21]. Their work showed that incoherent Airy accelerating beams can accelerate up to the same extent as their coherent counterpart with similar parameters even at comparatively small transverse coherence length (of the order of a wavelength). The intensity pattern mode occurring in their two-dimensional incoherent accelerating beams is still an Airy-like one, which is not suitable to generate hollow accelerating beams. To obtain a hollow accelerating beam, we choose to superpose two statistically independent (coherent) accelerating modes. The two coherent modes have specially modulated angular spectra and accelerate along the same circular path. We show that the incoherent beam thus obtained shares the same acceleration trajectory as that of each single mode as it should be; and its intensity, the sum of the intensities of two coherent modes, exhibits a hollow intensity profile. The word ‘hollow’ here means zero intensity at center in the cross section transverse to the circular accelerating path. Although the beam acquires a non-paraxial accelerating behavior, showing a large bending angle close to 90°, this donut-like intensity pattern is established at the expense of not being shape-preserved. The paraxial properties of previously demonstrated hollow accelerating beams may limit their uses in some applications. Our incoherent non-paraxial hollow accelerating beams, on the other hand, may be useful for some applications like particles manipulation.
We begin with the angular spectrum representation of the electric field of a + z-propagating wave packet as
Here A(θ, ϕ) is the angular spectrum of the field E(x), k = (ksinθcosϕ, ksinθsinϕ, kcosθ) is the wave vector with k denoting the wavenumber, and a harmonic time dependence exp(−iωt) is implied. Since Eq. (1) is the exact solution of the Maxwell equations, the transversity of the field E(x) requires that k∙A = 0. As a result, the angular spectrum function A(θ, ϕ) accepts a general form as A(θ, ϕ) = eθA1 (θ, ϕ) + eϕA2(θ, ϕ), where (eθ, eϕ) are two unit vectors along polar angle directions, and A1 and A2 are two arbitrary (scalar) functions of (θ, ϕ). Our objective is to look for a solution of A(θ, ϕ) that exhibits a non-paraxial acceleration along a circular path of radius R0 while maintaining a hollow (donut) transverse intensity pattern during propagation. The method to be used is similar to what we adopted in our previous work of generating curved optical tubes like torus and knot by piecing together a continuum of suitably transformed (annular) focal spots in a 4Pi focusing system [22]. We note that a hollow beam accelerating along a circular path is equivalent to an optical torus with one half cut off. However, in a 4Pi focusing system, the polar angle θ takes the value from 0 to π; while in the problem considered here it runs from 0 to π/2. Despite of this distinction, we may handle the problem first in a 4Pi focusing system and then, after obtaining the solution of A(θ, ϕ), cut off the θ > π/2 part so that the method can be completely transplanted from there. In [23–25], the translation technique was also employed to obtain some curve-like focusing fields in a single lens system. Those focused fields were created by translating a series of fundamental focal spots (as a building unit) δE0(x) to points of a given curve. For the focusing of a plane wave, δE0(x) is a small spherical focal spot centered at the focus with the corresponding angular spectrum δA0 = 1 (omitting the polarization vector u). Denote by δE(x, x0) the translation of δE0(x) from the focus to a given point x0 on the curve. The angular spectrum of δE(x, x0) is simply δA(x0) = exp(−ik∙x0)δA0(θ, ϕ). Thus the angular spectrum A(θ, ϕ) of the whole field is the integral of δA(x0) (may be multiplied by a dynamics phase factor) over the whole curve. Clearly, the focusing fields obtained in such a way exhibits a curve-like structure. For the generation of a hollow beam accelerating along a circular path of radius R0 considered here, the natural choice of the building unit δE0(x) is a annular focal spot, the angular spectrum of which has a vortex phase of charge m, i.e., δA0(θ, ϕ) = uexp(imϕ). The curve to be covered by these building units transformed is of course a semicircle in some plane containing the z axis, say, xz-plane. However, the method of translation used above fails when we attempt to construct such a hollow beam by moving the building units to cover this semicircle bit by bit. The reason for this is that an annular focal spot δE0(x) has a symmetry axis on which the intensity vanishes, while the cross section of the expected hollow beam also has a local symmetry axis, the tangent vector at each point of central-axis curve. Only when the translated annular focal spot has its symmetry axis coincide with the tangent vector, do we obtain a hollow (donut-like) beam. This means that, before translation, we need a rotation operation so that the resulted building unit δE(x, x0) has its symmetry axis along the tangent vector direction as depicted in Fig. 1.
Fig. 1 Sketch of a hollow accelerating beam and the coordinate transforms involved in constructing it.
Mathematically, the above procedure is expressed as follows. Let x0(ϕ0) represent the semicircle parameterized by ϕ0 taking value from −π/2 to π/2. As shown in Fig. 1, the building unit δE0(x) is to be sent to the point x0(ϕ0) for a given ϕ0 with tangent vector t(ϕ0). As stated above, we need first perform a rotation R so that R(δE0(x)) is orientated along t(ϕ0). Obviously, the required rotation is the one around the y-axis through an angle ϕ0. After rotation, δE0(x) is translated to x0(ϕ0) through a translation T, becoming δE(x, x0(ϕ0)) ≡ δE(x, ϕ0). Then we have δE(x, ϕ0) = TR(δE0). In the k-space, R and T induce two corresponding operations acting on δA0(θ, ϕ) [ = uexp(imϕ)] to give δA(ϕ0). For the translation T, the induced action is equivalent to imposing a phase factor exp[−ik∙ x0(ϕ0)], while the counterpart of R(ϕ0) is just the inverse R(−ϕ0) since the scalar product k∙x is invariant under the orthogonal transform R [see Eq. (1)]. Then δA(ϕ0) = exp[−ik∙ x0(ϕ0)]R(−ϕ0)δA0(θ, ϕ). Under the action of R(−ϕ0), on one hand the arguments (θ,ϕ) of δA0 are brought to the new angular variables (θ1,ϕ1), inducing a map fR: (θ,ϕ)→(θ1,ϕ1) = fR(θ,ϕ); on the other hand, the components of δA0(θ, ϕ) are transformed into the linear combination of each other. To facilitate the discussion, we ignore the effect of the latter and let δA0(θ, ϕ) be polarized along the azimuthal direction, i.e., δA0 ≡ eϕδA0(θ, ϕ) = eϕexp(imϕ) and δA(ϕ0) = eϕexp[−ik∙ x0(ϕ0) + imΦ(θ,ϕ)], where the function Φ(θ,ϕ)is defined through the inverse fR−1: (θ,ϕ)→(Θ,Φ). The map fR−1(θ,ϕ) is actually induced by the rotation R(ϕ0), whose form is determined as follows. When the rotation R(ϕ0) is applied, the point (u,v,w) = (sinθcosϕ, sinθsinϕ, cosθ) on the unit sphere is sent to (sinΘcosΦ, sinΘsinΦ, cosΘ), which is expressed in the matrix form as
After some algebra, it is shown thatHere the notation ‘arg’ means taking the argument of complex numbers. As indicated above, the angular spectrum A(θ, ϕ) of the whole field would be the integral of δA(ϕ0) over the accelerating path, i.e. a semicircle of radius R0. However, we have to add a dynamic phase factor exp(iβR0ϕ0) to guarantee the propagating property of the field, where β is a propagating constant. Furthermore, we use an extra factor α(k, ϕ0) for the purpose of further modulation. Combining all of these factors, we finally reach the angular spectrum A(θ, ϕ) for the generation of the circular accelerating beam, which isHere, the variable Φ is expressed as a function of (θ, ϕ) through the relation (3). As mentioned early, the angular spectrum A(θ, ϕ) thus obtained is a one in the 4Pi focusing system, where the polar angle θ runs from 0 to π. To conform to the case of accelerating beam here, we simply cut away the θ > π/2 part. However, this operation seriously weakens the focusing ability compared to the 4Pi case as will be shown below.As an illuminating example, we consider an accelerating trajectory of semicircle contained in the xz-plane centered at the focus with radius R0 = 50/π in units of wavelength λ = 1μm. Furthermore, we set β = 0.5k. For the value of m, we consider two cases: m =+ 4 and −4. We denote by A+(θ, ϕ) and A−(θ, ϕ) the angular spectra with m =+ 4 and −4, respectively and the corresponding fields by E+(x) and E−(x). For the purposes of discussion, all the lengths involved below are measured in units of wavelength λ. In Fig. 2(a), we plot the side view (xz-plane) of the intensity I+ = |E+(x)|2. We see that the intensity pattern consists mainly of two concentric semi-circles, the central axis of which has a radius approximately
Fig. 2 Intensity distributions of I+ = |(E)+(x)|2and I− = |(E)−(x)|2 of two complementary accelerating beams and their incoherent superposition. and (b) Side views of I+ and I−. Cross sections at ϕ0 = 0°, 15°, 45°,75° for I+:(c1)- (c4),I−: (d1)-(d4) and their sum I = I+ + I-: (e1)-e4). All the lengths are in units of wavelength.
equal to the prescribed value R0( = 50/π), implying clearly that the beam is accelerating along the prearranged trajectory—a semi-circle . The non-paraxial property is guaranteed by the fact that the two rings have appreciable intensity even at large angles close to 90°. The donut-like intensity pattern is observed in the region inside the two rings where, especially on the central axis line, the intensity value is much smaller than that on either of the semi-circles. Such a pattern is in accordance with what we expect from a non-paraxial hollow accelerating beam, at least in the xz-plane. We notice that the beam is obviously not shape preserving. The spacing between the two rings at large angles is much wider than the value at small angles while the intensities on the circles decrease with increasing the angle. The side view of the intensity I− = |E−(x)|2 is shown Fig. 2(b), which is quite similar to I+. Compared to the 4Pi case [22], where the shape preserving can be easily obtained, the lack of backward propagating spectral components here downgrades the quality of the beam. This becomes worse when we check the cross section (ϕ0-plane) intensity pattern. Figures 2(c1)-2(c4) display four slices respectively at ϕ0 = 0°, 15°, 45°,75° of three dimensional volume intensity data I+. Here, we skip the intensity patterns in the ϕ0 <0 planes since they are symmetrical (about the y-axis) to those of the corresponding ϕ0 > 0 planes, that is, I+(ρ, ϕ0, y) = I+(ρ, −ϕ0, −y). This can be established from Eqs. (1) and (4) by direct calculation. We see that at ϕ0 = 0° and 15°, each cross section intensity exhibits an annular pattern with main ring centered at ρ = R0 surrounded by a series of side annulus, assembling what a higher-order Bessel beam has. Thus a well-defined donut shape is obtained both at ϕ0 = 0° and at ϕ0 = 15° planes. At ϕ0 = 45°, however, the cross section pattern breaks down into a semi-ring opening upward. At ϕ0 = 75°, this breaking becomes worse. As a result, the presence of intensity pattern breaking at large angles rejects the possibility of hollow accelerating beam in non-paraxial regime, at least for the angular spectrum A+(θ,ϕ) thus constructed. How about A−(θ,ϕ)? In Figs. 2(d1)-2(d4), the corresponding cross section patterns of the intensity I− are shown, where the intensity pattern breaking is also observed at large angles. However,we see that I− at large angles shows symmetry (about y-axis) to I+. In view of this, the field E−(x) may be regarded as the complementary field of E+(x) in some sense, which suggests that we may form a non-paraxial hollow accelerating beam by incoherently superposing these two fields. The reason for incoherent superposition lies in the fact that under tight focusing, two input illuminations with orthogonal polarized states will produce two focused fields that are coherent, leading to unexpected intensity pattern. However, for two (completely) incoherent illuminations, the intensity of the focused field is simply the sum of their respective focusing field intensities. With this in mind, we combine the two fields E+(x) and E−(x) incoherently to obtain a new field that we denote by E(x) with intensity I(x) = I+(x) + I−(x). Figures 2(e1)-2(e4) show the cross section patterns of I(x) at ϕ0 = 0°, 15°, 45° and 75°, respectively.
We see that the pattern has been complemented and a donut-like pattern, although somewhat irregular, is created. According to the cross section patterns and their symmetry (about the y-axis) between ϕ0 > 0 and ϕ0 < 0, we can infer that a (incoherent) hollow accelerating beam is formed, which can bend itself up to an angle at least larger than 75°. A criticism of this accelerating beam may be its uniformity in intensity along ϕ0. From Figs. 2(e1)-2(e4), we see that the energy in the vicinity of donut at ϕ0 = 75° is much smaller than that at ϕ0 = 0°. To quantify this uniformity, we introduce the averaged cross section intensity Iavg(ϕ0) that is defined as
Here D is the region of interest and S(D) is its area. For the region of interest D, we set it to be a square of 9.37 × 9.37 centered at (ρ, y) = (R0, 0), as indicated by the dashed box in Fig. 2(e4). In Fig. 3, the averaged cross section intensity Iavg (dashed line: uncorrected) is plotted as a function of ϕ0. Overall, Iavg shows a decreasing trend with increasing the values of ϕ0. At ϕ0 = 45°, Iavg has decreased to about 0.62 relative to Iavg (ϕ0 = 0°), while at ϕ0 = 75°, Iavg further goes down to 0.42. In the worst situation where ϕ0 = 85°, Iavg ~0.3. If a uniform accelerating beam in the sense of average intensity is desired, something must be done. The most straightforward method is to let the magnitude |α| of the modulating factor α appearing in the integral (4) increase as ϕ0 increases. The magnitude |α| simply determine the contribution of the angular spectrum to the fields at ϕ0 plane. In the following, we adopt a simple but working iterative method to improve the intensity uniformity by modulating |α|. At each iteration, we simply assign 1/[Iavg (ϕ0)]1/2 to |α| and then carry out the next step. The process terminates if the tolerance or the maximum number of iterations is reached. The curve with squares in Fig. 3 gives the resulting magnitude |α |. Note that we require |α | with argument ϕ0 only within ϕ0 < 85° to be adjusted, that is, |α(ϕ 0≥ 85°)| = 1. Larger values of ϕ 0 will cause |α(ϕ 0≥ 85°)| to have a very much large value. With this |α(ϕ0)|, the corrected Iavg (solid line in Fig. 3) exhibit a good uniformity within ϕ 0≤ ~70°. Even at ϕ 0 = 75° and 85°, Iavg ~0.9 and 0.65, respectively, indicating a large improvement compared to the uncorrected one.Fig. 3 Average intensities with magnitude correction (solid) and without magnitude correction (dashed) as a function of ϕ0. The line with squares describes the magnitude correction factor |α|.
Given thisϕ 0 dependent magnitude |α(ϕ0)|, we obtain finally the magnitude and phase of the angular spectrum A+(kx, ky) (neglecting the polarization vector eϕ), as shown in Figs. (4a) and (4b), where the corresponding quantities are limited to a disk of radius k. The angular spectrum A−(kx, ky) of the complementary field E−(x) is related to A+(kx, ky) through the relation A−(kx, ky) = [A+(kx, −ky)]*, which follows from the integral (4) by a change of variables. The cross section intensities of the beam at ϕ 0 = 0°, 15°, 30°, 45°, 75°, and 80° are plotted in Figs. (4c)-(4h). We see that at large angles (ϕ0 = 75° and 80°), the corrected beam shows larger values of cross section intensity over those of the uncorrected beam in Fig. 2. As ϕ 0 increases from 0° to 80°, we see that the pattern shape evolves from an annulus at small angles into irregular form at large angles. Although irregular, the shape is still hollow in the sense that the intensity is zero near the center. Obviously the shape invariance is not linked to the accelerating beam considered here. In Fig. (4i), the side view (xz-plane) of the beam intensity is displayed, where the accelerating dynamics and zero intensity at the central axis are easily verified. Furthermore, we see that at large angles the intensity has appreciable values, suggesting that the uniformity along the ϕ0 direction is improved. Finally we give the three-dimensional intensity visualization of the beam as illustrated in Fig. 4(j). As predicted, we see clearly that a non-paraxial hollow accelerating beam like a half-torus is created accompanied by some hyperboloid-resembling side lobes on the inner side of the torus. In the integral (4), we notice that there is a phase gradient along the circular accelerating curve, implying that the energy flow of the fields may also follow the circle. This property of energy flow may be of much interest in some applications like optical manipulation.
Fig. 4 (a) and (b) Magnitude and phase of the angular spectrum (A)+(kx, ky) of one of two complementary coherent beams used to form an incoherent hollow accelerating beam.(c)-(h) Cross section intensity patterns of the resulting incoherent accelerating beam taken at ϕ 0 = 0°, 15°,30°,45°,75°, and 80°. (i) Side view of the incoherent beam. (h) 3D view of the incoherent beam.
In conclusion, we have proposed a hollow accelerating beam formed by incoherently superposing two well-designed (coherent) accelerating beams complementary to each other. On one hand, the obtained beam has a non-paraxial accelerating property, following its way along a circular trajectory up to an angle close to 90°; on the other hand, the cross section (transverse to the accelerating curve) intensity exhibits a hollow pattern at the cost of not being shape-invariant although at large angles, the cross section intensity pattern becomes an irregular donut. Furthermore the presence of the phase gradient along the circular trajectory may cause the energy to flow along the curve. These properties may help such proposed hollow accelerating beams find their applications in optical trapping: confining particles to a particular curve and transporting them along the curve. The suggested hollow accelerating beam is to be formed by incoherently combining two coherent accelerating beams. An alternative to such a beam is to use temporal modulation of the phase mask as demonstrated in [21]. Specific to the situation here, we suggest the following process. Noting that the angular spectrum in Eq. (4) is azimuthal polarized, indicating that the input field at the entrance plane of the objective lens is actually azimuthal polarized [26,27]. We may alternatingly impose two phase profiles on a spatial light modulator (SLM) to generate the desired amplitudes A+(kx, ky) and A−(kx, ky) in sequence (in linear polarization), followed by a polarization converter to convert them into azimuthal polarization. If the SLM switches between the two phase profiles faster than the characteristic response time involved in an application, e.g., optical particles manipulation, the particles will subject to the optical trap of a hollow accelerating beam in an average sense. Although only the circular trajectory is considered here, the creation of other accelerating paths presents no difficulty since the method imposes no limit on the central-axis curve. In addition, we may introduce a more rigorous optimization method to improve the beam’s intensity uniformity and shape regularity. All these issues are worth of further considerations.
Funding
National Basic Research Program of China (973 Program) (2012CB921900); National Natural Science Foundation of China (NSFC) (11474352, 61377008 and 81427802).
References and links
1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 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(21), 213901 (2007). [CrossRef] [PubMed]
4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]
5. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef] [PubMed]
6. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
7. 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(15), 2883–2885 (2011). [CrossRef] [PubMed]
8. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009). [CrossRef] [PubMed]
9. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]
10. T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014). [CrossRef] [PubMed]
11. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014). [CrossRef] [PubMed]
12. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012). [CrossRef] [PubMed]
13. 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(14), 2820–2822 (2012). [CrossRef] [PubMed]
14. 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(20), 203902 (2012). [CrossRef] [PubMed]
15. 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(10), 1736–1738 (2012). [CrossRef] [PubMed]
16. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21(12), 13917–13929 (2013). [CrossRef] [PubMed]
17. I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37(23), 5003–5005 (2012). [CrossRef] [PubMed]
18. J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories,” Opt. Lett. 38(4), 498–500 (2013). [CrossRef] [PubMed]
19. S. Yan, M. Li, B. Yao, X. Yu, M. Lei, D. Dan, Y. Yang, J. Min, and T. Peng, “Accelerating nondiffracting beams,” Phys. Lett. A 379(12–13), 983–987 (2015). [CrossRef]
20. J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5, 12086 (2015). [CrossRef] [PubMed]
21. Y. Lumer, Y. Liang, R. Schley, I. Kaminer, E. Greenfield, D. Song, X. Zhang, J. Xu, Z. Chen, and M. Segev, “Incoherent self-accelerating beams,” Optica 2(10), 886–892 (2015). [CrossRef]
22. S. Yan, X. Yu, M. Li, and B. Yao, “Curved optical tubes in a 4Pi focusing system,” Opt. Express 23(17), 22890–22897 (2015). [CrossRef] [PubMed]
23. S. H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18(7), 6988–6993 (2010). [CrossRef] [PubMed]
24. E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express 19(7), 5833–5838 (2011). [CrossRef] [PubMed]
25. J. A. Rodrigo, T. Alieva, E. Abramochkin, and I. Castro, “Shaping of light beams along curves in three dimensions,” Opt. Express 21(18), 20544–20555 (2013). [CrossRef] [PubMed]
26. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]
27. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
