Fiber-based Bessel beams with controllablediffraction-resistant distance
Paul Steinvurzel,* Khwanchai Tantiwanichapan, Masao Goto, and Siddharth RamachandranBoston University, Department of Electrical and Computer Engineering,
The zero-order Bessel beam supports a wavelength-scalecentral bright spot with the remarkable properties of pro-pagating without diffracting over distances many timeslarger than the Rayleigh range and self-healing behindobstructions or scatterers [1]. This robust, long, narrowpencil of light has been exploited in a number of interest-ing ways, such as forming 2-D optical line traps [2,3],depth sectioning microscopy [4–6], and micromachining[7,8]. The two commonly used methods for generatingBessel-like beams are annular apertures and axicons[1,9]. Computer-generated holograms are effective forcreating Bessel-like beams [10] with the benefit ofenabling very precise and tunable control of the beamparameters [11]. Dynamically tunable Bessel-like beamshave also been demonstrated using liquid core axiconlenses [12,13].Optical fiber-based methods for generating Bessel-like
beams are desirable, not only for the usual reasons thatthey are robust, compact, and provide alignment-free op-eration, but also because they enable remote deliveryand so make practical Bessel beam endoscopic micro-scopy [14]. Analogous to the free-space methods men-tioned above, Bessel-like beams have been generatedin fiber by focusing a ring mode with a lensed fiber tip[15] or by fabricating a microaxicon directly on the fibercore [14,16,17]. These methods, however, provide a lim-ited design space for tailoring the beam parameters andcannot be easily adapted for dynamic tuning. On-axis il-lumination of a large core multimode fiber can provide aBessel-like output via multipath interference [18], thoughwith a strong degree of axial variation in the near fieldand inherently strong wavelength-dependent behavior.Our technique for generating fiber Bessel-like beams
exploits the fact that individual guided modes of a glasscylinder are inherently Bessel-like. We experimentallydemonstrate this by selectively exciting LP0;m modeswith a long period fiber grating (LPG) [19]. By using pro-pagating modes of a fiber that have been shown to beinherently stable [20], we retain a key advantage uniqueto optical fibers—the ability to shape light, in time or fre-quency, via well-known linear and nonlinear waveguideeffects [21]. Choosing the mode order allows one to
effectively choose the Bessel beam cone angle andthereby precisely control the propagation distance andwidth of the center spot. Since the fiber gratings used togenerate our Bessel-like beams are wavelength-specificdevices, our demonstration points to the feasibility ofwavelength-dependent excitation of different mode or-ders, leading to a fiber device whose output has a tunablepropagation-invariant length scale. This leads to the intri-guing possibility of constructing endoscopes that havetunable depths of focus for the excitation beam.
We UV write 1–1:5 cm long LPGs in a H2-loadedhigh-NA single-mode fiber (SMBD0980B, OFS) using afrequency-doubled Arþ laser and a translation stage
Fig. 1. (Color online) (a) Schematic diagram of a LPGcoupler generating a high-order cladding mode. (b) Experimen-tally measured grating spectrum generating a LP0;10 mode (leftaxis) and output spectrum of probe laser (right axis). (c) Experi-mental setup for characterizing Bessel-like beam propagationand experimentally recorded near-field image of the LP0;10mode.
where the exposure dosage is optimized to provide>99%conversion efficiency from the fundamental mode. Wevary the grating pitch from 100 to 200 μm, which yieldscoupling to LP0;5 − LP0;15 modes around 1:064 μm. A re-presentative grating spectrum and the spectrum ofthe probe laser used for characterization are shown inFig. 1(b). As shown in Fig. 1(c), we launch light froma laser diode tuned to the grating resonance, imagethe fiber output on a 12 bit InGaAs CCD camera, andrecord the intensity pattern as the fiber is moved outof the focus of the imaging objective. The camera re-sponse is linear, but we effectively get a range of only9bits with the uncooled sensor, so with the exceptionof the LP0;5 mode data, we take scans both at low power,where maximum intensity across the scan is unsaturated,and higher power, where we can discern more details atlong propagation distances, and scale and stitch the dataaccordingly.In Fig. 2 we show representative contour plots of
x cuts of the recorded intensity profile as a function ofpropagation distance, along with corresponding beampropagation method simulation results and representa-tive raw images from which the experimental data arederived. The modes used for the launch field in the simu-lation are calculated analytically. In addition to the strik-ing agreement we obtain between simulation andexperiment, the data show that changing the mode orderfrom LP0;5 to LP0;15 allows us to tune the maximum pro-pagation distance zmax (defined as the point where thecenter intensity drops to e−1 of the global maximum cen-ter intensity) from 1:44mm down to 0:49mm, while theaverage center spot FWHM varies from 15:1 μm down to3:6 μm. By contrast, the Gaussian-like LP0;1 mode of this
fiber has a FWHM of 2:1 μm and a Rayleigh range of39 μm, 12–36× smaller than zmax.
This dependence on mode order can be qualitativelyexplained with a geometric model. Since a Bessel beamis formed by interfering plane waves on a cone with coneangle θ, it can be shown that tanðθÞ ∼ R=zmax, where R isthe radius of the full beam, and, by the same argument,that tanðθÞ ¼ kt=kz, where kt and kz are the transverseand longitudinal wave vectors, respectively, as shownin Fig. 3(a). For a fiber cladding mode, kz is just the pro-pagation constant β, and kt is the transverse wave vectorin the fiber cladding, where the majority of the opticalpower is guided. As the mode order increases, kt=βincreases, so the effective cone angle increases, and wethus expect both zmax and the center FWHM to decrease,as is the case in Fig. 2.
We may further quantitatively estimate zmax ∼ Rβ=kt,where is R now the fiber radius, and, assuming a J0transverse field dependence, the center intensityFWHM ∼ 2:25=kt. In Figs. 3(b) and 3(c), we compare thissimple model with experiment and simulation for theLP0;8 mode, and do obtain good agreement. The axialvariations are not well suppressed compared to thosegenerated in optimized holograms [11] or microaxicons[14,16,17] but are comparable to other fiber-based meth-ods [15,18]. In our embodiment, this can be improvedby choosing a different NA for the core and claddingand will be the subject of future work. In Figs. 3(d)and 3(e) we compare zmax and the FWHM extracted fromall experimental and simulation data with the geometric
Fig. 2. (Color online) Contour plots for free-space propagationof the LP0;5 (left) and LP0;15 (right) cladding modes. Top rowshows simulation results, middle row shows experimental results,and bottom row shows representative experimental 2-D images atthe positions indicated. Animations of full 3-D experimental dataare in Media 1 (3759KB) (LP0;5) and Media 2 (2467KB) (LP0;15).Bottom row images are on a linear scale normalized to maximumpixel value in each image, whereas animations are normalized tothe maximum pixel value across the entire image stack and scaledto Γ ¼ 0:5 for enhanced contrast.
Fig. 3. (Color online) (a) Schematic diagram for a geometricmodel of a LP0;m mode as plane waves on a cone. (b), (c) Com-parison between experiment, simulation, and the geometricmodel of the evolution of the center spot maximum intensityand FWHM as a function of propagation distance for the LP0;8mode. (d), (e) Comparison between experiment, simulation,and the geometric model of zmax and FWHM as a function ofradial mode order.
model, where the FWHM data are averaged over theinterval ð0; zmaxÞ. We again obtain good agreement inall cases.Changing the cone angle via mode order also affects
the self-healing properties of our fiber-generated Besselbeams. We magnify the Bessel beam to a radius ∼2:5 cmonto a glass slide with metal obstructions, reimage it witha second objective lens, and record the beam profile asthe camera is moved back from the slide. For the LP0;5mode [Figs. 4(b) and 4(c), Media 3, 3649KB], the centerspot does not reform after 23 cm of travel owing to thesmaller cone angle of the beam and so behaves similarto a conventional Gaussian beam that would create ashadow behind an opaque obstruction, at least over thispropagation range. The LP0;15 mode [Figs. 4(d) and 4(e),Media 4, 2972KB] magnified to the same size onto thesame blocking element reforms its center spot ∼7–9 cmbehind the block, illustrating a unique property of Besselbeams and confirming that higher-order modes withmore rings better approximate free-space Bessel beams.To summarize, we have demonstrated that LP0;m fiber
cladding modes behave like diffraction-resistant, self-healing Bessel beams in free space. We showed howthe propagation-invariant range and center spot widthdepend on mode order and explained the trend quan-titatively with a simple geometric model. LPGs are inher-ently sensitive to fiber bends or temperature fluctuations,so a practical embodiment would be written in a double
clad fiber that can stably propagate LP0;m modes [20],while the grating itself would be isolated from perturba-tions. In addition to providing excellent conversion effi-ciency, beam control, and remote delivery, our methodadmits more complex grating and waveguide designs,so we can achieve conversion over large (>100 nm)bandwidth [22], which would then make our methodattractive for endoscopic coherence tomography [14],or we may write multiple gratings in one fiber and gen-erate different mode orders at different wavelengthsyielding a single device with tunable depth of focus.
We thank N. Božinović, P. Mak, and D. Schimpf fortheir contributions to this research. This work was donewith the support of the Office of Naval Research (ONR),grants N00014-11-1-0133 and N00014-11-1-0098.
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Fig. 4. (Color online) (a) Schematic diagram of self-healingfiber Bessel beam experiment. Images show the obstructedLP0;5 mode at the (b) focus and (c) 10 cm behind the obstruc-tion. Equivalent images for the LP0;15 mode are shown in (d) and(e). Animations of full 3-D experimental data shown are inMedia 3 (3649KB) (LP0;5) and Media 4 (2972KB) (LP0;15).(b)–(e) are shown on a linear scale normalized to the maximumpixel value in each image, whereas the animations are normal-ized to the maximum pixel value across the entire image stack,and Media 4 is scaled to Γ ¼ 0:5 for enhanced contrast.
