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Deliverable Report
Deliverable No: D2.12
Deliverable Title: OAM waveguide coupling Grant Agreement number: 255914 Project acronym: PHORBITECH Project title: A Toolbox for Photon Orbital Angular Momentum Technology Project website address: www.phorbitech.eu Name, title and organisation of the scientific representative of deliverable’s lead beneficiary (task leader): Dr. Mark Thompson University of Bristol (UNIVBRIS) Bristol, UK Deliverable table Deliverable no. D2.12 Deliverable name OAM waveguide coupling WP no. 2 Lead beneficiary no. 4 (UNIVBRIS) Nature P Dissemination level PU Delivery date from Annex I Month 36 Actual delivery date 27 September 2013
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D2.12) OAM waveguide coupling: Excerpt from Annex I: Proof-of-concept OAM waveguides equipped with couplers and multiplexers for free-space - waveguide coupling. A short report or scientific publications will illustrate the employed methods and resulting performances. Development of integrated q-plates by liquid crystal infiltration of cylindric capillary cavities. The prototypes quality and the extent to which their perfomances reach our target goals will be assessed and validated by the steering committee. [month 36]
We demonstrated on-chip reconfigurable generation of light’s orbital angular momentum (OAM) in silicon nanophotonic waveguide circuits. The wave-front of the generated beams has been characterized using commercial spatial light modulators (SLM). The vector OAM beams were firstly decomposed into two scalar OAM beams, right-hand circularly polarized (RHCP) OAM beams and left-hand circularly polarized (LHCP) OAM beams [1]. Then, two scalar OAM beams were separately measured using SLM which enables generation of computer-controlled holograms (Figure 1) [2]. For example, when the topological charge of a vector OAM beam was -3, phase holograms with +4 and +2 dislocations inversely converted the LHCP and RHCP-OAM beams into the fundamental Gaussian beams, respectively (as shown in Fig1.(c) and (d)), qualitatively indicating well-defined and high-purity OAM states. Quantitative analysis of pure OAM states and superposition of OAM states would be carried out using SLM in the future.
Figure1. (a) Scanning electron microscopy (SEM) image of an integrated OAM emitter. (b) Computer-controlled holograms with dislocations (here, l=2) for measuring the topological charge of vector OAM beams. When topological charge of the vector OAM beam was -3, holograms with various dislocations l were used to measure the LHCP (c) and RHCP (d) scalar OAM beams. Only when l=+4 and l=+2, two scalar OAM beams are converted into Gaussian beams (highlighted), indicating the emitted vector beams carrying an OAM of -3ħ per photon. Besides the main work carried out in UNIVBRIS, work towards the other goals of the present deliverable has been carried out in UNAP in collaboration with UROM and with other external collaborators. In particular, a theoretical investigation on the feasibility of some kind of integrated q-plates, suitable to couple in a controlled way different polarization and spatial modes of a multi-mode waveguide has been started but is still in progress. On the experimental side, some first attempts of realizing integrated q-plates using liquid-crystal-infiltrated capillaries hitherto have been carried out, but they failed, as we could not obtain clean structures which could be used for wave-guiding. Success in this area will require substantially more resources than those planned for this task within PHORBITECH. The effort on the combination of the OAM and wave-guiding concepts has anyway led to some related theoretical results reported in scientific publications that are delivered here (see attached publications in the following pages). References [1] X. Cai, J. Wang, M."J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J."L. O’Brien, M."G. Thompson, and S. Yu, Science 338, 363 (2012).
[2] M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, No. 5, 35 (2004).
[3] V. R. Almeida and M. Lipson, Opt. Lett. 29, 2387 (2004). [4] L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
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PHORBITECH contribution to this deliverable From the UNIVBRIS partner, this work is presented in part as a contribution to the PHORBITECH project, with efforts from Jianwei Wang, Xinlun Cai, Mark Thompson and Jeremy O'Brien. It should be noted that this work also contains substantial contributions from non-PHORBITECH partners from the University of Bristol (Siyuan Yu and Dave Phillips) and the University of Glasgow (Michael Strain and Marc Sorel). PHORBITECH has also supported the work of Lorenzo Marrucci, Sergei Slussarenko, Vincenzo D’Ambrosio and Fabio Sciarrino. PHORBITECH contributors to this deliverable UNIVBRIS: Jianwei Wang, Xinlun Cai, Jeremy O'Brien and Mark Thompson UROM: Vincenzo D’Ambrosio, Fabio Sciarrino UNAP: Lorenzo Marrucci, Sergei Slussarenko PROTOTYPE'VALIDATION.'The'validated'prototype'is'the'integrated'vortex'beam'emitter'in'the'version'reported' in' the' Science' paper' mentioned' above.' The' quality' of' the' device' has' been' assessed' by' the'PHORBITECH' steering' committee' in' its' meeting' held' in' Glasgow' on' June' 6,' 2012,' after' a' detailed'presentation' of' the' device' features' and' performances.' The' committee' unanimously' judged' that' the'prototype' represents' the' first' implementation' of' a' fully' novel' concept,' and'as' such' it'meets' the' target'goals'set' in'the'proposal'(the'members'of' the'committee'who'are'directly' involved' in'this'work'did'not'participate'in'this'final'assessment).'No'specific'quantitative'assessment'of'the'device'performances'was'carried' out' in' this' evaluation,' because,' given' the' concept' novelty,' there' is' no' previous' technology' to'compare'with.!
Publications included in this deliverable 1. “Light confinement via periodic modulation of the refractive index”, A. Alberucci, L. Marrucci, G. Assanto, New
Journal of Physics 15, 083013 (2013).
2. “Optical vortices in antiguides”, L. Marrucci, N. F. Smyth, G. Assanto, Optics Letters 38, 1618-1620 (2013).
Light confinement via periodic modulation
of the refractive index
A Alberucci
1,4, L Marrucci
2,3and G Assanto
1
1 Nonlinear Optics and OptoElectronics Laboratory, University Roma Tre,Via della Vasca Navale 84, I-00146 Rome, Italy2 Dipartimento di Fisica, Universita di Napoli Federico II, Napoli, Italy3 CNR-SPIN, Complesso Universitario di Monte Sant’Angelo, Napoli, ItalyE-mail: [email protected]
New Journal of Physics 15 (2013) 083013 (20pp)Received 7 May 2013Published 6 August 2013Online at http://www.njp.org/doi:10.1088/1367-2630/15/8/083013
Abstract. We investigate, both theoretically and numerically, light confine-ment in dielectric structures with a transverse refractive index distribution pe-riodically modulated in the longitudinal coordinate. We demonstrate that lightcan be guided even in the balanced limit when the average refractive index con-trast vanishes in the direction of propagation, a dynamic trapping phenomenonanalogous to the Kapitza effect in quantum mechanics. Finally, with referenceto segmented waveguides with an unbalanced index modulation, we address theinterplay of dynamic and static confinements.
4 Author to whom any correspondence should be addressed.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal
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New Journal of Physics 15 (2013) 0830131367-2630/13/083013+20$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
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Contents
1. Introduction 2
2. Theoretical background 3
2.1. Dependence of the photonic Kapitza effect on the modulation profile . . . . . . 52.2. Physics of the effective index profile . . . . . . . . . . . . . . . . . . . . . . . 5
3. Quasi-modes 6
3.1. Gaussian trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. Numerical results 9
4.1. Dependence on the index profile U (x) . . . . . . . . . . . . . . . . . . . . . . 134.2. Dependence on longitudinal modulation f (z) . . . . . . . . . . . . . . . . . . 144.3. Dependence on input section . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5. Interplay of dynamic and static confinement 17
6. Conclusions 19
Acknowledgments 19
References 20
1. Introduction
Periodic structures are essential in photonics, as they support various phenomena that are usefulin controlling and managing light signals and beams. The periodic modulation of the refractiveindex in the propagation direction allows the realization of, for instance, filters and coatings,diffractive and coupling gratings, distributed feedback reflectors and resonators, and quasi-phasematching for parametric generation [1–5]. An index periodicity in the transverse plane is at thebasis of waveguide arrays for discrete diffraction [6]; periodic arrangements in two or threedimensions entail the realization of photonic crystals [7].
In this paper, we focus on refractive index structures featuring a bell-shaped profile inthe transverse plane and periodically modulated along the direction z of light propagation.A longitudinal modulation on light propagation was previously addressed in the case ofwaveguide arrays with reference to the linear refractive index [8] and the nonlinearity[9, 10]; experiments were reported on nonlinear inhibition of tunneling [11] as well as onquasi-Bloch oscillations [12]. Regarding single-waveguide structures, similar geometries werestudied in the context of segmented waveguides, that is, light-confining sections alternatingalong z with transversely homogeneous portions where light could diffract [2, 13–16]. Itwas demonstrated that, in most cases, segmented waveguides can be modeled and operate ascontinuous waveguides with a transverse index profile given by the spatial averaging along themodulated direction z [17].
The aim of the work hereby is to extend and generalize the treatment of z-periodicoptical waveguides to those with a zero mean longitudinal modulation, that is, an indexmodulation of alternating signs along z. In the case of zero mean contrast, one would intuitivelyexpect light to diffract owing to the alternation of index wells (focusing regions) and ofindex barriers (defocusing regions) of finite transverse extent. However, in the context ofquantum mechanics, it is understood that the fast-scale particle motion driven by a rapidly
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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oscillating potential (in time) gives rise to an additional time-independent effective potentialthat can, in turn, induce a slow-scale dynamics of the particle [18]. This effect, known asthe Kapitza effect from the original work of the Russian Nobel laureate on a mechanicpendulum [19], was exploited by the Nobel laureate Paul [20] to realize particle traps. Sinceparaxial light propagation in the harmonic regime and the motion of quantum particles areboth governed by the Schrodinger equation with the spatial coordinate z taking the role of anequivalent time [21], the analogy between optical propagation and the Kapitza effect in quantummechanics suggests the insurgence of an effective refractive index distribution even in zero-mean-contrast periodically segmented waveguides, a distribution uniform along z and possiblyable to transversely trap light. If an oscillating potential in quantum mechanics gives rise to asteady effective potential proportional to the square of the gradient of the periodic modulation[18, 22], in optics a bell-shaped transverse distribution of a z periodic refractive index results ina double-humped trapping well. Optical analogies of the Kapitza effect have been theoreticallydiscussed, limited to waveguide arrays [23] or transversely invariant dielectric stacks [24]. In thefirst case, discrete diffraction is inhibited by a longitudinal modulation of the coupling strengthbetween standard waveguides [23]. In the second, a (rather large) periodic modulation of thedielectric susceptibility can suppress diffraction of transverse-magnetic waves by minimizingthe longitudinal field component [24]. At variance with these previous reports, the approach wediscuss here entails light trapping solely via a photonic equivalent of the Kapitza effect: waveconfinement is polarization independent and scales fully with wavelength.
We consider transversely limited (waveguide-like) index structures periodically modulatedand balanced along z, i.e. with a vanishing mean index contrast in the propagation direction.Initially we treat the problem analytically and later verify our theoretical results using the beampropagation method (BPM), the latter also allowing us to assess the validity of the effectivepotential approximation. Finally, we extend our study to unbalanced periodic waveguides witha nonzero mean index contrast, addressing the interplay between static confinement due to theaverage distribution (bell shaped in the transverse plane) and dynamic Kapitza-like confinementstemming from the modulation.
2. Theoretical background
For the sake of simplicity we refer to planar guided-wave structures, i.e. (1+1)D geometries.Under the paraxial approximation, lightwave propagation is governed by the Schrodinger-likeequation
2ik0n0@ A@z
+@2 A@x2
+ 2n0k201n(x, z)A = 0, (1)
where A is the field envelope, n0 is the unperturbed refractive index, k0 is the vacuumwavenumber and 1n is the index profile in the transverse space, i.e. along x . The use ofequation (1) to describe light dynamics implies discarding the back-reflected wave components(i.e. propagating along �z), always present in a periodic structure owing to longitudinal indexvariations. Such approximation is usually valid unless Bragg resonances occur [13].
We take the index well to be factorizable, i.e. 1n(x, z) = f (z)U (x) and the ansatz
A(x, z) = �(x, z) eik0U (x)R z
z0f (z0) dz0
. (2)
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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We normalize to unity the maximum absolute value of f (z). By substituting equation (2) intoequation (1) we obtain [18]
2ik0n0@�
@z+
@2�
@x2+ 2ik0 g(z)
✓@U@x
@�
@x+
12
@2U@x2
�
◆� k2
0 g2(z)✓
@U@x
◆2
� = 0, (3)
where we introduce g(z) = R zz0
f (z0) dz0. By dealing with a periodic f (z) = f (z + 3), we can
expand it in a Fourier series, f (z) =P1l=�1 fl exp (2⇡ ilz/3). We assume f = 1
3
R 3
0 f (z) dz =f0 = 0, i.e. f (z) with a zero mean. It is straightforward to get g(z) = G(z) � G(z0), whereG(z) = 3
2⇡ i
P1l=�1
fll exp (2⇡ ilz/3). We note that g = 0 only if G(z0) = 0. Applying the
Fourier transform operator to equation (3), since the spectra of both g(z) and its square g2(z)consist of series of Dirac functions, equation (3) relates the spectrum �(�) = R
� exp (i�z) dz toits replicas centered in �m = 2⇡m/3 (m 2 Z). If the width 1� of � is smaller than �1, sidebandcopies of � do not interact with the baseband (more precisely their overlap is negligible because� is necessarily nonzero for all the spatial frequencies �). To derive a quantitative condition, wechoose a bell-shaped profile for field �. After approximating the wavefunction with a Gaussianof waist w0, the anti-aliasing condition reads
w0 >3
⇡. (4)
Equation (4) can be intuitively interpreted in terms of alternating focusing/defocusing regions:narrow wavepackets (beams) undergo strong spreading when f (z) < 0; when light reaches thenext focusing region along z, the beam width is too large and lensing is insufficient to trap andconfine the field. When equation (4) is satisfied, the application of the extra condition g = 0 toequation (3) yields
2ik0n0@�
@z+
@2�
@x2� k2
0g2(z)✓
@U@x
◆2
� = 0. (5)
The term �g2(z)�
@U@x
�2plays the role of an effective index difference 1neff. The condition
g = 0, easily satisfied by a proper choice of z0, is required to zero the contribution from thethird term in equation (3) (the role of the initial phase of the potential is well addressed in [25]).In physical terms, we expect that the quasi-modes of equation (1), if they exist, feature phasefronts with a periodically varying curvature (see equation (2)), as determined by the focusing ordefocusing character of the local index profile [17]. Equation (5) predicts that an effective indexdistribution
1neff = � 32
4⇡ 2
1X
l=�1
fl f�l
l2
!✓@U@x
◆2
(6)
acts on the wavepacket. According to equation (6) the effective photonic potential isproportional to the square of the modulation period 3. Thus, for short 3, the longitudinal indexvariations are too fast and light is not affected by the modulation. When 3 gets longer thedynamic effects increase indefinitely, clearly an unphysical result: in fact, equation (6) is validonly if condition (4) is satisfied. Summarizing, we expect dynamic (Kapitza-like) trapping to bemaximum for a finite value of the modulation period 3max.
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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Figure 1. (a) Plot of f (z) versus the normalized propagation coordinate z/3for a flat-top modulation. (b) Behavior of g2 (given by equation (8)) for f (z)as plotted in (a) versus d/3; solid and dashed lines correspond to the seriescomputed up to l = 50 and 1, respectively.
2.1. Dependence of the photonic Kapitza effect on the modulation profile
Equation (6) states the dependence of the effective potential 1neff on the longitudinalmodulation f (z) via the Fourier coefficients fl . To address such dependence, we first considerthe simplest case of a sinusoidal modulation f (z). Equation (6) yields
g2 = 32
8⇡ 2, (7)
and it is also z0 = 3/4 in order to ensure G(z0) = 0. Next, we consider a flat-top modulation,featuring two segments of length d with opposite amplitudes per period (see figure 1(a)). Weobtain
g2 = 232
⇡ 4
1X
l=1
1l4
sin2
✓⇡l2
◆sin2
✓⇡ld3
◆. (8)
When d = 3/2, the modulating function f (z) is a square wave; then equation (8) becomesg2 = �
232/⇡ 4� P1
m=0 1/(2m + 1)4. The behavior of g2 is plotted in figure 1. The last step isfinding out how the initial section z0 depends on d and 3. By setting G(z0 = 0) we get
z0 = 3 � d2
+ m3
2(m 2 Z). (9)
2.2. Physics of the effective index profile
The appearance of the effective index contrast 1neff can be physically understood from thesuperposition principle, in analogy to classical confinement being explained as a transversephase modulation due to an inhomogeneous refractive index. Taking the ansatz
A = c eik(z)x eiR z
0 E dz0(10)
with c an arbitrary constant, expanding 1n in a power series truncated to the linear term (i.e.1n(x, z) = 1n0(z) + 1n1(z)x), and substituting (10) into equation (1) we get
k(z) = k0
Z z
01n1(z0) dz0, (11)
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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E(z) = k2(z)2k0n0
� k01n0(z), (12)
where we choose k(0) = 0 in equation (11). Owing to the periodicity of 1n(z), the field Awill be periodic as well; thus we can set A(3) = A(0) ei'. The phase difference ' accounts forthe Kapitza effect acting on the wavepacket. The solution of equations (11) and (12) and itssubstitution in equation (10) provide
' =Z 3
0E dz0 =
Z 3
0
✓k2(z0)2k0n0
� k01n0(z0)◆
dz0. (13)
The integral of 1n0 vanishes due to the assumption of a zero mean for f (z). Conversely, theintegral along z of the equivalent kinetic energy k2/(2k0n0) is nonzero, as the light momentumk is periodically modulated by the local index gradient 1n1 = f (t)@U/@x . Summarizing, forthe phase delay ' over a single period we find the expression
' = k0
2n0
✓@U@x
◆2 Z 3
0g2(z) dz. (14)
Equation (14) predicts the same effective index profile of equation (5). We conclude that theKapitza effect is due to the longitudinal modulation of the equivalent kinetic energy induced bythe periodicity [26]; for well-chosen transverse profiles U (x), the transverse phase modulationcan compensate diffraction. Noteworthy, in obtaining equation (14) we consider a periodicA, the latter assumption clearly invalid for long modulation periods as diffraction can induceappreciable changes in wave amplitude, consistently with condition (4).
3. Quasi-modes
In this section, we want to examine those cases in which the effective potential can support lighttrapping. The easiest configuration corresponds to bell-shaped U (x): for instance, we can takesuper-Gaussian functions U (x) = U0 exp (�x2p/w
2pU ) (p 2 N). In this case, the effective index
1neff takes an inverted W-like transverse profile
1neff = �32 p2U 20
⇡2
1X
l=�1
fl f�l
l2
!x2(2p�1)
w4pU
exp
�2x2p
w2pU
!
. (15)
The effective refractive index profiles 1neff are drawn in figure 2 for three values of theparameter p; for large p the super-Gaussian tends to a buried slab-like waveguide. As is wellknown, this structure does not support guided modes because the evanescent profiles at |x | ! 1lead to a leaky behavior even in the central lobe of the guide [27]. Such types of geometries inoptics support leaky quasi-modes (corresponding to metastable states in quantum mechanics),actually consisting of a superposition of diffractive (free) modes [27].
We can compute the quasi-modes of the structure by using an index profile with the centrallobe only, i.e. retaining the index distribution between the two local minima (see figure 2(b));on the external regions we set the effective index to its global minimum. This approach worksuntil the overlap of the calculated mode with the external portions of the effective index wellcan be neglected.
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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Figure 2. (a) Profile of U (x) versus x and (b) corresponding effective refractiveindex contrast 1neff; we normalized the quantities to their maximum absolutevalue. The different lines in each panel correspond to p = 1 (blue), p = 4 (black)and p = 8 (red), respectively.
3.1. Gaussian trap
Let us take p = 1 in equation (15) and a sinusoidal oscillation for f (z). In this case, theeigenvalue problem stemming from equation (5) can be scaled with respect to the normalizedcoordinate x 0 = x/wU . Taking the ansatz �(x, z) = u(x) exp (i1�effz), equation (5) provides
2k0n0w2U1�effu = @2u
@x 02 � k203
2U 20
2⇡2F(x 0)u, (16)
where F(x 0) = x 02 e�2x 02rect
⇥x 0/(2xm)
⇤+
⇥H(x 0 � xm) + H(�(x 0 + xm))
⇤x2
m e�2x2m, with xm the
position of the local minimum of 1neff for positive x 0 and H the Heaviside step function.According to equation (16), in the normalized reference system the shape of the mode does
not depend on the width wU of the index profile, if U (x) is Gaussian; moreover, the effectiveindex variation 1Neff = 1�eff/k0 depends quadratically on wU .
The modal profiles computed via a standard numerical procedure are plotted in figure 3.As predicted, the larger the modulation period 3, the stronger the transverse confinement. Atthe same time, light trapping via the Kapitza effect improves as the depth of the index well U0
increases; unlike in conventional waveguides, the trapping strength is proportional to the squareof U0. The trend of 1Neff versus 3 is graphed in figure 4(a). Firstly, we note that 1Neff isnegative, corresponding to a propagation constant for mode u that is lower than n0k0. Secondly,1Neff decreases as 3 grows, i.e. as confinement gets stronger. The sign and the trend of 1Neff
are opposite to the case of static confinement, in both continuous and segmented waveguides.Finally, the absolute value of 1Neff becomes larger as U0 increases. Figure 4(b) shows the
modal width wmode = 2qR
x2|�|2 dx/R |�|2 dx versus 3. For short periods the transverse size
goes to infinity due to the absence of a trapping potential, whereas for large 3 the width tendsto the diffraction-limited value �. Equation (16) shows that the focusing strength depends on theproduct U03, as confirmed by the numerical results in figures 3 and 4.
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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Figure 3. Numerically computed fundamental quasi-modes for p = 1 versusx/wU when (a) U0 = 0.05, (b) 0.1 and (c) 0.5, respectively. The curvescorrespond to period 3 = 100, 50, 30, 20, 10 and 8 µm from the narrowestto the largest profile; in panels (a) and (b) the results for 3 = 8 and 8, 10 µmrespectively, are not shown as they are larger than 130, the size of our numericalwindow in normalized units. Here � = 1 µm.
Figure 4. (a) 1Neff versus 3 and (b) width of the fundamental mode normalizedto wU for U0 = 0.5 (blue line), U0 = 0.1 (black line) and U0 = 0.05 (red lines);in (a) and (b) the well depth decreases from the bottom to top curves. The dashedline in (b) represents wmode = wu/2. Here � = 1 µm.
The solutions in figure 3 are calculated neglecting the effects of the lateral lobes shownin figure 2(b). The mode of a W-shaped guide, however, is inherently leaky; thus we needto evaluate power losses of the wavepackets in figure 3 as they propagate. To this extent, weintroduce the loss coefficient ↵ via �(x = 0, z) = �(x = 0, z = 0) exp (�↵z) and define theattenuation length Ldamp of the quasi-mode as 1/↵ [27]. From the governing equation (5),we define a propagation length znorm = z/w2
U , similar to the normalization employed inthe eigenvalue problem (16). Otherwise stated, if Ldamp = L1 for wU = w1, then it is L2 =(w2/w1)
2 L1 for w2. Noteworthy, the damping length Ldamp depends quadratically on the waist(approximately equal to wU in the limit of strong confinement), in analogy to the Rayleighdistance; hence, the ratio between the diffractive losses due to a periodic index profile and thosedue to standard diffraction is constant: in the effective potential approximation the waveguidingdoes not depend on the width wU of the index trap. Ldamp can be numerically calculated witha BPM algorithm, accounting for the whole index landscape. Specifically, we consider at the
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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Figure 5. BPM computation of light propagation in the plane (x/wU , z) whenthe input field is the eigenmode computed from equation (16); the medium isassumed to be z-invariant with an effective index profile 1neff. The waveguideparameters in the top panels are U0 = 0.05 (a), 0.1 (b) and 0.5 (c), with 3 =50 µm in all of them; at the bottom 3 = 5 (d), 20 (e) and 100 µm (f), withU0 = 0.5 fixed. (g) Loss coefficient ↵ versus 3, each line corresponding to thedata plotted in figure 4(b). Here � = 1 µm and wU = 10 µm.
input the mode obtained from equation (16); then we solve equation (5) versus z and estimatethe mode attenuation.
The results are shown in figure 5: as expected, the diffraction losses diminish when thedepth U0 of the index well (panels (a)–(c)) or the period 3 increase (panels (d)–(f)). Figure 5(g)summarizes the results by graphing ↵ versus 3. We note that the same ↵ is obtained for differentU0 if the product 3U0 is conserved, as discussed above; thus, the curve for U0 = 0.1 assumes ageneral character. A sharp transition is observed in ↵ when wmode ⇡ wU/2, i.e. the quasi-modecomputed via equation (16) extends beyond the central lobe, in turn forcing non-negligiblecoupling to the radiation modes with larger transverse wavevector. This means that our approachis valid for wmode < wU/2, that is, the region below the dashed line in figure 4(b). This conditiondoes not yet guarantee light confinement, since equation (4) has not been accounted for. We dealwith it in the next section.
4. Numerical results
Hereby we solve the full equation (1) in order to check the validity of the effective equation (5)for dynamic light trapping, verifying condition (4).
We consider the propagation of the quasi-modes in figure 3 for a fixed U0 while varying thetrap width wU , i.e. we take quasi-modes belonging to the existence curves in figure 4(b). We firstconsider the quasi-modes (top line in figure 4(b)) not satisfying the condition wmode < wU/2 andthus undergoing large diffraction losses, consistently with figure 5(g). The results in figure 6demonstrate that diffraction losses cannot be neglected over distances comparable with theRayleigh length, because dynamic confinement does not take place, as predicted. At variance
New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
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Figure 6. Intensity evolution in the plane xz (top) and corresponding profile atthe input section z = 0 (solid red line, solution of (16)) and at the output (blackdashed line) (middle); the bottom panels show the field amplitude in x = 0 versusz, normalized with respect to the initial value in z = 0. The trap width wU is3 (a), 10 (b), 20 (c) and 40 µm (d), respectively. Here � = 1 µm, 3 = 100 µmand U0 = 0.05.
with the solutions of equation (5), the light dynamics depends on wU , as indicated by thefact that Ldampw
�2U is no longer conserved. The trend of the wavepacket peak versus z for
the narrowest trap (column a, wU = 3 µm) differs qualitatively from the ones (columns b–d)computed for wider traps. For large wU the intensity distribution can be normalized with respectto znorm = z/w2
U , analogously to equation (5), as the Rayleigh length is much shorter than theperiod 3. The ratio between the Rayleigh length and 3 governs the longitudinal oscillations inthe field peak, as well.
The strong dependence from wU is in agreement with the physical mechanism behind theKapitza effect: if diffraction is too strong (small wU ), the light can no longer be confined aftera defocusing section. Mathematically, the spectrum of � is large enough to induce aliasing;therefore equation (5) is not valid anymore, i.e. condition (4) is broken.
From the results above, it is straightforward to determine when light guiding via the Kapitzaeffect can occur and, if so, its dependence on wU . To this extent, figure 7 plots the existencecurves for the quasi-modes computed from equation (16), allowing us to pinpoint pairs ofparameters 3 and wU that allow Kapitza light confinement. The dashed line is the upper boundof the validity region for equation (16), as demonstrated by the results in figure 5. We haveto apply equation (4) to assess when the simplified equation (5) is a good approximation forthe original problem (1). In the plane of figure 7, condition (4) reads wmode/wU > 3/(⇡wU ):light confinement can occur in the region above the line starting from the origin and endingon the straight line wmode/wU = 0.5, with slope depending on the trap width wU . Dynamicconfinement is expected to occur in a triangular region, polygons ABC or ADC for wU1 orwU2, respectively (see figure 7). Wider index wells provide a larger parameter region where
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Figure 7. Qualitative assessment of dynamic confinement. The solid lines are theexistence curves in the plane (3, wmode/wU ) as obtained from equation (16) forthree index depths U0, such that U01 > U02 > U03. The dashed line representingwmode/wU = 0.5 is the upper bound of the validity region of the existence curves.The dashed-dotted lines represent the anti-aliasing condition (4) for two trapwidths wU1 < wU2, respectively.
dynamic confinement can take place. Moreover, for a given amplitude U0 of the index well,the range of 3 allowing dynamic confinement can be found by computing the intersection ofthe corresponding existence curve with the straight line wmode/wU = 0.5 (i.e. the smallest 3ensuring confinement) and with the line segment representing condition (4) (i.e. the largest 3yielding confinement without significant oscillations in radius). With reference to figure 7, letus take U0 = U02: for wU = wU1 confinement occurs only at point B, whereas for wU = wU2 thewavepacket gets trapped in all the points belonging to the arc BE.
To validate the approach, we simulated light propagation in a periodic index well whilevarying the periodicity of f (z), as well as the width and the peak value of U (x). Typical resultsare presented in figure 8. For wU = 10 µm (upper panels) low loss dynamic confinement isnever achieved for a trap depth U0 up to 0.5. In fact, for short periods (figures 8(a) and (c))the modal width overcomes wU/2 (figure 7); thus the field tails couple energy toward the outerregions. For larger periods (figures 8(b), (d) and (e)) condition (4) is not valid: wherever thelocal index well is defocusing light diffracts outward, forming complicated patterns in (x, z)and diffusing photons across x . For wU = 40 µm diffraction is reduced and light confinementvia the Kapitza effect takes place for all the used parameters (figures 8(f)–(j)). We note thatfor U0 = 0.1, 3 = 50 µm (figure 8(a)) and for U0 = 0.5, 3 = 10 µm (figure 8(c)) the quasi-mode retains nearly perfectly its shape over several Rayleigh lengths, that is, diffraction lossesare negligible. When the period increases, aliasing comes into play, distorting the modalprofile and generating oscillations in transverse size. Such oscillations are stronger for larger3 due to the shift (in parameter space (wmode/wU , 3)) toward the instability region (figure 7),eventually losing confinement, with dynamics analogous to the case wU = 10 µm previouslydiscussed.
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Figure 8. Intensity evolution in the plane xz (images) and behavior of the peak|A(x = 0, z)|/|A(x = 0, z = 0)| versus z (panels next to each image) when aquasi-mode from equation (16) is launched at the input. Light propagation iscalculated for U0 = 0.1, 3 = 50 µm (a), (f), U0 = 0.1, 3 = 100 µm (b), (g),U0 = 0.5, 3 = 10 µm (c), (h), U0 = 0.5, 3 = 50 µm (d), (i), U0 = 0.5, 3 =100 µm (e), (j) and wU = 10 µm (a)–(e) and wU = 40 µm (f)–(j), respectively.Here � = 1 µm.
Table 1. Calculated 3 for various waveguide parameters.wU (µm)
U0 10 20 30 40 50
0.1 38 51 62 72 810.5 16 24 31 37 43
To end this section, we assess the correctness of equation (4) resorting to BPM simulations.We define the period 3(wU , U0) = ⇡wmode, i.e. the value of 3 satisfying equation (4). Inphysical terms, 3 is the largest period ensuring dynamic light trapping when both wU andU0 are kept fixed; thus, in agreement with equation (6), at 3 light trapping is the strongest. Inaddition, we conveniently define 3sup(wU ) as the maximum of 3 for a fixed trap width wu ,with the extra constraint wmode 6 0.5wU (in figure 7 such a point for wU1 is indicated by B).The use of equation (4) easily provides 3sup = ⇡
2 wU . A direct comparison between 3 and 3sup
indicates whether the dynamic effect can be achieved for the given pair (U0, wU ): if 3 > 3sup
Kapitza-like confinement is inhibited by diffraction losses, if 3 < 3sup light trapping can occur.Table 1 is obtained from the data in figure 4. For wU = 10 and 40 µm, we get 3sup ⇡ 16
and 63 µm, respectively. The theoretical predictions can be compared with the full numericalsimulations in figure 8. We start with the case wU = 10 µm. For U0 = 0.1 (figures 8(a) and (b))
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light is not confined, consistently with table 1. For U0 = 0.5 there are small diffraction lossesfor 3 ⇡ 3sup = 16 µm (figure 8(c)): the confinement is borderline as this is the limit condition3 = 3sup (corresponding to point B in figure 7), where both a strong overlap of the modewith the edges of the effective index well and aliasing are simultaneously present. Due tothe joint action of the two detrimental effects, the best light localization is reached for 3 ⇡20 µm 6= 3sup. We now turn to the case wU = 40 µm: looking up table 1, light should notbe guided for U0 = 0.1; in spite of this, figures 8(f) and (g) show a collimated wavepacketthanks to the large Ldamp compared with the Rayleigh length. For U0 = 0.5 light confinementshould occur, as confirmed by figures 8(h) and (i): further numerical simulations (not shown)prove that the narrowest quasi-mode, jointly with the smallest oscillations in radius, is excitedfor 3 ⇡ 35 µm, in perfect agreement with the theoretical predictions. The numerical resultsalso demonstrate that transverse confinement occurs even for 3 > 3, but with the quasi-modeundergoing appreciable periodic variations of its width (see figures 8(i) and (j)). The existence oftrapped breathing waves for 3 > 3 agrees with our theory: smooth transition in the character oflight propagation can be expected when condition (4) is slightly missed, the latter correspondingto weakly overlapping replicas of �.
4.1. Dependence on the index profile U (x)
Up to now in our simulations we have considered a Gaussian U (x), but light trapping via theKapitza effect can be assessed for various U (x) using figure 7. A non-Gaussian U (x) impliesdifferent existence curves for the quasi-modes and different diffraction losses ↵ for a givenmodal width as well; the latter modify the applicability range of equation (16), rewritten for thenew U (x). Furthermore, if the modal width is fixed, a different U (x) affects the profile u of thequasi-mode, with small changes in condition (4); considering the numerical results discussed insection 4, the latter changes are expected to be negligible.
In section 3, we calculated the effective index well for a super-Gaussian U (x) (seeequation (15)), as this ansatz permits us to address the role of the abruptness of U (x)in the efficiency of dynamic confinement. Substitution of equation (15) into equation (5)demonstrates that, in complete analogy with the Gaussian case p = 1, for every value ofp light propagation depends on the normalized transverse (longitudinal) coordinate x/wU
(znorm = z/w2U ). Figure 9(a) shows sample solutions u of the equivalent of equation (16) as
p varies, for fixed 3 and U0 . When p is small, the optical modes resemble a Gaussian function,whereas for large p they are similar to sinusoidal branches, as in slab waveguides. The latter isconfirmed by figure 9(b) showing the degree of confinement versus 3. In fact, when the modalwidth wmode is smaller than wU (i.e. for large 3), trapping improves when p is smaller dueto shrinking of the region with a flat 1neff. In the opposite limit of modal widths comparablewith wU , light confinement improves with p owing to sharper peaks near the edges of 1neff
(from equation (15) the peak of the effective index well is roughly proportional to p2). Thisinterpretation of the p dependence for dynamic light trapping fully accounts for the behaviorof 1neff in figure 9(c), as well. Figure 9(d) graphs the diffraction losses associated with thequasi-modes: for a fixed p the coefficient ↵ has the same qualitative behavior of the Gaussiancase p = 1, with ↵ diminishing for larger p.
The degree of confinement of the modes computed in figure 9 was also investigated viaBPM, as shown in figure 10. As p increases, the maximum 3 ensuring trapping becomes larger(see the amplitude peak versus z in figures 10(a)–(d)): in fact, the higher the p, the flatter the
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Figure 9. (a) Quasi-mode profiles u versus x/wU for 3 = 100 µm and p = 2(red solid line, narrowest profile), p = 4 (black solid line, intermediate width)and p = 8 (blue solid line, widest profile); the dashed lines are inverted replicasof the index wells with depth normalized to 1. (b) Normalized width of thequasi-mode wmode/wU , (c) changes in effective index 1Neff and (d) diffractionlosses ↵ versus period 3 for p = 2 (red line with symbols), p = 4 (black linewith symbols) and p = 8 (blue line with symbols); the dashed lines correspondto the widest quasi-mode with negligible losses for a fixed p. In (d) the losscoefficient was calculated for wU = 5 µm. Here U0 = 0.5 and � = 1 µm in allthe simulations.
existence curve of the modes for large 3 (see figure 9(b)). At the same time, the oscillationsalong z get bigger with p, even for 3 = 3, the latter ascribable to a larger spectrum � for a givenwidth. Moreover, even if the average peak appears nearly unaltered along z, small losses occurin the defocusing regions. The presence of radiated light can be inferred from the calculatedwidth of the wavepacket along z (figures 10(e)–(h)), with radiation increasing with p, as well.Finally, figures 10(i)–(l) display the computed intensity evolution in the plane xz for p = 8 andfour 3. Despite the stronger diffraction losses with respect to the Gaussian case p = 1, light isstill confined for 3 up to 33sup, i.e. the Kapitza effect is effective on a larger interval of 3.
4.2. Dependence on longitudinal modulation f (z)
After investigating the role of the transverse profile U (x) on dynamic confinement, we studyhow the Kapitza effect depends on the modulation f (z) for a given periodicity. Equation (6)states that, once U (x) is fixed, the transverse profile of the confining potential does not changewith f (z), the latter acting on the amplitude of the effective well. In this section, we take aGaussian U (x) and consider various f (z).
The comparison between different f (z) has to be carried out for a fixed depth U0, the latterimposed by technological limitations. Comparing, for example, equations (7) (sine modulation)and (8) for d/3 = 0.5 (square wave) with U0 kept constant, it is clear that the overall amplitudeof 1neff for the square wave is about 16/⇡2 times larger than in the sinusoidal case. Thus, the
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Figure 10. Upper and middle rows: normalized amplitude peak (a)–(d) andmode width (e)–(h) versus z for p = 1 (a), (e), p = 2 (b), (f), p = 4 (c), (g) andp = 8 (d), (h), respectively; black and red lines correspond to 3 = 3 and 33sup,respectively (for the chosen parameters 3 < 3sup is valid). Bottom: contour plotsof light intensity evolution in the plane xz for p = 8 and period 3 equal to 3 (i),3sup (j), 23sup (k) and 33sup (l), respectively. Here wU = 20 µm, U0 = 0.5 and� = 1 µm.
quasi-modes are exactly the same if we compare the square wave with U0 = U 0 with a sine wavefeaturing U0 = (4/⇡)U 0. Analogous considerations are valid for flat-top modulations when therole played by the duty cycle d/3 has to be accounted for (see figure 1(b)). BPM simulations(not shown) indicate that when wavepackets undergo transverse confinement, identical quasi-modes at the input evolve in a similar manner along z, regardless of the shape of f (z).
To confirm the statements above, we performed numerical simulations as summarized infigure 11 for various modulations f (z), keeping U (x) constant. From the direct comparisonof figures 11(a)–(c) with figures 11(d)–(f), the dynamics of light propagation with period 3(see figure 10) indicates a lower 1neff for the sine case than for the square wave case (note that infigure 11(f) the wavepacket breaks up due to the outstanding narrowness of the quasi-mode). Thecomparison between the middle and bottom panels is consistent with the results in figure 1(b),that is, for d/3 = 0.1 the effective index well is weaker than for d/3 = 0.5. For d/3 = 0.1 thewavepacket undergoes appreciable diffraction losses for 3 = 30 µm (figure 11(g)), whereas for3 = 90 µm it is still trapped (figure 11(i)) because condition (4) is satisfied, at variance withfigure 11(f).
4.3. Dependence on input section
As pointed out in section 2.1, the phase fronts of the quasi-modes found via equation (16)exhibit a periodic curvature versus z (see equation (2)), i.e. the transverse phase is constant onlyin specific sections z = const. Otherwise stated, a periodic guide, unlike standard (longitudinallyuniform) structures ensuring static confinement, does not possess longitudinal invariance with
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Figure 11. Comparison between different longitudinal modulations f (z).Intensity evolution for sinusoidal f (z) (a)–(c), for flat-top f (z) with d/3 = 0.5(d)–(f) and d/3 = 0.1 (g)–(i), respectively. The period 3 is either 30 µm (a), (d),(g) or 60 µm (b), (e), (h) or 90 µm (c), (f), (i), respectively. Here wU = 20 µm,U0 = 0.5 and � = 1 µm.
Figure 12. Light propagation for a quasi-mode launched at different z0. HerewU = 20 µm, 3 = 30 µm, d/3 = 0.3, U0 = 0.5 and � = 1 µm.
respect to z; hence, in order to minimize coupling/insertion losses, the quasi-mode computedvia equation (16) needs be launched in specific z0, the latter depending on the form of theperiodic f (z). For flat-top modulation, we previously demonstrated that equation (9) is valid;hereby we test the theoretical prediction by means of BPM simulations. The numerical resultsfor flat-top modulation with d/3 = 0.3 are visible in figure 12. As predicted by equation (9),optimum input coupling is achieved for z0 = 0.253 and 0.753. For the other values of z0 thediffraction losses are apparent, together with an appreciable breathing of the modal profileversus z. Additional simulations (not shown here) for different ratios d/3 as well as in thesinusoidal case validate our theoretical approach.
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Figure 13. Modal profiles, normalized to their peak, computed via equation (18)for � = 1 (a), 0.1 (b) and 0.01 (c), respectively. Each line corresponds to aperiod 3, given by 1, 20, 40, 60 and 100 µm, from widest to narrowest profiles,respectively. We note that in the case 3 = 1 µm the Kapitza effect is almostuninfluent.
5. Interplay of dynamic and static confinement
So far we assumed no static confinement, as we considered periodic index distributions with anet zero average contrast. In this section we study the interplay between static and dynamicconfinement in unbalanced z-periodic structures. To this extent, in equation (1) we take arefractive index of the form
1n(x, z) = ⇥� + f (z)
⇤U (x), (17)
where the average of f (z) is still zero. The real coefficient � weighs static versus dynamic indexwells. We stress that equation (17) is not the most general, as we assume the same transverseprofile (U (x)) for both the static and the periodic components of 1n; nevertheless, thisdoes not hamper the general conclusions while it models realistic geometries (e.g. segmentedwaveguides [13–16]).
By looking at equations (2) and (3), it appears that a static component is equivalentto introducing in equation (3) new terms proportional to z, thus rendering inappropriate theapproach used for f = 0. We can argue that a static (cw) component introduces an extra phasemodulation across x , which in turn adds to the phase stemming from the Kapitza effect, until thecw portion of 1n appreciably modifies the field profile over a single period 3. Mathematically,we compute the modes of the structure via
2k0n01�effu = @2u@x2
+ k20
⇥2� n0U (x) + 1neff(x)
⇤u (18)
with 1neff still expressed by equation (6). Hereafter, for conciseness we set U (x) Gaussian withU0 = 0.5 and wU = 20 µm, and f (z) sinusoidal; thus, equation (15) for p = 1 describing theeffective dynamic potential due to the Kapitza effect is valid.
Figure 13 displays the numerical solutions of (18) for three values of � . When � = 1,the cw and the periodic components have the same amplitude; thus the index contrast nevergoes negative, as in segmented waveguides: according to figure 13(a) the Kapitza effect doesnot affect the modal profile. Conversely, for lower � (figures 13(b) and (c)) the Kapitza trappingbecomes relevant: the modal profile u undergoes appreciable variations as the period 3 changes;in particular it shrinks for large 3 in agreement with equation (6).
Next we analyze the shape and stability of the modes using BPM simulations. We startwith the case � ⌧ 1 , i.e. when the dynamic phase is relevant according to figure 14. We choose� = 0.01 to maximize the dynamic effects. Since the quasi-modes supported by the Kapitza
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Figure 14. Top row: static (red solid line) and complete (static plus dynamiceffects, dashed black line) potential versus x for � = 0.01. Middle row: intensityevolution in the plane xz when the input field is a solution of equation (18) for� = 0.01. Bottom: as in the middle but the mode from equation (18) is computedneglecting the Kapitza effect. From left to right, 3 is 30, 40, 60 and 80 µm,respectively. Here wU = 20 µm, U0 = 0.5 and � = 1 µm.
Figure 15. Propagation of quasi-modes for � = 1. The periods 3 are 30 (a),40 (b), 60 (c), 80 (d), 120 (e) and 150 µm (f), respectively.
effect are not perfectly invariant along z, but undergo oscillations and eventually break up forlarge 3 (see figure 8), we compare their evolution when the dynamic phase is included (middlerow in figure 14) and when it is neglected (bottom row in figure 14). The results show clearly thatmodes are more localized when the Kapitza effect is accounted for, demonstrating the validityof equation (18). As we found for � = 0 (see figure 11), no localization occurs for large 3 ascondition (4) is not satisfied.
We now examine the case � = 1. The modes are slightly affected by the Kapitza effect(figure 13(a)), but we need to check whether they break up for large 3, as occurs for smallor vanishing � . Figure 15 shows the typical behavior for various periods 3. For small 3, theperiodic modulation induces only small oscillations along z; for large enough 3, diffraction
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in the low index regions is no longer compensated by focusing in the high index regions,resulting in strong coupling to radiation. Noteworthy, for even larger 3 the wavepacket acquiresonce again a localized structure, even if undergoing strong oscillations along z due to thesimultaneous excitation of several modes. Further increases in period yield loss of confinement.This is not observed for small � and can be ascribed to resonance effects between multimodalinterference in the guide and longitudinal modulation of refractive index.
6. Conclusions
We have discussed light propagation in guided-wave dielectric structures subject to a periodiclongitudinal modulation of the refractive index distribution. While existing models of segmentedwaveguides state that light propagates according to an average index well along z, yielding noconfinement for a vanishing average index contrast, we found that confinement can take placeeven in the balanced case owing to a transverse modulation of the wavevector, similarly to theKapitza effect in quantum mechanics. Using BPM simulations we discussed light propagationversus geometric parameters, including the profile of the index well versus x and the form ofthe longitudinal modulation along z. We introduced a graphic method to roughly estimate lighttrapping and its range of validity within the Kapitza model. In particular, wide wavepacketsare not confined due to strong coupling with diffractive losses to radiation, whereas verynarrow modes break up due to aliasing phenomena. Remarkably enough, several conceptsstemming from our analysis in optics also apply to quantum mechanics where, to the best ofour knowledge, they are a novelty.
Although the experimental observation and demonstration of the proposed new mechanismfor light trapping might be a challenge (particularly in anisotropic dielectrics), table 1 indicatesthat dynamic confinement can be accessed for index changes in the range ±0.1–0.5 intransverse regions of tens of micrometers. These levels of index contrast could be obtainedin semiconductor heterostructures (e.g. AlGaAs composites [28]), in voltage biased electro-optic crystals or liquid crystals [29]. In the latter, the refractive index profile could beeasily controlled—both transversely and longitudinally—by applying external electric fieldsor defining non-uniform boundary conditions [30–33]. The required periods of the longitudinalmodulation are well within those achievable with lithographic techniques. Finally, we studiedthe interplay between dynamic confinement (due to the periodic modulation of the index) andstatic confinement (due to a cw index contrast), analyzing the role of their mutual interplayversus waveguide parameters.
We believe our findings pave the way to a new family of optical waveguides and photonicdevices with exotic features and large tunability; moreover, these results can trigger theinvestigation of chaos in wave mechanics. The generalization to nonlinear cases can unveil newkinds of solitons and solitary waves based on the Kapitza effect.
Acknowledgments
AA thanks Regione Lazio for financial support. LM acknowledges funding from the EuropeanCommission, within the FET-Open Program of the 7th Framework Programme, under grant no.255914, PHORBITECH.
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New Journal of Physics 15 (2013) 083013 (http://www.njp.org/)
Optical vortices in antiguidesLorenzo Marrucci,1,2,* Noel F. Smyth,3 and Gaetano Assanto4
1Dipartimento di Fisica, Università di Napoli “Federico II,” Complesso di Monte S. Angelo, Napoli 80126, Italy2CNR-SPIN, Complesso Universitario di Monte S. Angelo, Napoli 80126, Italy
3School of Mathematics, University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK4NooEL—Nonlinear Optics and OptoElectronics Lab, University “Roma Tre,” Rome 00146, Italy
*Corresponding author: [email protected]
Received March 6, 2013; accepted April 1, 2013;posted April 8, 2013 (Doc. ID 186557); published May 7, 2013
We address the question of whether an optical vortex can be trapped in a dielectric structure with a core of a lowerrefractive index than the cladding—namely an antiguide. Extensive numerical simulations seem to indicate that thisinverse trapping of a vortex is not possible, at least in straightforward implementations. Yet, the interaction of avortex beam with a curved antiguide produces interesting effects, namely a small but finite displacement of theoptical energy center-of-mass and the creation of a symmetrical vortex–antivortex pair on the exterior of theantiguide. © 2013 Optical Society of AmericaOCIS codes: (050.4865) Optical vortices; (130.2790) Guided waves; (260.6042) Singular optics.http://dx.doi.org/10.1364/OL.38.001618
Analogies between different physical systems can be animportant inspiration for conceiving new effects. Amongthem, swapping the roles of light and matter has beenrather fruitful. For example, it is well known that in trans-parent media, light can be trapped in regions of a higherrefractive index, such as in the core of standard wave-guides [1]. Such trapping occurs in the transverse coor-dinates only, relative to the waveguide. Exchanging theroles of light and matter, we are naturally led to the con-cept of optical tweezers, which can trap small materialparticles in regions of higher light intensity [2,3]. In somesense, both effects stem from the same fundamental in-teraction between light and matter, that is, photon elasticscattering.Optical tweezers work when the particles have a
refractive index higher than the surrounding fluid. Con-versely, when the particles have a lower index, they arerepelled by the most intense regions of a bell-shapedbeam. On the other hand, such particles can be trappedby means of doughnut beams, e.g., vortex beams with acentral hole in the intensity distribution [4].Now, exchanging the roles of light and matter, one
might speculate that an optical vortex could be trapped(in the transverse plane) by a dielectric cylindrical struc-ture having a core of lower refractive index than the clad-ding, i.e., an antiguide for brevity. If the antiguide iscurved, then, one might be able to bend the vortex pathand steer the light surrounding the vortex as well, sincethe doughnut is associated with a form of continuous ro-tation of optical energy around the vortex axis where thephase singularity is located. Such a rotation might tie theoptical field of the doughnut to the axial singularity andforce the light to follow the waveguide, at least in part, bybeing refracted sideways. In this Letter, we aim at testingthese ideas using numerical experiments.Light beams endowed with a vortex on axis carry
orbital angular momentum (OAM) and have been thesubject of intense studies in the last 20 years [5]. In par-ticular, their interaction with inhomogeneous, aniso-tropic, or nonlinear optical media has been attractingincreasing attention [6–16]. One of the simplest examplesof such vortex beams is the Laguerre–Gauss mode with
the radial number p ! 0, which has the following expres-sion in the focal plane:
u"r; θ# ! arjmje−r2
w2$imθ: (1)
Here "r; θ# are polar coordinates in the transverse plane"x; y#, z is the propagation coordinate, w is the beamwaist, a is an amplitude constant (unimportant in a linearanalysis), and m is the integer winding number or vortexcharge.m also quantifies the OAM per photon carried bythe beam in units of ℏ.
For the wave propagation, we adopt the slowly varyingenvelope approximation. In dimensionless coordinates,we solve the paraxial Helmholtz equation:
i∂u∂z
$12∇
2Tu$ 2G"x; y; z#u ! 0: (2)
Here u is the complex envelope of the wave, ∇2T the
transverse Laplacian in x and y and G a function deter-mined by the refractive index distribution n"x; y; z# defin-ing the guide. A longitudinal length scale L, which can bechosen arbitrarily for normalization, gives the units of thez coordinate. The transverse length LT !
!!!!!!!!!!!!!!!!!!!!!!!Lλ∕"2n0π#
p
gives the units of the transverse x and y coordinatesand the waist w, with λ the vacuum wavelength and n0the asymptotic refractive index of the surroundingmedium (cladding). The function G giving the waveguideindex profile is
G !π"n2
− n20#L
2n0λ≈ π"n − n0#
Lλ; (3)
where the approximate equality holds for small indexvariations.
In our simulations, we launch a beam defined byEq. (1) at the input plane z ! 0 into a curved graded-index waveguide with the Gaussian index distribution:
G"x; y; z# ! Ae−y2$%x−S"z#&2
W2 ; (4)
1618 OPTICS LETTERS / Vol. 38, No. 10 / May 15, 2013
0146-9592/13/101618-03$15.00/0 © 2013 Optical Society of America
where S!z" # X $ R −
!!!!!!!!!!!!!!!!R2
− z2p
is the waveguide axisposition versus z. The function S!z" defines the wave-guide bend shape, which in dimensionless units is a cir-cular arc lying within the plane !x; z" of radius ofcurvature R and initial position X . The constant A givesthe maximum waveguide depth in terms of the refractiveindex change and length scale L. For negative A the guideanticore has a lower refractive index than the surround-ing, infinitely extended, cladding.We first investigate the case m # 1, which is known to
be the most stable case, because the vortex has the mini-mum possible charge (and OAM). Hence, it can neitherspontaneously split into separate vortices of smallercharge, nor can it vanish owing to the conservation ofthe total topological charge. In a standard waveguide(positive A) the vortex can be confined by the refractivepotential, consistent with previous studies on vortexstabilization by means of nonlocal spatial solitons[6,10,11,17]. In a curved positive waveguide, the vortexpath can bend, undergoing profile distortions, vortex–antivortex pair generation, and radiation losses relatedto the curvature and to the waveguide core size relativeto the input beam. However, it retains its global phasesingularity located in the main beam, as illustrated, forexample, in Fig. 1. The vortex behavior in an antiguideis displayed in Fig. 2. As the vortex propagates in theplane !x; y # 0; z" an intensity hole appears to followthe antiguide axis, suggesting that the vortex is indeedbeing guided. However, a secondary hole, less visible,propagates straight [Fig. 1(a)]. This is more visible inthe output intensity distribution, illustrated in Fig. 1(c).To distinguish between these, one needs to look at theoptical phase distribution in the transverse plane !x; y"[Fig. 1(d)]. Clearly, the vortex is associated with the hole
which propagates straight without significant transversedisplacement with respect to the input. The second hole,following the bend, is due to light expelled from the anti-guide. Therefore, there is no significant vortex trappingby the antiguide. Nevertheless, some interesting observa-tions are in order.
First, the antiguide is found to push the light energylocated on the right-hand side of the core, i.e., the sidetoward which the guide is curved. This can be clearlyappreciated in the output intensity distribution, shownin Fig. 2(c) and is quantified in Fig. 2(b), which graphsthe variation of intensity peak position and of theweighed average position of the light, as defined by
hxi #R∞−∞
R∞−∞ xjuj2dxdyR∞
−∞R∞−∞ juj2dxdy
: !5"
The latter is the center-of-mass of the intensity distribu-tion. The intensity peak essentially follows the antiguide,keeping just to the right of the core. The center-of-massposition is displaced by a much smaller, but finiteamount.
Second, in the output section shown in Figs. 2(c) and2(d), there is a pair of oppositely charged vortices(m # $1 and m # −1) generated by the refractive dis-turbance, located at x ≈ 50 (close to the waveguide core)and y ≈%20. It is important to note, albeit less clear, thecollateral generation of oppositely charged vortices isalso visible in Fig. 1(d) for the case of a positivewaveguide.
Very similar behavior is observed for higher chargevortices. A representative example with m # 3 is shownin Fig. 3, but different m > 1 give qualitatively similarresults. In comparison with the m # 1 case, the main
Fig. 1. Interaction of an optical vortex with chargem # 1 anda Gaussian positive waveguide. (a) Intensity evolution in alongitudinal cross-section (xz plane), with intensity-scale infalse colors. (b) Displacement Δx # x!z" − X of the waveguidecenter (dashed–dotted blue line), light intensity peak (solid redline) and average center (dotted green line) versus z. The loca-tion of the numerical maximum switches periodically betweenthe symmetrical peaks on the two sides of the waveguidecenter. For z > 230 the maximum location is not stable dueto the strong radiated energy. (c) Intensity distribution in atransverse cross-section at the exit plane z # 300. (d) Opticalphase distribution in the same plane (phase-scale in false col-ors). The simulation parameters are: A # 1, w # 10, W # 3,R # 500, X # −50.
Fig. 2. Interaction of an optical vortex with chargem # 1witha Gaussian antiguide. (a) Intensity evolution in a longitudinalcross-section (xz plane), with intensity scale in false colors.(b) Displacement Δx # x!z" − X of the waveguide center(dashed–dotted blue line), intensity peak (red solid line), andaverage center (dotted green line) versus z (the latter is re-scaled by a factor ×10). The red curve for small z is omittedbecause it switches randomly between the two symmetricalpeaks on the two sides of the waveguide center. (c) Intensitydistribution in a transverse cross-section at the exit planez # 300. (d) Optical phase distribution in the same plane withphase scale in false colors. The simulation parameters are:A # −1, w # 10, W # 3, R # 500, and X # −50.
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difference is that the central vortex, although not trappedby the waveguide, is perturbed enough to split into jmjsingle charge vortices, which remain located in the cen-tral region. This is not unexpected, given the known in-stability of higher-order vortices.We performed a number of tests by varying parame-
ters, such as the ratio w∕W between beam waist andguide size and the guide bend radius of curvature Rand found no evidence of regimes in which the vortexis guided. This is not a formal proof negating the possibil-ity of vortices being guided by an antiguide; very largeradii Rmight indeed allow for vortex bending but, in sucha limit, the long propagation distances required to ob-serve a significant displacement would also result inlarge diffraction of the bright doughnut components,making the guiding effect nearly irrelevant. A possibleroute to counteract diffraction and bend vortices mightconsist of using positive waveguides with large R. Weplan to explore this and other possibilities in future work.In conclusion, our numerical experiments indicate that
trapping an optical vortex by an antiguide is not possible,at least in those practical situations in which the trappingwould be strong enough to operate over sufficiently shortdistances. As a consequence of the antiguide-vortex in-teraction, a small transverse shift of the optical energyand the generation of a vortex–antivortex pair on the
antiguide exterior are induced, but no bend of the vortexpath occurs. Higher-order vortices end up splitting intosmaller vortices, due to their known structural instability.Further investigation will be required to assess thegeneral validity of these results or identify exceptionsin limiting cases.
This work was supported by the Royal Society ofLondon, under Grant IE111560, and by the EuropeanCommission within the FET-Open Program of the7th Framework Programme, under Grant No. 255914,PHORBITECH.
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Fig. 3. Same as Fig. 2, but for an optical vortex with chargem ! 3.
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