3482 Vol. 45, No. 13 / 1 July 2020 /Optics Letters Letter

Light beams with volume superoscillationsThomas Zacharias AND Alon Bahabad*Department of Physical Electronics, School of Electrical Engineering, Fleischman Faculty of Engineering, Tel-Aviv University,Tel-Aviv 69978, Israel*Corresponding author: alonb@eng.tau.ac.il

Received 2 April 2020; revised 15 May 2020; accepted 18 May 2020; posted 19 May 2020 (Doc. ID 394270); published 22 June 2020

Using a superposition of shifted Bessel beams with differentlongitudinal wave vectors and orbital angular momenta, werealize an optical beam having simultaneous axial, angular,and radial focusing narrower than the Fourier limit. Ourfindings can be useful for optical particle manipulationand high-resolution microscopy. © 2020 Optical Society ofAmerica

https://doi.org/10.1364/OL.394270

Introduction. In superoscillation, a band-limited signal locallyoscillates faster than its fastest Fourier component [1] allowingus to focus light below the diffraction limit, which is useful forapplications such as microscopy and optical tweezers [2,3].The roots of superoscillations can be traced to optics [4] andmore recently to quantum weak measurements [5]. Followinga proposition to employ superoscillations for super-resolution[6], many works followed in optics regarding spatial sub-Fourierfocusing and microscopy [7–10], sub-Fourier focusing in thetemporal domain [11,12], narrow-band frequency conversion[13], and also optical tweezers [14]. Recently, the complemen-tary phenomenon for superoscillations, namely suboscillations,where a lower bound limited signal locally oscillates slower thanits slowest Fourier component, was discovered and began to beemployed in optics [15–17]. Most works employing superoscil-lations for spatial light focusing do so in the plane transverseto the optical axis. In a previous work [18], we demonstratedsub-Fourier axial focusing. While there have been works aimingto achieve sub-wavelength dimensions in all directions (in a vol-ume), they have either been in the creation of three-dimensional(3D) hollow spots [19,20], or in creating optical features byusing spatiotemporal waveforms [21]. In this work, using onlyspatial modes of a continuous wave laser, we create a superposi-tion of optical Bessel beams [22] that produces an optical featurewhose size can be controlled independently in three dimensions(axial, angular, and radial) and can be made narrower than theFourier limit along either some or all directions simultaneously.

Theory. A known canonical form for superoscillation is thefollowing [5,6]:

fso(x )= (cos(k0x )+ ia sin(k0x ))N, (1)

where k0 is a fundamental spatial frequency, a > 1 ∈R is thesuperoscillatory tuning parameter, and N ∈N+. It is easilyverified that though this function is spectrally limited in theband [−Nk0, Nk0], its local frequency around x = 0 is a times

faster than its fastest Fourier component Nk0. This is also truefor the real part of fso(x ), which has the following cosine Fourierseries for N = 3 [18]:

Re{ fso(x )} =C1 cos(k0x )+C3 cos(3k0x ),

where C1 = 3(1− a2)/4 ; C3 = (1+ 3a2)/4. (2)

This is a superposition of plane wave modes oscillating atharmonics+1,−1,+3, and−3 of the fundamental.

For the following, we also need the form of an ideal Besselbeam in polar coordinates [22]:

E (r , φ, z)= A0 exp(ikzz)Jn(kr r ) exp(±inφ), (3)

where A0 is an arbitrary amplitude, kz and kr are longitudinaland transverse wavenumber components, respectively, Jn(x )is the nth-order Bessel function, and exp(±inφ) is the angularmomentum term of the beam.

Angular superoscillating beam. Experimentally, anapproximation to a zero-order Bessel beam can be created atthe front focal plane of a lens by setting an annular aperturein its back focal plane [22]. Adding angular phase (nφ) to theaperture creates the corresponding nth-order Bessel beam.Using the identity, J−n(r )= (−1)n Jn(r ), it is possible tosuperpose two Bessel beams originating from a single annularaperture with opposite orbital angular momenta±n to realize asinusoidal intensity pattern along the angular (azimuthal) direc-tion. Further, the asymptotic form of J1(kr r ) and −J3(kr r )converges when the argument of the functions is much largerthan |α2

− 0.25| with α being the order of the Bessel function.Therefore, by applying +1, −1, +3, and −3 orbital angularmomentum charges on an annular aperture accompanied by theamplitudes given with Eq. (2), it is possible to create a beam thatis superoscillating in the angular direction:

ERing(r , φ, z)=3∑

n=−3, n∈{odd}

An exp(ikzz)Jn(kr r ) exp(inφ)

∼=r�8.75/kr

A0 exp(ikzz)J1(kr r )× [Cang1 cos(φ)+Cang3 cos(3φ)],

(4)

where Cang1,Cang3 are the Fourier coefficients from Eq. (2)corresponding to the angular superoscillation tuning parametera = aang.

0146-9592/20/133482-04 Journal © 2020Optical Society of America

Letter Vol. 45, No. 13 / 1 July 2020 /Optics Letters 3483

0 50 100 150 200 250 300x[ m]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Am

plitu

de [

a.u.

]J

1(k

r1

ra)

J1(k

r2

rb)

J1(k

r3

rc)

J1(k

r4

rd)

J0(k

r0

re)

Fig. 1. Shifted Bessel functions. Bessel functions shifted alongx , resulting in aligned lobes around which the amplitudes areapproximately equal. {ra ,b,c ,d ,e } correspond to shifted coordinates.{kr1,r2,r3,r4,r0 } correspond to radial wave vectors of the Bessel beamsoriginating from the four annular rings {R1,2,3,4,0}.

Angular + Axial superoscillating beam. A set of four annu-lar rings will contribute four different values of kz, which canrealize axial superoscillation [18]. Assume that each annularring is also modulated to produce angular superoscillationas described in the preceding subsection. However, the netinterference of the beams originating from all four rings is notsuperoscillating in the axial direction as each contains a differentradial wave vector, which cannot be factored out. To solve this,we shift the Bessel beams in space such that at a given range ofthe radial coordinate, the radial spatial frequencies are aboutthe same (see Fig. 1). The required shift is found numerically.This alignment is done for lobes of the Bessel beams, whichare far enough from the on-axis position so that the conditionJ1(kr r )=−J3(kr r ) is true for all beams (to ensure angularsuperoscillation). The beam shifts are realized experimentallyby adding appropriate linear phase to the rings generating theBessel beams. The overall field in the region of interest (ROI),which is the volume around the aligned lobes delimited by theclosest surrounding surface of zero intensity, is

Enet({r , φ, z} ∈ ROI)=3∑

m=−3,m∈{odd}

{Am J1(kr1ra ) exp(ikzm z)

× [Cang1 cos(φ)+Cang3 cos(3φ)]},(5)

where ra = |r− a| with a= a x̂ representing the shift. The rel-ative shifts of the Bessel beams originating from different ringsare verified numerically to be small enough that the differencebetween the angular coordinates φ is negligible in the ROI(<0.06 rad), allowing us to use the same coordinate φ for allbeams.

Setting the appropriate values of Am , Eq. (5) is simplified to

Enet({r , φ, z} ∈ ROI)= A0 J1(kr1ra ) exp(ikz0 z)[Ca x1 cos(1kzz)

+Ca x3 cos(31kzz)][Cang1 cos(φ)+Cang3 cos(3φ)],(6)

where kz0 is the mean longitudinal spatial frequency. This fieldsuperoscillates simultaneously in the angular and axial direc-tions while the degree of superoscillation in each direction canbe independently controlled by setting the value of the super-oscillatory tuning parameter “a” for each direction: aang and aax.These, from Eq. (2), also set the Fourier coefficients in the axialand angular directions {Cax, Cang}.

Angular + Axial + Radial superoscillating beam. Anotherknown superoscillating function is [13,23]

f cosso (x )= (cos(k0x )− a)2, (7)

with k0 a fundamental spatial frequency and 0< a < 1 a super-oscillatory tuning parameter. As Bessel functions are bandlimited, we propose a similarly superoscillating Bessel-basedfunction,

f Besso (r )= (Jn(kr r )− a)2. (8)

The fastest Fourier component of Jn(kr r ) is the transversewavenumber kr [24], so the fastest Fourier component off Besso (r ) is 2kr . By introducing the superoscillatory tuning

parameter, a ∈ (0, 1), and setting it to a value such that the localoscillation is faster than the oscillation of the fastest Fouriercomponent of f Bes

so (r ), i.e., 2kr , it is possible to achieve asuperoscillating Bessel-based function. Thus, a Bessel-basedsuperoscillating beam in the radial direction can be obtainedby interfering a Bessel beam with a plane wave whose relativeamplitude can be used as the radial superoscillatory tuningparameter arad. That is, the intensity of the field,

f radso (r , φ, z)= A0 exp(ikzz)Jn(kr r ) exp(±inφ)

+ A1 exp(ik′z), (9)

at z= 0 andφ = 0 is proportional to Eq. (8) and so is superoscil-lating in the radial direction for certain A0, A1. Here k′ is thewave vector of the plane wave.

However, trying to incorporate this radial superoscillation tothe previous ones by adding a plane wave to the field describedby Eq. (6) would destroy the axial superoscillation due to theadded oscillation (k′) along z. To solve this, instead of a planewave, we use a zeroth-order Bessel beam with a longitudinalwavenumber kz0 whose central lobe is aligned to the ROI. Thecentral lobe is regarded as a slow enough oscillation such thatit serves as a pseudo-DC superoscillatory tuning parameter.Overall, the intensity within the ROI is now

I ({r , φ, z} ∈ ROI)∝ |J1(kr1ra )[Cax1 cos(1kzz)

+Cax3 cos(31kzz)] − arad J0(kr0re )|2,

(10)

where re = |r− e| and e= e x̂ represent the shift of the centrallobe. This beam is superoscillating in its field in the axial andazimuthal directions and in intensity in all directions, includ-ing the radial one. The amplitude arad can be set in order tocontrol the degree of radial superoscillation. One offshoot ofintroducing radial superoscillations is that our beam superoscil-lates in intensity in the axial and angular directions even if aangand aax are both 1 as can be understood by a careful inspectionof Eq. (10). As the degree of radial superoscillation increases,so does the degree of the axial and angular superoscillations.

3484 Vol. 45, No. 13 / 1 July 2020 /Optics Letters Letter

Fig. 2. Experimental setup. BE, beam expander; SLM, spatiallight modulator; MST, moving stage; M, mirror; L1, L2, L3 are lenses.d1 + d2 equals the lens L3 focal length whose focal plane is at Z0. Inset,a realization of one of the phase-only masks used in the experiment.It is hard to observe the ring R0 because of the low intensity beam itcontributes.

We note, however, that once the radial width of the superoscil-lating feature has been set to the desired value using arad, wecan further use aax and aang to control the size of our feature inthe axial and angular directions, respectively, and independ-ently. Hence, we are able to obtain an optical feature with anintensity, which is narrower than the Fourier limit whose sizecan be controlled in the axial, angular, and radial directions asrequired.

Experiment. The experimental setup (Fig. 2; also see [18])consists of a 532 nm continuous wave laser (Quantum Ventus532 Solo) and a reflective phase-only spatial light modulator(Holoeye Pluto SLM) on which we mount our optical masks.The masks (see inset of Fig. 2) have been encoded with therequired amplitude and phase profiles using a known phase-encoding technique [25], and the desired beam is projected intothe first diffraction order. The laser beam is expanded and colli-mated before the SLM, reflected off it, and Fourier transformedusing a 20 cm focal lens with a numerical aperture of 0.0635.A CMOS camera (Ophir Spiricon SP620U) was used to imagethe intensity of the generated first-diffraction-order beam atdifferent planes in the vicinity of the focal plane using a steppermotor linear stage (Newport FCL200).

We generated a collection of phase-only diffraction maskscorresponding to different values of the superoscillatory tuningparameters aax, aang, and arad. Each mask is equipped with aset of five annular rings, four of which (R1,2,3,4) contributeto the axial and angular superoscillations, and the ring (R0)generates the pseudo-DC field for controlling the degree ofradial superoscillation. Linear phase shifts are applied on therings R1,2,3,4,0, so the corresponding Bessel beams are shifted by{a , b, c , d , e } = {29.7, 37.1, 43.1, 47.9, 126.6} µm in the x̂direction, respectively.

The rings, R1,2,3,4, contribute a linear set of kz values,with 1kz = 85.63 [m−1

], and the ring R0 produces a beamwith a longitudinal wave vector at the center of the kz spec-trum produced by the other four rings. This gave us radii ofR{1,2,3,4,0} = {2.6, 2.81, 3.01, 3.2, 2.91} mm. These valueswere chosen to reduce the difference between the correspondingradial wave vectors, increasing the range in x over which the

Fig. 3. Experimental results. Columns 1, 2, and 3 plot the intensity line-outs with the axial, angular, and radial directions, respectively, aroundthe ROI. The dashed lines enclose the superoscillating feature, and the dotted line represents the Fourier limited case. Column 4 shows the measuredtransverse intensity distribution of the beam at the focal plane. The larger circle marks the angular coordinates, and the smaller circle marks the ROI.Inset, a magnified view of the ROI is also presented. The different rows correspond to different values of the superoscillatory tuning parameters:(a) {aax, aang, arad} = {1, 1, 0}, (b) {aax, aang, arad} = {1, 1.5, 0.5}, and (c) {aax, aang, arad} = {1.2, 1.5, 0.5}.

Letter Vol. 45, No. 13 / 1 July 2020 /Optics Letters 3485

Table 1. Dimensions of the Optical Feature

{aax, aang, arad} 1z [mm] 1φ [rad] 1r [µm]

(a) {1, 1, 0} 12 0.7 19.8(b) {1, 1.5, 0.5} 8.5 0.46 13.3(c) {1.2, 1.5, 0.5} 6.5 0.46 13.3(d) Fourier limited case 12 1.04 16.6

different Bessels overlap (see Fig. 1), while incorporating a1kzthat is large enough to ensure visible beats in the axial direction.

Figure 3 shows intensity line-outs around the ROI alongthe axial, angular, and radial directions and the image of theintensity distribution in the focal plane for different valuesof aax, aang, and arad. Outside the ROI, the beams inter-fere arbitrarily and are of no interest to us. The beams shiftsaligns the lobes at R = 126.6 [µm] from the center along thex axis. The larger white circle over the intensity distribution,φ = φ(r = 126.6 µm), denotes the angular coordinate forwhich we design the intensity to be superoscillating. The smallercircle marks the ROI in which our approximations are valid. Allexperimental results are compared and agree quite well with theanalytical expectations (based on Eq. 10) and the beam propa-gation simulations (based on numerical solver of the Fresneldiffraction integral). The dip seen in the angular intensity plotsfor nonzero values of arad atφ = 0 is due to the zero-order Besselintroduced for controlling the radial superoscillation. Thiscould have been avoided if we used a plane wave instead of theBessel beam, but that would have destroyed the axial super-oscillation intensity pattern as discussed earlier. Still, the opticalfeature along the angular direction (from zero node to zeronode) is narrower than the Fourier limit and can be controlledusing aang, per our design.

Table 1 shows the dimensions of the optical features alongthe axial, angular, and radial directions (delimited by blackdashed lines in Fig. 3) for different values of the tuning param-eters. Case (d) in Table 1 shows the theoretically calculateddimensions of the Fourier limited optical feature, which corre-sponds to single mode oscillations at frequencies 31kz, 3, andmax{kr0,r1,r2,r3,r4} = kr4 , along the axial, angular, and radialdirections, respectively (see Eq. 10).

Note the differences in case (a) and (d) in Table 1. It is notsurprising that case (a) in Table 1 is radially wider as there areadditional slower radial wave vectors. However, in the angulardirection it is narrower, but this is due to the shifting of theBessel beams. Also notice that the beam is narrower in theaxial direction in Table 1 in case (b) compared to case (a) eventhough aax has not increased between the two. The reason forthis, as mentioned earlier, is that the introduction of radialsuperoscillation also causes superoscillations in the angular andaxial directions. On increasing the value of aax to 1.2 in case (c),while keeping the other parameters the same, we can see fromTable 1 that the feature gets narrower in the axial direction withits width unaffected in the angular and radial directions. We canthus completely control the dimensions of our isolated opticalfeature by first setting arad and then controlling aang and aax.

We can define the volume of our optical feature as the product1zloc × R1φloc ×1r loc, where 1zloc, R1φloc, and 1r loccorrespond to the physical length between the nodes in the axial,angular, and radial directions, respectively. Of the measure-ments taken, the volume of the smallest optical feature [case (c )in Table 1 and Fig. 3] was found to be 19.1% of the volume

related to the Fourier limited case. It is also clear from Fig. 3 thatas the superoscillatory parameters increase, the correspondingintensity of the sub-Fourier feature decreases, which is a defin-ing characteristic of super- and suboscillating functions. Forextremely large values of the superoscillatory parameters, thesuperoscillatory feature becomes sensitive to phase noise andmay be easily destroyed [26]. Aware of these constraints, we havechosen our tuning parameters accordingly.

Conclusion. We experimentally constructed, measured, andobserved an optical volume superoscillatory feature confined ina finite isolated volume, which is 19.1% of the volume related tothe Fourier limited case. This was realized using a superpositionof zero-, first-, and third-order shifted Bessel beams with orbitalangular momentum charges to create an optical feature whosedimensions can be controlled beyond the Fourier limit. Thisfinding can find potential applications in optical and electronbeam shaping, high-resolution microscopy, optical particlemanipulation, and nonlinear optics.

Acknowledgment. The authors would like to thankBarak Hadad and Yaniv Eliezer for fruitful discussions.

Disclosures. The authors declare no conflicts of interest.

REFERENCES1. M. Berry, Quantum Coherence and Reality (World Scientific, 1994),

pp. 55–65.2. G. Chen, Z.-Q.Wen, and C.-W. Qiu, Light Sci. Appl. 8, 1 (2019).3. G. Gbur, Nanophotonics 8, 205 (2019).4. G. T. Di Francia, Nuovo Cimento 9, 426 (1952).5. Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351

(1988).6. M. Berry and S. Popescu, J. Phys. A 39, 6965 (2006).7. E. T. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis,

and N. I. Zheludev, Nat. Mater. 11, 432 (2012).8. A. M.Wong andG. V. Eleftheriades, Sci. Rep. 3, 1715 (2013).9. E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and

M. Segev, Opt. Express 21, 13425 (2013).10. Y. Eliezer and A. Bahabad, ACS Photon. 3, 1053 (2016).11. Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, Phys.

Rev. Lett. 119, 043903 (2017).12. Y. Eliezer, B. K. Singh, L. Hareli, A. Bahabad, and A. Arie, Opt.

Express 26, 4933 (2018).13. R. Remez and A. Arie, Optica 2, 472 (2015).14. B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, Light Sci. Appl. 6,

e17050 (2017).15. Y. Eliezer and A. Bahabad, Optica 4, 440 (2017).16. Y. Eliezer, T. Zacharias, and A. Bahabad, Optica. 7, 72 (2020).17. I. Chremmos, Y. Chen, and G. Fikioris, J. Phys. A 50, 345203 (2017).18. T. Zacharias, B. Hadad, A. Bahabad, and Y. Eliezer, Opt. Lett. 42,

3205 (2017).19. Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang,

G. Liang, Y. Liu, and G. Chen, Opt. Express 26, 7866 (2018).20. Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and

G. Chen, J. Phys. D 52, 415103 (2019).21. A. M.Wong andG. V. Eleftheriades, Phys. Rev. B 95, 075148 (2017).22. D. McGloin and K. Dholakia, Contemp. Phys. 46, 15 (2005).23. M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C.

Struppa, J. Tollaksen, E. T. Rogers, F. Qin, M. Hong, X. Luo, R. Remez,A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades,Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu,A. Kempf, E. Katzav, andM. Schwartz, J. Opt. 21, 053002 (2019).

24. M. Abramowitz, Handbook of Mathematical Functions, withFormulas, Graphs, andMathematical Tables (Dover, 1974).

25. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, Opt.Lett. 38, 3546 (2013).

26. M. Berry, J. Phys. A 50, 025003 (2016).