aa r X i v : . [ phy s i c s . op ti c s ] M a y Generalized Spatial Talbot Effect based on All-dielectric Metasurfaces
Shulabh Gupta ∗ Department of Electronics, Carleton University1125, Colonel by Drive, Ottawa, Ontario, Canada.
A generalized spatial Talbot effect is proposed where the period of the input aperture is scaledby a non-integer real value, as opposed to the integer-only factor in a conventional Talbot effect.This is achieved by allowing and engineering the phase discontinuity distributions in space usingmetasurfaces, in conjunction with free-space propagation. The introduction of such abrupt phasediscontinuities thereby enables non-integer scalings of the aperture periodicities. Specific implemen-tations using Huygens’ metasurfaces are proposed and their operation to achieve non-integer scalingof the input aperture period is demonstrated using numerical results based on Fourier propagation.
I. INTRODUCTION
Talbot effect was originally observed by Henry Fox Tal-bot and described in his seminal work in [1]. When aplane wave is incident on a periodic aperture, the imageof the aperture is self-replicated at specific discrete lo-cations away from the aperture. At other distances inbetween, the aperture is self-imaged with smaller periodresulting in higher repetition of the periodic illumina-tions. This effect is known as the spatial Talbot effect.Talbot effect has found extensive applications in imaging,optical communication, optical computing, and opticalinterconnection, to name a few and a more comprehen-sive review can be found in [2]. The Talbot effect hasalso been translated to temporally periodic signals. Ina temporal Talbot effect, the space-time duality is ex-ploited to achieve self-imaging of periodic pulse trains,where their repetition rates are increased using simplephase-only filtering techniques [3][4].While Talbot effect has conventionally been used inrepetition-rate multiplication of either the periodic spa-tial aperture or the periodic pulse trains, they have beenextended to achieve repetition-rate division as well re-cently [5][6]. However, to the best of my knowledge, theperiod of the aperture is either multiplied or divided byan integer factor only. In this work, this concept of pe-riod scaling is generalized to non-integer factors, wherethe input periodic aperture can be imaged with eithera higher or a lower repetition rate by any arbitrary realnumber. The specific contributions of this work are: 1)Proposal of a generalized Talbot effect described in thespatial domain for arbitrary scaling of aperture periodic-ities. 2) Specific implementations of generalized Talbotsystem using metasurfaces [7][8], which are 2D array ofsubwavelength particles to provide abrupt phase discon-tinuities in space. ∗ [email protected] II. PRINCIPLEA. Conventional Talbot Effect G ENERALIZED T ALBOT E FFECT C ONVENTIONAL T ALBOT E FFECT M ETASURFACE O UTPUT P LANE I NPUT W AVE F REE -S PACE P ROPAGATION
PSfrag replacements | ψ ( x, y, || ψ ( x, y, | | ψ T ( x, y, ∆ z ) || ψ T ( x, y, ∆ z ) | ψ ( x, y, z ) xy z Periodic Aperture
Λ = mX where m ∈ R Λ = X /m where m ∈ I , ≥ X ∆ zT ( x, y ) FIG. 1. Generalized spatial Talbot effect where the period ofthe 2D array of sources is scaled by an arbitrary real number m in the output plane, as opposed to integer-only values inthe conventional Talbot effect. Consider an aperture consisting of a 2D array of holeswith period Λ = X at z = 0, as shown in Fig. 1(a),illuminated by a plane wave of frequency ω (or wave-length λ ). Such an aperture acts as a 2D array of pointsources, which then radiate along the z − axis. Due todiffraction, the periodic sources interfere with each otherforming complex diffraction patterns. However, at inte-ger multiples of distances ∆ z = X /λ = z T , known asthe Talbot distance, the input aperture distribution isre-constructed with the same periodicity as a result offree-space interference. This phenomenon is called theinteger Talbot effect. At other distances z = z T /m , thefields are self-imaged with m − times the spatial repeti-tion rate of the input aperture, as shown in the bottomof Fig. 1(a), i.e. Λ = X /m . This phenomenon is calledthe fractional Talbot effect. Therefore, in a conventionalspatial Talbot effect, the periodicity of the input apertureis always increased by an integer number m , i.e. Λ= X /m z }| { ψ T ( x, y, z T /m ) = ψ T ( x, y, | {z } Λ= X ∗ h ( x, y ) , (1)where h ( x, y ) is the impulse response of free-space. B. Generalized Talbot Effect
The conventional Talbot effect is observed under theassumption that there is no phase discontinuity acrossthe aperture so that ψ T ( x, y,
0) = | ψ T ( x, y, | . Thisassumption consequently restricts the repetition-rate in-crease factor m to integer values only. However, if theaperture is allowed to feature abrupt phase discontinu-ities across it, this restriction of integer-only values of m is lifted. This can be understood by considering two frac-tional Talbot distances z = z T /p and z = z T /q suchthat ψ T ( x, y, z T /p ) = | ψ T ( x, y, | ∗ h ( x, y ) (2) ψ T ( x, y, z T /q ) = | ψ T ( x, y, | ∗ h ( x, y ) . (3)Alternatively, the above equations can be written in acomplex spatial frequency domain as˜ ψ T ( k x , k y , z T /p ) = | ˜ ψ T ( k x , k y , | × ˜ H ( k x , k y ) (4)˜ ψ T ( k x , k y , z T /q ) = | ˜ ψ T ( k x , k y , | × ˜ H ( k x , k y ) . (5)Substituting | ˜ ψ T ( x, y, | from the second equation intothe first and using ˜ H ( k x , k y , z ) = e − ik z z , one get˜ ψ T ( k x , k y , z T /p ) = ˜ ψ T ( k x , k y , z T /q ) exp (cid:20) − ik z z T (cid:18) p − q (cid:19)(cid:21) (6)In this case considered, p < q . Inverse Fourier transform-ing the above equation leads to ψ T ( x, y, ∆ z ) = ψ T ( x, y, ∗ h ( x, y, ∆ z ) , (7)where ∆ z = z T (1 /p − /q ), where z = z T /q is chosen asa new reference plane. This equation can then be furtherwritten as Λ= X /p z }| { ψ T ( x, y, ∆ z ) = [ | ψ T ( x, y, | | {z } Λ= X /q × Metasurface z }| { T ( x, y ) ] ∗ h ( x, y, ∆ z ) , (8)where T ( x, y ) = ∠ ψ T ( x, y,
0) = ∠ ψ T ( x, y, z T /q ). Thisequation tells us that an input aperture field with a pe-riod Λ = X /q when multiplied with a phase function T ( x, y ) results in another periodic field at z = ∆ z witha period Λ = X /p . In other words, the period has beenscaled by a non-integer value m = q/p between the in-put and the output planes. Such a system is referredhere to as the generalized spatial Talbot system and thecorresponding effect as the generalized Talbot effect, asillustrated in the bottom of Fig. 1(b). The phase func-tion T ( x, y ) represents a spatial phase discontinuity pro-file which enables the scaling of the repetition rate ofthe input aperture fields by a non-integer value. Sucha phase discontinuity can be easily introduced using ametasurface, as will be shown in Sec. IV.In summary, when the input aperture field with theperiod Λ = X is spatially cascaded with a metasurfacewith transmittance T ( x, y ) = ψ T ( x, y, z T /q ), and propa-gated by a distance z = z T (1 /p − /q ), the output fielddistribution exhibits a scaled period Λ = ( q/p ) X . III. METASURFACE TRANSMITTANCEFUNCTIONS
As described in the previous section, the metasurfaceof Fig. 1, is a phase-only function, and it mimics thephase distributions of the fractional Talbot self-images,i.e. T ( x, y ) = exp { i ∠ ψ T ( x, y, z T /m ) } . In this section,the closed-form expressions of these phase distributionswill be developed.Let us assume for simplicity, a one-dimensional peri-odic array of sources with the amplitude distribution ψ in ( x ) = + ∞ X a = −∞ δ ( x − aX ) . (9)After a propagation through free-space along z − axis,the output fields are given by ψ ( x ) = ψ in ( x ) ∗ h ( x, z ),where the impulse response h ( x ) = exp( − iπx /λz ), un-der paraxial conditions (time convention used here is e jωt ). Using this equation with (9), and simplifying theconvolution integral, we get ψ ( x ) = e − i πx λz + ∞ X a = −∞ exp (cid:0) − iπa n (cid:1) exp (cid:18) in aπX x (cid:19) , (10)where X /λz = n assumed to be an integer, i.e. n ∈ -100 0 100-100-50050100 -20-15-10-50 -100 0 100-100-50050100 -1-0.500.51 -100 0 100-100-50050100 -20-15-10-50 -100 0 100-100-50050100 -20-15-10-50 -100 0 100-100-50050100 -1-0.500.51 -100 0 100-100-50050100 -20-15-10-50 I NPUT F IELD M ETASURFACE F REE - SPACE P ROPAGATION O UTPUT F IELD
PSfrag replacements y ( µ m ) y ( µ m ) y ( µ m ) y ( µ m ) y ( µ m ) y ( µ m ) x ( µ m) x ( µ m) x ( µ m) x ( µ m) x ( µ m) x ( µ m) | ψ ( x, y, | (dB) | ψ ( x, y, | (dB) ∠ T ( x, y ) ( π rad) ∠ T ( x, y ) ( π rad) | ψ ( x, y, z ) | (dB) | ψ ( x, y, ∆ z ) | (dB) | ψ ( x, y, ∆ z ) | (dB) / / − .
15 dB − . z = (1 / − / z T ∆ z = (1 / − / z T m = 4 / . m = 3 / . FIG. 2. Computed fields at the output plane when a periodic array of sources are phased following (13a) and (13b), to achievea repetition rate scaling by a factor of m = 1 .
33 and m = 1 .
5, respectively. Here X = 100 µ m and the individual sources areassumed to be Gaussian functions ψ ( x, y ) = exp[ − ( x + y ) / w ], with w = 5 µ m. The design frequency is 250 THz. Only asmall part of the overall aperture is shown for clarity. I . This corresponds to propagation distance z = z T /n ,where z T = X /λ , known as the Talbot distance.1. Case 1: when is n even , exp (cid:0) − iπa n (cid:1) = 1, andthus (10) reduces to ψ ( x ) = X n + ∞ X a = −∞ δ (cid:18) x − a X n (cid:19) exp (cid:18) − i πa n (cid:19) , (11)2. Case 2: when n is odd , (10) reduces after straight-forward manipulation to ψ ( x ) = X n + ∞ X a = −∞ δ (cid:18) x − a X n (cid:19) [1 − e jπa ] exp (cid:18) − i πa n (cid:19) (12)These equations are derived using the identity P ∞ a = −∞ exp(2 πjax/X ) = X P ∞ a = −∞ δ ( x − aX ).Equations (11) and (12) reveals that the period of theoutput field ψ ( x ) is smaller by a factor of n comparedto the input field in both cases, as expected at the frac-tional Talbot distance z = z T /n . Repeating the sameprocedure for a 2D array of sources, the complex phaseof the output self-imaged patterns at the location ( a, b )on the aperture can be verified to be [9] φ ( a, b ) = − π ( a + b ) n (13a) φ ( a, b ) = − π [(2 a + 1) + (2 b + 1) ]4 n (13b)These equations can be used to determine the complexphase of the fields at any fractional Talbot distance lyingbetween [0 , z T ], and thus can be used to construct themetasurface transmittance function T ( x, y ) for the caseof generalized Talbot effect as described in Sec. II. Itshould be noted that while the above phase functions aredeveloped only for the special case of z ∈ [0 , z T ], similarprocedure can be carried out to cover z ∈ [ z T , z T ] andso on [9].Figure 2 shows two examples, where the above princi-pal is applied to achieve scaling the spatial period of theinput field by a non-integer factor. In the first example,the input field has a period of 25 µ m and the desired in-crease in the period m = 4 / q/p , so that the outputΛ = (4 / × µ m. The metasurface transfer function T ( x, y ) is then constructed using (13a) with n = q = 4as shown in the middle of Fig. 2. The output of themetasurface is then free-space propagated by a distance∆ z = (1 /p − /q ) z T leading the output fields, as shownon the right of Fig. 2, and as expected an increase theperiod by a factor of 4 / /
2. It should be noted that while these ex-amples only illustrate a period increase by a non-integerfactor, similar demonstrations can be easily made for areduction as well by utilizing fraction Talbot distancesbetween [ z T , z T ]. IV. METASURFACE IMPLEMENTATION
Metasurfaces are two dimensional arrays of sub-wavelength electromagnetic scatterers, which are the di-mensional reduction of a more general volumetric meta-material structures. By engineering the electromagneticproperties of the scattering particles, the metasurface canbe used to manipulate and engineer the spatial wave-front of the incident waves. By this way, they provide apowerful tool to transform incident fields into specifiedtransmitted and reflected fields [10]. More specifically,metasurfaces can either impart amplitude transforma-tions, phase transformations or both, making them appli-cable in diverse range of applications involving lensing,imaging [7][8], field transformations [11], cloaking [12]and holograming [13], to name a few. Therefore, consid-ering their versatile field transformation properties andtheir electrically thin dimensions, they are ideally suitedto provide abrupt phase discontinuities in free-space re-quired in the proposed generalized Talbot effect.PSfrag replacements p m r ΛΛ 2 r tn n h x y z | R | = 0 (a)
245 250 255-25-20-15-10-50 245 250 255-25-20-15-10-50 P HASE
ONLY T RANSMITTANCE P ERFECT
REFLECTOR
PSfrag replacements frequency f (THz)frequency f (THz) | T | , | R | ( d B ) | T | , | R | ( d B ) pp , m m r = 27 nm κ = 0 . r = 55 nm κ = 0 . (b) FIG. 3. Huygens’ source metasurface based on all-dielectricresonators. a) Unit cell periodic in x − and y − direc-tions. b) Typical amplitude transmission (solid) and reflec-tion (dashed) for the two cases when the two dipole mo-ments p and m are frequency aligned, and not frequencyaligned, respectively. Design parameters: r = 300 nm, t = 220 nm, Λ = 666 nm, n h = 1 .
66 (Silica) and n = 3 . δ = 0 .
001 (Silicon).
To provide the needed phase-only filtering characteris-tics, the metasurface must exhibit ideally a unit ampli-tude transmission without any reflections, i.e. | T ( x, y ) | = 1 ∀ ∠ T ( x, y ) ∈ [0 , π ] and reflectance | R ( x, y ) | = 0.These specifications can be conveniently achieved usinga so-called Huygens’ metasurface . A huygens’ configu-rations consists of an orthogonally placed electric andmagnetic dipole moments [14], p and m , respectively,as shown in Fig. 3(a), resulting in a complete cancel-lation of backscattering as a result of destructive inter-ference of the fields generated by the two dipolar mo-ments. Metasurface consisting of such scattering parti-cles is perfectly matched to free-space and thus has zeroreflections, i.e. | R ( x, y ) | = 0. Under lossless conditions,a Huygen’s metasurfaces acts as an all-pass surface, with | T ( x, y ) | = 1 and ∠ T ( x, y ) = φ ∈ [0 , π ].A practical Huygen’s metasurface is conveniently real-ized using all-dielectric resonator arrays which naturallyproduce orthogonal p and m with lower losses comparedto their plasmonic counterparts [15]. A good review ona such all-dielectric metasurfaces can be found in [16]. Ageneralized unit cell of an all-dielectric Huygens’ meta-surface used in this work is shown in Fig. 3(a), consistingof a high-dielectric holey elliptical puck embedded in ahost medium of a lower refractive index n h . The puckhas an ellipticity of κ and the hole inside the puck hasthe elliptical shape as well, but rotated by 90 ◦ . This con-figuration is particularly useful because its transmissionphase at a fixed frequency, can be conveniently tuned byvarying the inner radius r and κ only, without affectingthe thickness, lattice size of the unit cell and the outerradius r , and simultaneously maintaining a good matchto free space.Fig. 3(b) shows a typical response of such a unit cellfor two sets of parameters r and κ , whereby in the firstcase, the two dipole moments p and m are properly ex-cited at the same design frequency (250 THz in this ex-ample). This results in an optimal interaction of the twodipoles resulting a near-perfect transmission of the wave,as expected from a Huygens’ source. This situation cor-responds to phase-only transmission response to be usedshortly in the generalized Talbot system. The secondcase, however shows the mis-aligned dipoles resulting ina strong reflection from the unit cell. This situation thuscorresponds to a near-perfect reflector.Using these two configurations, a metasurface aperturecan now be constructed to demonstrate a generalized Tal-bot effect. Let us take an example where the required pe-riod scaling at the output plane is m = 1 . q/p = 3 / q = 3 is odd, the discrete phase values are firstcomputed using (13b). Next, the metasurface unit cellof Fig. 3(a) is designed to approximate these phase val-ues. Fig. 4(a) shows the transmission and the phase ofthree such unit cell designs. The reflection in all casesis < −
10 dB which is sufficiently low in typical practi-cal situations. Using these transmission responses andthe perfect reflector unit cell configuration, a metasur-face aperture is formed as shown in Fig. 4(b). Thiscompletes the metasurface design. A plane wave inci-denting on this aperture, and propagating by a distance∆ z = (1 /p − /q ) z T = (1 / − / µ m) /λ (250 THz), -100 0 100-100-50050100 -20-15-10-50
245 250 255-3-2.5-2-1.5-1-0.5 -100 0 100-100-50050100 -2.2-2-1.8-1.6-1.4-1.2-1 -100 0 100-100-50050100 -20-15-10-50
245 250 255-2-1.5-1-0.50 P ERFECT R EFLECTOR (a) (b) (c)
PSfrag replacements r (nm) κ frequency f (THz)frequency f (THz) | T ( x, y ) | (dB) ∠ T ( x, y ) ( π rad) l og | T | ( d B ) P h a s e ∠ T ( π r a d ) | ψ ( x, y, ∆ z ) | (dB) x µ m x µ m x µ m y µ m y µ m y µ m
20 log | R | < −
10 dB
FIG. 4. Demonstration of the generalized Talbot effect using an all-dielectric metasurface, to achieve a period scaling by afactor of m = 1 .
5, as an example. a) FEM-HFSS simulated transmission and phase responses of three different metasurfaceunit cells to approximate the required discrete phases. b) Amplitude and phase transmittance of the metasurface apertureusing the unit cells of (a). c) The output fields at ∆ z = (1 /p − /q ) z T under plane-wave excitation of the metasurface aperturecomputed using Fourier propagation, i.e. ψ ( x, y, ∆ z ) = ψ ( x, y, ∗ h ( x, y ). Only a small part of the overall aperture is shownfor clarity. transforms into the output fields as shown in Fig. 4(c). Asexpected and required, the output period is now 50 µ m,and thus is m = 1 . V. CONCLUSIONS
A generalized spatial Talbot effect has been proposedwhere the period of the input aperture is scaled by a non-integer real value, as opposed to the integer-only factor ina conventional Talbot effect. This has been achieved by engineering phase discontinuity distributions in space us-ing metasurfaces, in conjunction with free-space propaga-tion. Specific implementations using all-dielectric meta-surfaces has also been presented and non-integer scalingsof the input aperture has been demonstrated using nu-merical results based on Fourier propagation. While thegeneralized Talbot effect has been discussed here in thespace domain, the proposed principle is equally appli-cable in the time domain based on space-time duality,where in that case, the repetition rate of the periodicpulse trains may be scaled by a non-integer factor usingtemporal phase modulators. [1] H. F. Talbot, Philos. Mag. , 401 (1836).[2] J. Wen, Y. Zhang, and M. Xiao, Adv. Opt. Photon. ,83 (2013).[3] J. Aza˜na and M. A. Muriel, Appl. Opt. , 6700 (1999).[4] J. Aza˜na and M. A. Muriel, IEEE Journal of SelectedTopics in Quantum Electronics , 728 (2001).[5] R. Maram, J. V. Howe, M. Li, and J. Aza˜na, Nat. Comm. (2014).[6] L. Romero Cort´es, R. Maram, and J. Aza˜na,Phys. Rev. A , 041804 (2015).[7] C. Holloway, E. F. Kuester, J. Gordon, J. O’Hara,J. Booth, and D. Smith, Antennas and PropagationMagazine, IEEE , 10 (2012).[8] N. Yu and F. Capasso, Nature Materials (2014).[9] W. Wang and C. Zhou, Optical Engineering , 2564(2004). [10] K. Achouri, M. Salem, and C. Caloz, IEEE Trans. An-tennas Propag. , 2977 (2015).[11] S. A. Tretyakov, Philosophical Transactions of the RoyalSociety of London A: Mathematical, Physical and Engi-neering Sciences (2015).[12] Y. Yang, H. Wang, F. Yu, Z. Xu, and H. Chen, ScientificReports (2016).[13] G. Zheng, H. Muhlenbernd, M. Kenney, G. Li, T. Zent-graf, and S. Zhang, Nat. Nanotech. , 308?312 (2015).[14] M. Kerker, The Scattering of Light and Other Electro-magnetic Radiation (Academic Press, New York, 1969).[15] M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N.Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar,Advanced Optical Materials , 813 (2015).[16] S. Jahani and Z. Jacob, Nature Nanotechnology2