Nodal Areas and Structured Darkness
Garreth J. Ruane, Sergei Slussarenko, Lorenzo Marrucci, Grover A. Swartzlander Jr
NNodal Areas and Structured Darkness
Garreth J. Ruane, Sergei Slussarenko, Lorenzo Marrucci, and Grover A. Swartzlander, Jr. ∗ Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology,54 Lomb Memorial Drive, Rochester, NY 14623, USA Dipartimento di Fisica, Universit di Napoli Federico II,Complesso Universitario di Monte S. Angelo, Napoli, Italy (Dated: October 24, 2018)Generalized beams of light that are made to turn inside out, creating black nodal areas of totaldestructive interference are described. As an example we experimentally created an elliptical nodefrom a uniformly illuminated elliptical aperture by use of a lossless phase mask called a q -plate.We demonstrate how a modified phase retrieval algorithm may be used to design phase masksthat achieve this transformation for an arbitrary aperture shape when analytical methods are notavailable. This generic wave phenomenon may find uses in both optical and non-optical systems. PACS numbers: 42.30.Kq, 42.25.-p, 42.79.-e, 42.25.Lc
As a corollary to the principle that nature abhors a vac-uum, it may be said that wave function amplitudes aredisinclined to be zero-valued. Exceptions are infinitesi-mally narrow nodal lines and points in the cross-sectionof a coherent wave. The former occurs when the phase ofthe wave differs by π across two regions, such as the zerosbetween Airy rings [1] or when a wave has odd symmetry(e.g. E ( x > , y ) = − E ( x < , y )). Nodal points appearin the cross section when the phase azimuthally variesfrom 0 to 2 π (or an integer multiple of 2 π ) about thezero-valued point, forming a vortex wave [2]. An addi-tional exception was discovered in the form of a circularnodal area [3–5]. The phase is singular in all these casesbecause it is undefined. Zero-valued points in a systemand their asymptotic limit are intriguing states of na-ture [6] and attract considerable attention. For example,laser speckle patterns are dense with vortices [7, 8]. Wavefunction vortices may exhibit fluid-like dynamics in lin-ear and nonlinear systems [9–12]. The dark structureof optical vortices enables advanced imaging techniques[13, 14], low roughness laser machining [15], nano-scalephotolithography [16], and trapping of particles [17–21].What is more, circular and spherical harmonic vortexwave functions are associated with spin and orbital an-gular momentum [22, 23]. In optics this enables encodingof information on a single photon [24], enhanced fiber orfree space communication capacity [25–28], and an as-sortment of manifestations in quantum optics [29]. Herewe explore whether other nodal area shapes generally ex-ist, beyond the circular case. In particular we describehow light transmitted through a uniformly illuminatedaperture may, in essence, be turned inside out by useof a lossless phase element. Optical radiation protectionand structured light illumination for various optics andphotonics applications may be envisioned. What is more,the ability to project a nodal area to a distant plane mayfind applications beyond the field of optics, such as elec-tron waves or sound.The purpose of this Letter is to introduce the basic concept of nodal areas and to pose currently unansweredquestions about them. We conjectured that any aper-tured beam of finite support may be optically trans-formed with a lossless phase element such that supportof the final beam is the complement of the original. Toour knowledge a general analytical proof of this conjec-ture does not yet exist. In fact, an analytical approachto find the matched functions of the phase element andthe final beam containing a nodal area is an underde-termined problem. However, we provide numerical ex-amples showing single and disconnected apertures thathave been transformed into nodal areas. An analyticaltransformation that turns an apertured beam inside outwas first realized for a circular aperture and was the basisfor the construction of a high contrast coronagraph [3–5]. Wondering whether this was a unique manifestation,we investigated turning other apertured beams inside out[30]. Here we explore how to form nodal areas using alossless phase mask such as a so-called q -plate [31] andexperimentally verify that the circle is not the only case.The discovery of other closed structures admitting nodalareas would be useful in optics for high contrast imagingapplications, optical patterning, or light-matter interac-tions. We point out that the zero-amplitude state is anidealized concept that in practice can only be approachedasymptotically owing to limitations such as wave frontaberrations, non-paraxial waves, the degree of coherence[32], and quantum mechanics [33].A circular aperture of radius R can be transformed intoan isomorphic circular nodal area in a 4- f optical system,illustrated in Fig. 1 [4, 5]. The lens L1 focuses a uniformplane wave of incident light in the x, y plane on a vortexphase element in the x (cid:48) , y (cid:48) plane. Transmission of theoptical field through the vortex element is representedby the function where m is a nonzero even integer calledthe topological vortex charge, and θ (cid:48) is the azimuth inthe transverse x (cid:48) , y (cid:48) plane (i.e. tan( θ (cid:48) ) = y (cid:48) /x (cid:48) ). For ex-perimental convenience we use a polarization-dependentelement called a q -plate as the vortex element [31]. The a r X i v : . [ phy s i c s . op ti c s ] A p r FIG. 1. Optical configuration for producing a ring of firewith a q -plate (QP). Uniform laser light that is circularlypolarized by polarizing optics P1 enters an aperture AP atthe x, y plane. Lens L1 focuses the transmitted light ontoQP. Lens L2 forms the exit pupil at the x (cid:48)(cid:48) , y (cid:48)(cid:48) plane. Thesecond set of polarizing optics P2 is circularly cross-polarizedto P1. The ring of fire function appears at the x (cid:48)(cid:48) , y (cid:48)(cid:48) plane.A CCD detector array captures the image of the ring of fire. transformed aperture function in the x (cid:48)(cid:48) , y (cid:48)(cid:48) plane may beunderstood as a convolution of the circular aperture withthe Fourier transform of the vortex transmission func-tion [34], F T { t m ( x (cid:48) , y (cid:48) ) } = a m (1 /r (cid:48)(cid:48) ) exp ( imθ (cid:48)(cid:48) ) where r (cid:48)(cid:48) and θ (cid:48)(cid:48) are radial coordinates in the x (cid:48)(cid:48) , y (cid:48)(cid:48) plane, a m = ( − i ) m +1 mf /k , f is the focal length of the lenses, k = 2 π/λ , and λ is the wavelength. Rather thanimaging the aperture, this system remarkably diffractsall the light outside the aperture, resulting in a nodalarea where E ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) = 0 in the interior ( r (cid:48)(cid:48) < R ),and a “ring of fire” in the exterior ( r (cid:48)(cid:48) > R ). In thecase where m = 2, the exterior ring of fire function is E ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) = ( R/r (cid:48)(cid:48) ) exp( i θ (cid:48)(cid:48) ).A generalized circular nodal area is expected for anylinear superposition of vortex transmission functions hav-ing nonzero even values of the topological charge [34].Such a superposition, however, does not change the cir-cular size or shape of the ring of fire. We questionedwhether different aperture shapes could produce isomor-phically similar rings of fire. To our knowledge it is cur-rently unknown whether the desired lossless transmis-sion functions mathematically exist. In fact, it is dif-ficult to imagine how other apertures and phase trans-mission functions could also produce a nodal area basedon either a convolution or direct integral point of view.We may, however, formulate a phase retrieval approachto calculating the required transmission function. Con-sider a 4- f imaging system with aperture function A ( x, y )and transmission function t ( x (cid:48) , y (cid:48) ) = exp[ i Φ( x (cid:48) , y (cid:48) )].The field at the focal plane just before the phase el-ement is given by the Fourier transform of the aper-ture F ( x (cid:48) , y (cid:48) ) = F T { A ( x, y ) } . Similarly, the exit pupilfield is G ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) = F T { F ( x (cid:48) , y (cid:48) ) exp[ i Φ( x (cid:48) , y (cid:48) )] } . Wewish to calculate the phase function Φ( x (cid:48) , y (cid:48) ) necessaryto form an exit pupil field G ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) that is zero val-ued over the support of F T { F ( x (cid:48) , y (cid:48) ) } . We were unsuc-cessful at finding a mathematical mapping that main-tains phase-only transmission for seemingly simple aper-tures such as a hyper-ellipse defined by the boundary ( x/a ) p + ( y/b ) p = 1, where p is a positive real number.An important aperture function from a practical pointof view is an annulus because many telescopes have aCassegrain design containing a central obscuration. Re-flecting microscope objectives also share this design. Thediscovery of a nodal area for this system could provide apath toward the direct observation of exoplanets. Variousattempts to achieve extreme high-contrast astronomicalimaging have met with limited success on Cassegrains[30, 35–37]. Assuming a rotationally symmetric annu-lar aperture, we may constrain the phase transmissionfunction to have azimuthal modal symmetry as in thecircular case, but allow radial variation. That is, thephase function may be written Φ = mθ (cid:48) + µ ( r (cid:48) ), where µ ( r (cid:48) ) is the rotationally symmetric component. Theexit pupil field becomes an m th order Hankel transform G ( r (cid:48)(cid:48) , θ (cid:48)(cid:48) ) = e imθ (cid:48)(cid:48) H m { F ( r (cid:48) ) e iµ ( r (cid:48) ) } and has the form G ( r (cid:48)(cid:48) , θ (cid:48)(cid:48) ) = g ( r (cid:48)(cid:48) ) e iν ( r (cid:48)(cid:48) ) e imθ (cid:48)(cid:48) , where g ( r (cid:48)(cid:48) ) and ν ( r (cid:48)(cid:48) )are the rotationally symmetric components of the am-plitude and phase, respectively. We were unsuccessful infinding an analytic expression for µ ( r (cid:48) ), g ( r (cid:48)(cid:48) ), and ν ( r (cid:48)(cid:48) )in general. Reverting instead to numerical methods, wefound that a modified Gerchberg-Saxton (G-S) phase re-trieval algorithm was suitable for predicting phase-onlymasks that were matched to a given aperture to producearbitrary nodal areas [38]. This method takes advan-tage of analytically inspired phase elements that diffracta substantial amount light outside of the exit pupil. Wehave found that a “helical axicon” phase element of theform µ ( r (cid:48) ) = ± αr (cid:48) , where α > kR/f is a suitable ini-tial condition for the optimization algorithm [39]. Theapproximate ring of fire is the Fourier transform of theproduct of the initial phase function and the point spreadfunction (PSF) of the optical system. The light that ap-pears within the geometric exit pupil is numerically setto zero and the field is inverse Fourier transformed. Thisyields a new focal plane amplitude and phase. The ampli-tude is replaced with that of the PSF and a new approx-imate ring of fire is computed via the Fourier transform.This process is repeated until the power within the geo-metric exit pupil decreases to a predefined stopping con-dition. The algorithm returns the focal plane phase thattransforms the PSF into a nodal area at the exit pupil.However, there is no guarantee that the calculated phasemask is amenable to existing fabrication techniques. Byconstraining the G-S algorithm to a single modal symme-try, it is possible to reduce the complexity of the phasefunction.An annular aperture, as well as the matched phasefunction of the vortex element and the predicted nodalarea, are all shown in Fig. 2(a-c). This, and other ex-amples not reported here, tends to confirm our conjec-ture that a phase-only (lossless) mask can be found toproduce a nodal area having the shape of an arbitraryentrance pupil. We see in Fig. 2(b) that the phase func-tion maintains the circular symmetry of the annulus, but FIG. 2. (a) An annular aperture matched with (b) the nu-merically computed phase mask generates (c) a ring of fireoutside of the outer radius of the aperture. (d)-(f) Same as(a)-(c), but with a disconnected elliptical aperture pair, pro-duced dual rings of fire (f). that radial rings add complex structure (see Fig 2(a)).The nodal area in Fig. 2(c) has the same diameter as theouter diameter of the annulus. It too has complex radialstructure not seen in the ring of fire for a circle. Theseradial structures change for different values of the ratioof the outer and inner radii of the annulus (0.38 for Fig.2(a)).To further test our conjecture, we explored whethermultiple disconnected apertures could also form nodalareas. Multiple disconnected nodal areas could inspire anew branch of optics involving structured darkness. Asa first step in this direction we used the same proce-dure above, except the nodal area was defined as a pairof ellipses and the modal constraints were lifted. Some-what surprisingly, a phase-only mask that satisfied thiscondition could indeed be found, starting from an ellip-tical m = 2 vortex as an initial guess in the G-S algo-rithm. The aperture function, computed phase function,and nodal areas are shown in Fig. 2(d-f). Again we findradial structure and other complicated patterns in thephase function. The construction of such a phase ele-ment may be possible with lithographic patterning tech-niques [40]. Undoubtedly the computed structures areindicative of modal symmetries and relationships yet tobe analytically discovered.The simplest aperture beyond a circle is an ellipse. Wefound a skewed vortex transmission function that pro-duces an elliptical nodal area and corresponding ring offire: ˜ t m ( x (cid:48) , y (cid:48) ) = exp( im Φ), where Φ = arctan( by (cid:48) /ax (cid:48) ),with the major and minor axes of the ellipse representedby a and b (see Fig. 3(a)) [30]. We note that skewed vor-tices were explored in relation to an optical Magnus effect[41] and interesting propagation dynamics [42]. A phaseelement producing the elliptical vortex function ˜ t m may be constructed by means of a q -plate, a spatially varianthalf-wave retarder with fast axis orientation described by α ( x (cid:48) , y (cid:48) ) = q arctan( by (cid:48) /ax (cid:48) ) + α , where q = | m/ | and α are constants [31]. The transmission of the q -platemay be written in terms of Jones matrices in the circu-lar polarization basis as E R,L = e ± i α E L,R , where E R and E L are respectively the right and left hand circularpolarization components of the incident field. Note thateach output vortex component is orthogonal to the inputstate [40]. The circularly polarized field components inthe image plane of the aperture are found to be, for thecase m = 2 U R,L ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) = E L,R ρ (cid:48)(cid:48) exp ( ± i , (1)where ρ (cid:48)(cid:48) >
1, Θ = arctan( ay (cid:48)(cid:48) /bx (cid:48)(cid:48) ), and ρ (cid:48)(cid:48) = [( x (cid:48)(cid:48) /a ) +( y (cid:48)(cid:48) /b ) ] / . Both components are zero-valued if ρ (cid:48)(cid:48) < q -plate used in this demonstration was preparedby aligning a nematic liquid crystal (LC) material withthe orientation angle α ( x (cid:48) , y (cid:48) ) = arctan( by (cid:48) /ax (cid:48) ), i.e. q = 1. The desired planar alignment of the liquid crys-tals was induced using a photoalignment technique [43].We used 0.1% wt. solution of Brilliant Yellow (BY) dyein dimethylformamide (DMF) as the aligning surfactantand Indium-Tin-Oxide (ITO) coated glass substrates soto have the possibility of applying an external electricalfield to the LC film. As part of the fabrication process[44, 45], a polarized UV laser beam was expanded by aset of lenses, sent through a half-wave plate and focusedon a sample with a cylindrical lens of 75 mm focal length.Both the waveplate and the sample were attached to ro-tating mounts connected to a computer through step-motors. By live control of the relative step size of twomotorized mounts during exposure it was possible to im-press elliptical orientation patterns with angular depen-dence. Owing to the manufacturing process, a relativelysmall defect, 25 µ m in diameter, occurs at the center ofthe sample, which is small compared to the dimensions FIG. 3. Transmission through an elliptical q -plate. (a) Phaseof the elliptical vortex function Φ( x (cid:48) , y (cid:48) ) with m = 2 and a/b = 2. Lines of constant phase (dotted) are indicated insteps of π/
4. (b) Image of the fabricated elliptical q -plateplaced between orthogonal linear polarizers, showing the ex-pected irradiance pattern given by sin (2 α ), where α is shownin steps of π/ of the q -plate (20 mm ×
15 mm).Figure 3(a) shows the expected phase pattern im-pressed on left-handed circularly polarized light; thatis, an elliptical vortex function with topological charge m = 2 and aspect ratio a/b = 2. When the q -plate isplaced between two orthogonal linear polarizers, the ex-pected transmitted irradiance, sin (2 α ), agrees well withthe experimental measurement shown in Fig. 3(b). Anelliptical ring of fire pattern was realized for the first timein the laboratory using the optical system illustrated inFig. 1 with the elliptical vortex q -plate. An expanded,collimated, He-Ne laser beam ( λ = 632 . a = 1 .
50 mmand b = 0 .
75 mm as depicted in Fig. 4(a). This beamis expected to form a ring of fire with the light turnedinside out, as in the theoretical image of Fig. 4(b). Theelliptical beam was focused by lens L1 onto the q -plate.The LC was tuned to the half-wave phase retardationcondition for wavelength λ = 632 . . µ m along the x (cid:48) axis and 1030 µ m along the y (cid:48) axis; that is, the focal spot is at least 20 times larger thanthe ∼ µ m central deformation of the q -plate. LensesL1 and L2 both have a focal length of 1 m, but differ-ent diameters: 25 . . x (cid:48)(cid:48) , y (cid:48)(cid:48) plane. An image of the entrance pupilforms on the detector array when the q -plate is removedfrom the system, or when the center of the q -plate isdisplaced from the center of the beam, as shown in Fig.4(c). However when these centers coincide, an ellipticalring of fire appears as seen in Fig. 4(d). A circular po- FIG. 4. Experimental verification of an elliptical nodal area.Theoretical beam profiles in the image plane of an ellipti-cal aperture with (a) and without (b) the elliptical vortex q -plate. Corresponding measured irradiance profiles (c) and(d), respectively. The relative quality of the measure nodalarea (d) is 0.0024. The scale bar in (c) also applies to (d). larizer analyzer (P2) was used to remove light that mayhave been transmitted in the wrong polarization state,owing to irregularities in the q -plate. A dark frame wascaptured immediately before exposing the CCD detectorarray (SBIG STF-8300 cooled to 0 . ◦ C).A practical gauge of the quality of the experimentallymeasured nodal area is the fraction of the total beampower that is detected across the innermost half of thenodal region. We achieved an experimental value of 2 . × − , compared to the theoretical value of zero.In conclusion, we have shown the experimental trans-formation of a uniformly illuminated elliptical apertureinto a dark nodal area of equal size and shape using anelliptical vortex q -plate. To our knowledge, this repre-sents the first experimental demonstration of a noncircu-lar nodal area. In addition, our numerical results suggestthat it is possible to form nodal areas of arbitrary shapewith intricately designed phase elements. The nodal ar-eas are surrounded by a luminous ring of fire and maybe made up of numerous detached shapes. Structuringdarkness in an optical field may be a valuable approachto light pattern control for various applications. We an-ticipate that additional experimental and theoretical in-vestigations will help uncover the mathematical proper-ties of nodal areas as well as identify new applications ofstructured darkness.We thank Thomas Grimsley, and both Lindsay Quandtand Michael Rinkus, all from RIT for respectively fab-ricating the elliptical apertures and photographic work.This work was supported by the U.S. Army Research Of-fice under grant number W911NF1110333-60577PH andby the FET-Open Program within the 7th FrameworkProgramme of the European Commission, under GrantNo. 255914, Phorbitech. ∗ [email protected][1] G. B. Airy, Trans. Cambridge Phil. Soc. , 283 (1834).[2] J. F. Nye and M. V. Berry, Proc. R. Soc. A. , 165(1974).[3] D. Rouan, P. Riaud, A. Boccaletti, Y. Cl´enet, and A.Labeyrie, Publ. Astron. Soc. Pac. , 1479 (2000).[4] D. Mawet, P. Riaud, O. Absil, and J. Surdej, Astrophys.J. , 1191 (2005).[5] G. Foo, D. M. Palacios, and G. A. Swartzlander, Jr., Opt.Lett. , 3308 (2005).[6] M. V. Berry, Phys. Today , 10 (2002).[7] N. B. Baranova, B. Y. Zel’Dovich, A. V. Mamaev, N. F.Pilipetskiˇi, and V. V. Shkukov, Pis’ma Zh. Eksp. Teor.Fiz. , 206 (1981) [JETP Lett. , 195 (1981)].[8] N. Shvartsman and I. Freund, Phys. Rev. Lett. , 1008(1994).[9] L. Landau, Phys. Rev. , 356 (1941).[10] V. L. Ginzburg and L. P. Pitaevskii, Zh. Eks. Teor. Fiz. , 1240 (1958), [Sov. Phys. JETP , 858 (1958)].[11] L. P. Pitaevskii, Zh. Eks. Teor. Fiz. , 646 (1961), [Sov.Phys. JETP , 451 (1961)]. [12] D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., Phys.Rev. Lett. , 3399 (1997).[13] S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, Opt. Express , 689 (2005).[14] V. Westphal and S. W. Hell, Phys. Rev. Lett. , 143903(2005).[15] J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S.Tanda, and T. Omatsu, Opt. Express , 2144 (2010).[16] M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, S.M. Tan, and N. Hayashi, J. Micro/Nanolith. MEMSMOEMS , 293 (2004).[17] A. Ashkin, Biophys. J. , 569 (1992).[18] H. He, M. E. J. Friese, N. R. Heckenberg, and H.Rubinsztein-Dunlop, Phys. Rev. Lett. , 826 (1995).[19] K. T. Gahagan and G. A. Swartzlander Jr., Opt. Lett. , 827 (1996).[20] J. Arlt and M. J. Padgett, Opt. Lett. , 191 (2000).[21] J. E. Curtis and D. G. Grier, Opt. Lett. , 872 (2003).[22] C. Cohen-Tannoudji, B. Diu, and F. Lalo¨e, Quantum Me-chanics (Hermann, Paris, 1977).[23] L. Allen, S. M. Barnett, and M. J. Padgett,
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