Realization of doubly inhomogeneous waveplates for structuring of light beams
RRealization of doubly inhomogeneous waveplates for structuring of light beams
Radhakrishna B, ∗ Gururaj Kadiri, ∗ and G. Raghavan Materials Physics Division, Materials Science Group,Indira Gandhi Centre for Atomic Research, HBNI, Kalpakkam, 603102, India † Waveplates having spatially varying fast-axis orientation and retardance provide an elegant andeasy way to locally manipulate different attributes of light beams namely, polarization, amplitudeand phase, leading to the generation of exotic structured light beams. The fabrication of such doublyinhomogeneous waveplates (d-plates) is more complex, compared to that of singly inhomogeneouswaveplates (s-plates) having uniform retardance, which can be easily fabricated by different meanssuch as photoalignment of liquid crystals, metasurfaces etc. Here, exploiting the SU(2) formalism,we establish analytically that any d-plate can be equivalently implemented using a pair of quarter-wave s-plates and a half-wave s-plate. An important advantage of this method is that it gives theflexibility to realize a whole family of distinct d-plates using the same triplet of s-plates. To underlinethe scope of this method, we propose novel d-plates for spatially tailoring the phase and complexamplitudes of light beams. Towards complex amplitude shaping, we present a generic methodfor carving out higher-order eigenmodes of light using a d-plate in conjugation with a polarizer. Ageneralized q-plate-like gadget, for imparting a polarization-dependent phase profile to a scalar lightbeam, is proposed as a demonstration of phase-polarization interplay. For these two illustrations,the corresponding three-s-plate gadget is constructed, and its functioning is validated with extensivenumerical simulations. The main result and its illustrations are generic and agnostic to the way thes-plates are fabricated and we believe they carry the potential to push the current state of the artin interdisciplinary applications involving structured light beams.
I. INTRODUCTION
Waveplates have been indispensable workhorses in op-tics laboratories, extensively used for manipulating thestate of polarization (SoP) of light beams. Traditionally,they are made from optically anisotropic materials andcharacterized by a uniform retardance and a unique ori-entation of fast-axis confined to its plane, classified hereas homogeneous waveplates. One could also fabricatewaveplates with spatial variation in either retardance orfast-axis orientation or both, and are here termed as in-homogeneous waveplates. For notational convenience, weclassify the waveplates based on their inhomogeneity, intofour kinds, as summarized in the Tab. (I).
Kind Retardance
Γ ( r, φ ) fast-axis α ( r, φ ) Homogeneous Uniform UniformSingly inhomogeneous 1 Uniform Nonuniform(s-plate) 2 Nonuniform UniformDoubly inhomogeneous Nonuniform Nonuniform(d-plate)Table I. Classification of waveplates in the order of increasinginhomogeneity.
A light beam with a spatially uniform SoP in its trans-verse plane, referred to as scalar beam, through a homo-geneous waveplate exits as a scalar beam with a differentSoP. In this process, it also picks-up a kind of geometric ∗ These two authors contributed equally † Radhakrishna B: [email protected] phase, called the Pancharatnam-Berry (PB) phase [1, 2],whose magnitude depends on the retardance and fast-axis orientation of the waveplate, in addition to the inputSoP. On the other hand, a scalar beam through an inho-mogeneous waveplate would emerge out with a spatiallyvarying SoP and/or phase. Light beams with spatiallyvarying SoPs are termed as vector beams[3, 4]. Gener-ation of light beams with spatially varying phase, i.e.,wavefront shaping, is also possible using inhomogeneouswaveplates, based on the idea of PB-phase. For instancea well-known s-plate of the first kind is the q-plate[5],whose fast-axis varies linearly with the azimuthal angle.The standard q-plates have a retardance π and a scalarlight beam with circular polarization through it acquiresa helical wavefront, in addition to flipped helicity in itspolarization.Such tailoring of light beams, with spatial variationin SoP, amplitude and phase, leads to a bigger classof light beams called structured light[6–8]. In the re-cent past, structured light has found increased appli-cations. For instance, radially polarized light, a kindof vector beams, are shown to provide sharper focusingin comparison with linearly polarized light [9]. Lightbeams with helical phase are known to carry orbitalangular momentum[10, 11] and this concept has givenbirth to many novel phenomena in optics like: spin-orbit interactions[12], spin-Hall effect[13] and these haveshown promising applications too[14, 15]. Structuringof light has also contributed in, among others, opticaltrapping[16–18], material processing[19] and quantum in-formation tasks[20].The last couple of decades has therefore witnessed aflurry of activity towards designing inhomogeneous wave-plates aimed at the generation of structured light beams. a r X i v : . [ phy s i c s . op ti c s ] F e b These efforts have been particularly successful with re-spect to s-plates, leading to diverse methods of fabricat-ing them being established, prominent ones being thosebased on photoalignment of liquid crystals[21–23] andmetasurfaces[24–26]. The richness offered by d-plates inapplications is just beginning to be explored[27–31], theiradaptation being slow perhaps due to the challenges intheir fabrication. Use of metasurfaces for fabricating thed-plates is now picking up steam[32, 33], but it involvesprecise control of dimensions in the nano-meter lengthscale. Further, in this method the parameters of the d-plate get fixed at the time of fabrication and cannot bedynamically tuned. In liquid crystal based s-plates, onthe other hand, it is possible to achieve active controlof retardance through external means like applied volt-age [34, 35] or temperature[36], but, to the best of ourknowledge, there has been no literature on fabrication ofd-plates using liquid crystals.These limitations in current methods of fabricating d-plates forces one to seek alternate ways of realizing them.In this context, it is interesting to explore whether astack of s-plates can effectively function like a d-plate.Such effective waveplates, have earlier assisted in real-izing many functionalities which otherwise are not pos-sible through individual waveplates. For instance, earlywork of Pancharatnam demonstrated that a half-wave(HW) plate sandwiched between two identically orientedquarter-wave (QW) plates functions effectively as a tun-able retardance waveplate and has been used for real-izing achromatic waveplates[37, 38] and are even avail-able commercially [39]. Stacking of s-plates has alsoyielded many novel results. For instance, combination ofq-plates and waveplates are used for changing the topo-logical charges of q-plates[40, 41]. Passively tuning theretardance of q-plates is possible from a combination of q-plates wherein, the retardance is tuned by merely chang-ing the relative orientation of the involved q-plates[42].In this article, we first demonstrate that an arrange-ment involving a pair of identically oriented QW-s-plateswith a HW-s-plate placed in between (henceforth referredto as QHQ-s-plate) is effectively equivalent to a d-plate.Conversely, given a d-plate, we establish that there ex-ists a unique QHQ-s-plate equivalent to it and derivethe fast-axis orientations of those s-plates. While localmanipulation of SoPs of light using waveplates is well-studied, spatial structuring of its complex amplitude andphase is more involved and rarely discussed. Thereforeas a strong demonstration of this d-plate ≡ QHQ-s-plate equivalence, we theoretically propose novel d-plate basedgadgets for local manipulation of amplitude and phase oflight beams. For these two cases, we numerically simulatethe corresponding QHQ-s-plate and validate its abilityto mimic the d-plate. These examples are of significantinterest and complexity in their own realm, and havespawned a large amount of literature.The first illustration is that of tailoring the com-plex amplitude of light beams. Different vari-eties of light beams are studied in the literature[43], for instance Laguerre-Gaussian(LG) and Hermite-Gaussian(HG) beams[44], non-diffracting beams likeBessel beams[45, 46], accelerating beams[47] and soon. Each of these beams have found numerous ap-plications, for instance, in reducing the thermal noiseof gravitational wave detectors [48, 49], in STEDmicroscopy[50, 51], in optical tweezers[52, 53] Bose-Einstein-condensation[54] etc. Here, we demonstratethat starting from the fundamental Gaussian mode oflight, higher-order LG and HG beams can be generatedusing a combination of d-plate and a polarizer. Similarstrategy of realization has been employed in [30, 55], butrestricted to higher order LG-beams.The next illustration is meant at designing a d-platethat imparts a polarization-dependent phase profile tothe input scalar light beam. This, in a sense, generalizesthe notion of q-plate to arbitrary elliptical polarization(instead of circular polarization) and arbitrary wavefront(instead of helical wavefront). This example also servesto bring out the limitation of d-plates in affecting PB-phase based SoP transformations.The rest of the article is arranged as follows: SectionII presents theoretical analysis of our scheme for realiz-ing d-plate, and in section III we provide two distinctcase studies as illustrations. We conclude the article byaggregating the essential results in section IV.
II. THEORY
The SoP of a light beam refers to the direction ofthe time-varying electric field vector in its transverseplane. SoPs are described in different but equivalentways, prominent ones being Jones vector, Stokes vectorand as points on the Poincare sphere. The Jones vector | θ, ϕ (cid:105) of an SoP is a two-dimensional complex vector ofunit norm given by: | θ, ϕ (cid:105) = cos θ | L (cid:105) + e iϕ sin θ | R (cid:105) (1)where | L (cid:105) and | R (cid:105) are left and right circular polariza-tions respectively, which in the horizontal-vertical ba-sis, | H (cid:105) = (1 , T and | V (cid:105) = (0 , T , are chosen to be | L (cid:105) = √ ( | H (cid:105) + i | V (cid:105) ) and | R (cid:105) = √ ( | H (cid:105) − i | V (cid:105) ) and ≤ θ < π , ≤ ϕ < π . One could also characterize theSoP | θ, ϕ (cid:105) by a three-dimensional real unit vector S θ,ϕ ,called the Stokes vector whose explicit expression is S θ,ϕ = [sin θ cos ϕ, sin θ sin ϕ, cos θ ] T (2)The Jones and Stokes vector representations of a polar-ization state are connected by[56] S θ,ϕ = (cid:104) θ, ϕ | σ | θ, ϕ (cid:105) (3)where σ = ( σ x , σ y , σ z ) is a Pauli-spin vector with com-ponents: σ x = (cid:34) − (cid:35) , σ y = (cid:34) (cid:35) , and σ z = (cid:34) − ii (cid:35) (4)These SoPs can also be represented geometrically aspoints on the surface of a unit sphere, called the Poincaresphere. In this description, the SoP | θ, ϕ (cid:105) is mapped tothe point whose polar and azimuthal coordinates are θ and ϕ respectively.SoP of a light beam can be altered, without chang-ing its intensity, by use of anisotropic optical elementscalled waveplates. They are characterized by two pa-rameters: retardance Γ and orientation of the fast-axis α . They function by introducing a phase difference of Γ between the electric field component along the fast-axis α and its orthogonal direction. For instance QW-plateand HW-plate introduce a phase difference of Γ = π and π respectively. As waveplates merely introduce phase dif-ference between the orthogonal components, they do not alter the intensity of the light beam. Hence, their actionis mathematically described by norm-preserving matri-ces, which in Jones vector formalism are unitary matricescalled Jones matrices and in Stokes vector formalism areorthogonal matrices.By definition, Jones matrix of waveplates accounts forthe linear transformation between two Jones vectors ofSoPs. Jones vectors, however, are arbitrary up to a globalphase factor, and hence waveplates that transform theSoPs can be represented by unitary matrices, withoutregard to the determinant. Nevertheless, for correct han-dling of the associated PB phase change, it is preferableto represent Jones matrices as unitary matrices of unitdeterminant, i.e., SU (2) matrices [56]. Therefore, in therest of the article, the Jones matrices of waveplates aredescribed using SU(2) matrices.On the Poincare sphere, the action of a waveplate withparameters Γ and α , on an SoP | θ, ϕ (cid:105) is described as arotation of its Stokes vector S θ,ϕ , about the rotation axis (cos 2 α, sin 2 α, , by the angle Γ . The Jones matrix ofsuch a waveplate, denoted as W Γ ( α ) , in the {| H (cid:105) , | V (cid:105)} basis is given by: W Γ ( α ) = cos (cid:0) Γ2 (cid:1) I + i sin (cid:0) Γ2 (cid:1) (cos 2 α · σ x + sin 2 α · σ y ) (5) = (cid:34) cos (cid:0) Γ2 (cid:1) + i sin (cid:0) Γ2 (cid:1) cos 2 α i sin (cid:0) Γ2 (cid:1) sin 2 αi sin (cid:0) Γ2 (cid:1) sin 2 α cos (cid:0) Γ2 (cid:1) − i sin (cid:0) Γ2 (cid:1) cos 2 α (cid:35) ∈ SU (2) (6)where I is the × identity matrix. It may be notedthat W Γ ( α ) is a symmetric matrix, with diagonal ele-ments being complex conjugates of each other, and off-diagonal elements being purely imaginary.For a stack of waveplates, their combined action is de-scribed by a matrix obtained by multiplying the Jonesmatrix of each waveplates. While the product of Jonesmatrices of waveplates is also an SU(2) matrix, it neednot be a Jones matrix of the form of Eq. (6). However, ifthe resulting matrix is also of this form, then the sequenceof waveplates effectively functions like a single waveplate W Γ e ( α e ) , having an effective retardance Γ e and an ef-fective fast-axis orientation α e which can be determinedfrom the SU(2) matrix: Γ e = 2 cos − (cid:18) trace ( W Γ e ( α e )) (cid:19) , (7) α e = 12 tan − (cid:18) w w (cid:19) (8)where w jk is the imaginary part of [ W Γ e ( α e )] jk .As mentioned in the previous section, an arrangementof HW-plate sandwiched between two identically orientedQW-plates acts like a single waveplate, with effectiveretardance Γ e depending on the relative orientation be- tween the HW-plate and QW-plates, and effective fast-axis orientation α e inclined at an angle of π with re-spect to the QW-plates. This arrangement of waveplates,called here as QHQ-waveplate, enables realizing a tun-able retardance waveplate where the tuning is achievedby merely changing the relative orientation of the plates.The above Jones matrix formalism of waveplates canbe extended to s-plates and d-plates, except that now thematrix W Γ ( α ) is a function of the radial and azimuthalcoordinates ( r, φ ) of the plate, through Γ or α or both.For instance, the standard q-plate[5] is the most studieds-plate of the first kind, with retardance π , and fast-axisorientation α varying linearly with the azimuthal angleas α ( φ ) = qφ + α , where q is the topological chargeand α is the offset angle. Its Jones matrix is therefore afunction of azimuthal coordinate φ : W π ( φ ) = i ( σ x cos 2 ( qφ + α ) + σ y sin 2 ( qφ + α )) (9)Here we extend the notion of QHQ-waveplate to suchinhomogeneous waveplates and explore its effective be-havior. Since the QHQ-arrangement is possible only withs-plates of first kind, in the rest of the article, s-plate al-ways refers to the s-plate of first kind classified in theTab. (I), unless mentioned otherwise.Consider a HW-s-plate with fast-axis orientation α H ( r, φ ) placed in between two identically oriented QW-s-plates having fast-axis distribution α Q ( r, φ ) . We re-fer to this QHQ arrangement of s-plates as the “QHQ-s-plate”. The effective retardance Γ e ( r, φ ) and the effectivefast-axis orientation α e ( r, φ ) of the QHQ-s-plate is (seeappendix A): Γ e ( r, φ ) = 2 π + 4 ( α Q ( r, φ ) − α H ( r, φ )) , (10) α e ( r, φ ) = α Q ( r, φ ) + π (11) Figure 1. Schematic for realizing effective d-plates using QHQarrangement of s-plates (color-coded by the retardance andarrows indicating the fast-axis orientations). The different d-plates depicted on the top correspond to different orientationsof the same set of QHQ s-plates.
It should be noted that the effective retardance Γ e acquires spatial dependence, and thereby converts thethree s-plates into an effective d-plate. Conversely, andmore importantly, any desired d-plate with Γ e ( r, φ ) and α e ( r, φ ) can be uniquely realized as QHQ-s-plate with α Q ( r, φ ) and α H ( r, φ ) given by: α Q ( r, φ ) = α e ( r, φ ) − π , (12) α H ( r, φ ) = α e ( r, φ ) + ( π − Γ e ( r, φ )) (13)It should be emphasized that the QHQ-s-plate is com-pletely equivalent to the d-plate, not just in bringingabout the SoP transformations but also in capturing theassociated PB phases correctly. Moreover, an importantadvantage of realizing d-plate through this method is thata single set of QHQ s-plates can be employed for gener-ating a variety of d-plates. This is possible by merelychanging the orientations of three s-plates, constrainingto the QHQ arrangement.To elaborate, consider a QHQ-s-plate, having fast-axisvariation of QW-s-plate and HW-s-plate as α Q ( r, φ ) and α H ( r, φ ) respectively, set for realizing a d-plate with pa-rameters Γ e ( r, φ ) and α e ( r, φ ) . Rotating these s-platesby angles δ Q and δ H respectively about their axes resultsin new s-plates having α newQ ( r, φ ) and α newH ( r, φ ) , see Eq.(B4) of appendix (B). These new s-plates continue to re-main in the QHQ arrangement and therefore yeild a newd-plate, with parameters Γ newe ( r, φ ) and α newe ( r, φ ) givenby: Γ newe ( r, φ ) = 2 π + 4 ( α Q ( r, φ − δ Q ) − α H ( r, φ − δ H ) + ( δ Q − δ H )) (14) α newe ( r, φ ) = α Q ( r, φ − δ Q ) + δ Q + π (15)The five insets in Fig. (1) depict the distinct d-platesrealized using the same physical set of three s-plates butby rotating them constraining to QHQ arrangement.In the following section, we explore the possible appli-cations of these results towards spatial control of ampli-tude and phase of light beams. III. ILLUSTRATIONS OF STRUCTURING THELIGHT BEAM USING D-PLATESA. Tailoring the complex amplitude of light beams
In this subsection, a generic method for spatial tai-loring of the amplitude and phase of light beams, usingd-plates is proposed. As a demonstration of this tech-nique, realization of higher order LG beams and HGbeams starting from a fundamental Gaussian beam isdiscussed. We aim to tailor a desired complex electricfield E des ( r, φ ) , starting from a scalar light beam havingelectric field E in ( r, φ ) . We achieve this by transformingthe input scalar light beam using a d-plate, such thatone of the components of the emerging vector beam, ina particular orthogonal basis, is E des ( r, φ ) . This compo-nent can be extracted by projecting out the orthogonalcomponent. For concreteness, we work in {| H (cid:105) , | V (cid:105)} ba-sis, and let the SoP of the initial light beam E in ( r, φ ) be | H (cid:105) . The transformation briefed above can be achievedby an SU (2) transformation T ( r, φ ) : T ( r, φ ) ( E in ( r, φ ) | H (cid:105) ) = E des ( r, φ ) | H (cid:105) + E rem ( r, φ ) | V (cid:105) (16)where E rem ( r, φ ) is the remnant electric field, satisfyingthe relation: | E des | + | E rem | = | E in | (17)Assuming E in ( r, φ ) does not vanish within the regionof interest, T ( r, φ ) | H (cid:105) = E des ( r, φ ) E in ( r, φ ) | H (cid:105) + E rem ( r, φ ) E in ( r, φ ) | V (cid:105) (18)Given the action of T ( r, φ ) on | H (cid:105) , its action on verti-cal polarization | V (cid:105) gets fixed because of its SU (2) prop-erty: T ( r, φ ) | V (cid:105) = − (cid:18) E rem ( r, φ ) E in ( r, φ ) (cid:19) ∗ | H (cid:105) + (cid:18) E des ( r, φ ) E in ( r, φ ) (cid:19) ∗ | V (cid:105) (19)where ∗ denotes the complex conjugation. Hence, thematrix of T ( r, φ ) in the {| H (cid:105) , | V (cid:105)} basis is given by T ( r, φ ) = E des E in − (cid:16) E rem E in (cid:17) ∗ E rem E in (cid:16) E des E in (cid:17) ∗ (20)The SU (2) matrix T ( r, φ ) will correspond to theJones matrix of a d-plate, as in Eq. (6), pro-vided its off-diagonal elements are purely imaginary.The ratio appearing along the off-diagonal is E rem E in = | E rem || E in | e i ( δ rem − δ in ) , where δ rem and δ in are the respec-tive phases which, in general, can be spatially variant.As only the magnitude of E rem ( r, φ ) is fixed throughEq. (17) leaving us free with the choice of δ rem , theoff-diagonal entries can be rendered purely imaginary bysetting δ rem = δ in + π , so that the resulting matrix canthen be realized by a d-plate W Γ ( α ) . Placing a polar-izing beam splitter at the state determination stage (seeFig. (1)), yields the desired field in the horizontal armand the remnant field in the vertical arm.The required retardance Γ ( r, φ ) and fast-axis orienta-tion α ( r, φ ) of the d-plate are extracted from the result-ing matrix through Eqs. (7 and 8): cos Γ( r,φ )2 = | E des || E in | cos ( δ des − δ in ) , (21) tan 2 α ( r, φ ) = | E rem || E des | δ des − δ in ) (22)It follows from Eqs. (21 and 22) that for the case of δ des = δ in , the fast-axis orientation α ( r, φ ) equals π atall ( r, φ ) , reducing the d-plate to an s-plate of the secondkind (see Tab. I).Here, we apply this idea towards simulating LG andHG modes of higher order. The functional forms of thesemodes are readily available, for instance[44]. At the lo-cation of the beam waist ( z = 0) , they simplify to: E l,p ( r, φ ; A , w ) = A w (cid:32) √ rw (cid:33) | l | L | l | p (cid:18) r w (cid:19) e − r w e ilφ (23) E m,n ( x, y ; A , w ) = A w H m (cid:32) √ xw (cid:33) H n (cid:32) √ yw (cid:33) e − x y w (24)where L | l | p denotes the associated Laguerre polynomi-als with l and p being the azimuthal and radial indicesof the LG beam respectively; H m refers to the Hermitepolynomial of order m ; w is the beam waist and A is aconstant, indicating the power of the light beam.We chose the input beam as the horizontally polarizedfundamental Gaussian mode, since it is non-vanishing atall finite values of ( r, φ ) , thereby satisfying the conditionrequired for Eq. (18). Placing the d-plate at the locationof the input beam waist, we have E in = E , ( r, φ ; A in , w in ) (25)The LG and HG modes simulated are E des = E l,p ( r, φ ; A des , w des ) and E des = E m,n ( x, y ; A des , w des ) respectively. The possible values for A des and w des is re-stricted through Eq. (17) which demands | E des ( r, φ ) | ≤| E in ( r, φ ) | at all ( r, φ ) within the region of interest in thetransverse plane, which in turn dictate the conversion ef-ficiency of the d-plates.We now illustrate the working of this method by study-ing a few cases through numerical simulations. We con-sider ( l, p ) = (1 , and (2 , in case of LG modes, and incase of HG modes, ( m, n ) = (1 , and (2 , . The valuesof A des and w des in all four cases are taken to be A in and w in respectively. The retardance and fast-axis orien-tations of the d-plates required for this apodization arecalculated using Eqs. (21 and 22). The required fast-axisorientations of the QW-s-plates and HW-s-plate for re-alizing these d-plates are respectively determined usingEqs. (12 and 13) and are summarized in Fig. (2).To validate the conversion to higher order modes, evo-lution of light through the QHQ-s-plates is numerically Figure 2. Prototype of d-plates and the corresponding s-platesrequired for realizing the higher order LG and HG modesstarting from the fundamental Gaussian mode. The top rowdepicts the d-plate whose spatial distribution of retardance iscoded in color and its fast-axis orientations depicted by ar-rows. Second and third rows depict the fast-axis orientationsof the QW-s-plates and HW-s-plate respectively required forrealizing the above d-plates. The first and second columncorrespond to LG beam with ( l, p ) = (1 , and (2 , respec-tively, while the third and fourth columns correspond to HGbeam with ( m, n ) = (1 , and (2 , respectively. simulated, ignoring the diffraction and propagation ef-fects. The input scalar beam, as it emerges out of theQHQ-s-plate, gets converted into a vector beam throughEq. (6). The numerically simulated intensity and SoPdistributions in the transverse plane at the exit plane ofthese QHQ-s-plates is depicted in Fig. (3). The desiredmodes and remnant fields are contained in the horizon-tal and vertical component of these vector beams, whichcan be extracted by projecting them, for instance, usinga polarizing beam splitter. Figure (3) summarizes the in-tensity distributions, | E des | and | E res | for all the fourcases. The spatial distribution of intensity in the hor-izontal projection is consistent with that of the desiredmodes. The ratio of beam waists of output and inputbeams appears to be a crucial parameter in deciding theefficiency of conversion, as also observed in [30].To summarize this subsection, d-plate, together with astandard polarizer, can aid in sculpting any complex am-plitude light beams, out of the ubiquitous fundamentalGaussian beams. B. Arbitrary polarization-dependent wavefrontshaping
Waveplates, apart from manipulating the SoP, also im-part a total phase (=dynamic phase + geometric phase),which depends on the plate parameters and the involvedSoPs. Given a pair of SoPs, there always exists a wave-plate that transforms one state to other unitarily andpicking up a particular phase in the process, depending
Figure 3. Intensity profile of higher order LG and HG beamssimulated numerically using the QHQ arrangement of s-platesand polarizing beam splitters. Along with intensity, the toprow depicts SoPs distribution at the exit plane of the QHQarrangement of s-plates. Second and third rows are the inten-sities projected along horizontal and vertical directions cor-responding to the desired (cid:0) | E des | (cid:1) and remnant (cid:0) | E rem | (cid:1) intensities respectively. on the path of transformation. Owing to the spatial in-homogeneity of their fast-axis orientation, s-plates arecapable of imparting desired phase distribution onto theinput light beam. In these lines, the most studied s-platesare the q-plates which basically flip the handedness of theinput circular polarization, and impart a helical phase inthe process. In this subsection, we seek to extend thisfunctioning of q-plates to arbitrary SoPs instead of cir-cular polarizations, and to arbitrary phase distributionsinstead of helical phase. The mathematical treatment re-quired in the conversion of an arbitrary SoP into an SoPof flipped handedness, and with arbitrary phase distribu-tion is first discussed. A d-plate based gadget is designedto impart a polarization-dependent phase distribution tothe input light beam.In general, using a waveplate it is not possible to trans-form SoPs with a given phase, as the possible rotationaxes for transforming them are limited to two directions,thereby restricting the number of phases to two. Thisstems from the fact that, the midplane of the SoPs in-volved intersects the equatorial plane at only two pointson the Poincare sphere (which are antipodal), unless themidplane coincides with the equatorial plane (see Fig. 1in the ref. [57]). Such a case occurs when the SoPs in-volved are enantiogyres[58], that is, mirror reflections ofeach other about the equatorial plane of the Poincaresphere. Since the midplane of the enantiogyres coin-cides with the equatorial plane of the Poincare sphere,every fast-axis orientation α of the waveplate is a rota-tion axis for transforming the states. This enables ac-cessing every possible path in unitary transformation ofenantiogyres[29]. In other words, for a waveplate orientedat any angle α ∈ (0 , π ) , there always exists a retardance Γ depending on α , such that W Γ ( α ) takes an SoP to itsenantiogyre. Each one of these W Γ ( α ) generate a differ-ent phase between to π in the process.Conversely, given a pair of enantiogyres and a desiredspatial phase distribution, one could always conceive ofa d-plate W Γ ( α ) affecting such a transformation. In thissubsection, we provide a general prescription for con-structing these d-plates, and their realization using QHQarrangement of s-plates. This exercise demonstrates, tai-loring the wavefront of a light beam and also assist toachieve an arbitrary spin-orbit conversion.Towards this, we seek a unitary transformation U , thattransforms the SoP | θ, ϕ (cid:105) to its enantiogyre | π − θ, ϕ (cid:105) : U | θ, ϕ (cid:105) = e iψ ( r,φ ) | π − θ, ϕ (cid:105) (26)Being a unitary transformation, U naturally trans-forms | π − θ, π + ϕ (cid:105) (the orthogonal state of | θ, ϕ (cid:105) ) to | θ, π + ϕ (cid:105) (the orthogonal state of | π − θ, ϕ (cid:105) ): U | π − θ, π + ϕ (cid:105) = e iχ ( r,φ ) | θ, π + ϕ (cid:105) (27)where ψ ( r, φ ) and χ ( r, φ ) are arbitrary spatially varyingphases. The determinant of this unitary transformationis e i ( ψ + χ ) (see appendix C). Since, waveplates are math-ematically SU(2) (determinant= ), they alone cannotbring about this transformation. However any unitarytransformation U (2) is equivalent to U (1) · SU (2) [59, 60],hence the above transformation is possible by employinga phase plate ( U (1)) for inducing a phase of (cid:16) ψ + χ (cid:17) , inconjugation with a waveplate: U = e i ( ψ + χ ) I · W Γ ( α ) (28)where e i ( ψ + χ ) I indicates the action of phase plate. Therequired retardance Γ and fast-axis orientation α of thewaveplate depend both on the SoP parameters θ , ϕ andthe difference of the phases ψ , χ (see appendix C): cos Γ2 = sin θ cos ∆ (29) tan 2 α = sin (∆ − η ) (cid:112) sin ϕ + cos ϕ cos θ sin θ sin (∆ − ϕ ) + cos θ sin (∆ + ϕ ) (30)where ∆ ( r, φ ) = ψ − χ and η = tan − (cot ϕ cos θ ) .For transformation between special enantiogyres leftcircular polarization ( | , (cid:105) ) and right circular polariza-tion ( | π, (cid:105) ) , the retardance is uniformly π , independentof the phases, and the fast-axis orientation is of the form α ( r, φ ) = π + ∆2 , indicating that the necessary waveplateis a HW-s-plate. For any other enantiogyres, parameters Γ and α both acquire a spatial dependence through ∆ ,thereby demanding W Γ ( α ) to be a d-plate. As d-platecan be realized by the QHQ arrangement of s-plates, aug-menting this arrangement with a phase plate makes it possible to generate scalar beams having any desired spa-tial variation of phase.As a concrete illustration of this idea, we now studythe so-called J-plate[29, 33] proposed for converting anarbitrary SAM to OAM states of light. J-plate extendsthe notion of q-plates for circularly polarized light toarbitrary elliptically polarized light. For J-plates, thephase distribution of Eqs. (26 and 27) are of the form ψ ( r, φ ) = mφ and χ ( r, φ ) = nφ , where m and n are in-tegers. The phase plate required for this turns out to bea spiral phase plate of order m + n [61].Here we demonstrate the construction of such a J-plate for transforming the SoP (cid:12)(cid:12) π , (cid:11) to its enantiogyre, (cid:12)(cid:12) π , (cid:11) . The required retardance and fast-axis orienta-tions of Eqs. (29 and 30) in this case simplify to: cos Γ2 = cos ∆ √ (31) tan 2 α = − cot ∆ √ (32)where ∆ = m − n φ . Figure (4) shows the azimuthal varia-tion of Γ and α of the above equations, for ∆ = φ, φ and φ . The necessary fast axes variation of the QW-s-platesand HW-s-plates for realizing these using QHQ arrange-ment are also shown. The smooth azimuthal variation of α Q and α H of the s-plates indicate the ease of their fabri-cation. The jumps observed in the fast-axis variations arealways of magnitude π and arise due to numerical inver-sion of Eq. (32). Since orientation of fast-axis is modulo π , these jumps are experimentally inconsequential. Figure 4. Azimuthal variation in the retardance (Γ) and fast-axis orientation ( α ) of d-plates W Γ ( α ) of Eqs. (31 and 32), for m − n = 1 , , is plotted in the left column. For these d-plates,the corresponding azimuthal variation of fast axes orientationof QW-s-plates ( α Q ( φ )) and HW-s-plates ( α H ( φ )) requiredfor realizing them through QHQ arrangement are shown inthe right column. We now demonstrate the functioning of the QHQ ar-rangement towards realizing the J-plates. Figure (5)traces the simulated phase and polarization profile in thetransverse plane of the light beam as it exits through eachof the four plates (spiral phase plate and QHQ-s-plates).The input SoPs considered are (cid:12)(cid:12) π , (cid:11) and its orthogo-nal state (cid:12)(cid:12) π , π (cid:11) , shown as rows (A) and (B) respectively.Rows (1), (2) and (3) correspond to the J-plates hav-ing ( m, n ) =(3 , , (6 , and (9 , respectively. TheseJ-plates correspond to the d-plates of Fig. (4) togetherwith the spiral phase plate of order , and respec-tively. In these figures, the polarization profile along theradial direction is uniform and hence is suppressed for vi-sual clarity. The greyscale color coding depicts the phasein the transverse plane. In each of the six cases, eventhough the input scalar beam gets converted to vectorbeam in transit, the beam at the exit plane of the finalplate is once again scalar but with the intended phasedistribution. This demonstrates the utility of the QHQ-s-plates towards mimicking the J-plate. Figure 5. Polarization and phase profile of the light beamin its transverse plane, after emerging from the spiral phaseplate and the QHQ arrangement of s-plates, set for realizingthree different J-plates. The chosen polarization parametersfor the J-plate are θ = π and ϕ = 0 , and ( m, n ) being (3 , , (6 , and (9 , . Rows A corresponds to the SoP | π , (cid:105) androws B correspond to its orthogonal SoP | π , (cid:105) as inputs. To summarize this subsection, a generalized version ofq-plates, for affecting polarization-dependent phase pro-file to the input light beam is mathematically elucidatedthrough Eqs. (26 and 27). Parameters of the d-platethat assists this transformation are deduced in Eqs. (29and 30). A simple, albeit non-trivial illustration of thisidea has been demonstrated numerically in realizing the“J-plate” proposed recently in ref. [29].
IV. SUMMARY AND CONCLUSIONS
Optical elements which affect the polarization andphase of the light beam, without affecting its intensitybelong to the class of “waveplates”. They are charac-terized by their retardance and orientation of fast-axis,and have been classified here into different kinds based onthe spatial variation of these parameters. Waveplates ex-hibiting inhomogeneity in both their retardance and fast-axis orientation, referred to as d-plates here, have provedto be of immense utility in the recent years. The inter-play of phase and polarization of the light, as it traversesthrough the d-plate, can be gainfully exploited towardsrealizing exotic structured light beams.Fabrication of these d-plates, however, continues to bea formidable challenge, in contrast to singly inhomoge-neous waveplates (s-plates), whose manufacture has beenstandardized in multiple ways. In this article a novelmethod for realizing “effective” d-plate involving only s-plates, has been proposed. This is achieved by establish-ing the equivalence of a d-plate and a gadget involving aHW-s-plate placed between two identically oriented QW-s-plates, using SU(2) formalism. Given a d-plate, -withan arbitrary spatial variation in both retardance and fast-axis orientation,- the necessary HW-s-plate and QW-s-plates for realizing it have been identified. An importantadvantage of this approach is that, using the same set ofthree s-plates, a variety of inequivalent d-plates can be re-alized by mere change of their relative orientations. Themethodology is generic enough to be employed with anyof the s-plate fabrication techniques. Further, the abil-ity to dynamically tune the retardance of s-plates (foreg. with voltage controlled liquid-crystal based s-plates)allows for QHQ-s-plates to realize d-plates for differentwavelengths.The versatility and potential of this method has beenexhibited in two steps: firstly, specific d-plates requiredfor tailoring individual aspects of light have been iden-tified. For each of these d-plates, the required s-plateshave been identified and their equivalence with the d-plates has been analytically and numerically established.Another significant contribution of this article has beenin fashioning d-plates for (i) realizing an arbitrary com-plex field amplitude distribution and (ii) polarization-dependent phase manipulation.In the first application, starting from a known electricfield distribution, a generic polarimetric approach for re-alizing arbitrary complex electric field distribution hasbeen presented. As a concrete example, LG beams andHG beams of higher order have been sculpted from thehorizontally polarized fundamental Gaussian beam, us-ing a specially designed d-plate and a polarizer. Theratio of input and output beam waists is seen to play animportant role in deciding the conversion efficiency andneeds to be optimized.The second application presented here generalizes thefunctioning of q-plates. The phase-polarization interplayseen in waveplates has been exploited in designing a d-plate that functions like a polarization-dependent spa-tially varying phase plate. This d-plate imparts distinctphase distribution on the input light beam depending onits polarization.The d-plates discussed here, although of significant in-terest in themselves, are not exhaustive but merely cur-sory pointing to their rich class. QHQ-s-plates offer aneasy method of realizing any such d-plates, and we be-lieve they will greatly advance the experimental state ofthe art in structuring of light and application involvingstructured light beams.
Appendix A: Single effective waveplate.
The Jones matrices of HW-plate (retardance
Γ = π ) isgiven by (see Eq.(5)) W π ( α ) = i ( σ x cos 2 α + σ y sin 2 α ) and the Jones matrix of QW-plate (retardance Γ = π )is given by: W π ( α ) = √ ( I + i ( σ x cos 2 α + σ y sin 2 α )) In QHQ arrangement, an HW-plate oriented at anyangle say α H is sandwiched between two identically ori-ented QW-plates α Q . W = W π ( α Q ) · W π ( α H ) · W π ( α Q ) (A1)The resultant matrix W of QHQ arrangement can beevaluated by making use of the Pauli-matrices identitynamely: σ j · σ k = δ jk I + i(cid:15) jkl σ l , given by: W = (cid:34) − cos (2 α Q − α H ) − i sin (2 α Q − α H ) sin 2 α Q i sin (2 α Q − α H ) cos 2 α Q i sin (2 α Q − α H ) cos 2 α Q − cos (2 α Q − α H ) + i sin (2 α Q − α H ) sin 2 α Q (cid:35) (A2)The resultant matrix of QHQ arrangement, eq. (),is symmetric and has purely imaginary off-diagonal ele-ments, similar to the Eq. (6). This arrangement is there-fore equivalent to a single waveplate W Γ e ( α e ) , whose ef-fective retardance Γ e and an effective fast-axis orientation α e can be derived from Eqs. (7 and 8), and they are: Γ e = 2 π + 4 ( α Q − α H ) (A3) α e = α Q + π (A4) Appendix B: Rotating the waveplate
Consider a d-plate whose spatial variation in fast-axis orientation and retardance are respectively given by α ( r, φ ) and Γ ( r, φ ) . Mathematically, given the fast-axisorientation α , the fast-axis in the X-Y plane is given by ˆ k ( α ) = cos α ˆ e x + sin α ˆ e y , where ˆ e x and ˆ e y are unitvectors along the X and Y directions: ˆ k ( r, φ ) = cos α ( r, φ ) ˆ e x + sin α ( r, φ ) ˆ e y = (cid:34) cos α ( r, φ )sin α ( r, φ ) (cid:35) (B1)Now, consider rotating this plate by an angle, say δ .Because of this rotation, the field of vectors changes to ˆ k ( r, φ − δ ) . The rotated vectors in the old coordinates isgiven by ˆ k new ( r, φ ) = R ( δ ) · ˆ k ( r, φ − δ ) (B2) where R ( δ ) = (cid:34) cos δ sin δ − sin δ cos δ (cid:35) is the rotation matrixabout the Z -direction. Upon simplification, we have, ˆ k new ( r, φ ) = (cid:34) cos ( α ( r, φ − δ ) + δ )sin ( α ( r, φ − δ ) + δ ) (cid:35) (B3)From which the fast-axis orientation α new ( r, φ ) after therotation can be obtained as: α new ( r, φ ) = α ( r, φ − δ ) + δ (B4)Retardance being a scalar, transforms simply as. Γ new ( r, φ ) = Γ ( r, φ − δ ) (B5) Appendix C: Determining the Jones matrix
The Jones matrix of the unitary transformation in the {| H (cid:105) , | V (cid:105)} basis is given by: U {| H (cid:105) , | V (cid:105)} = (cid:34) (cid:104) H |U| H (cid:105) (cid:104) H |U| V (cid:105)(cid:104) H |U| H (cid:105) (cid:104) V |U| V (cid:105) (cid:35) (C1)Given the unitary transformation U of Eqs. (26 and27), its action on | H (cid:105) , | V (cid:105) can be identified by expressingthem in the {| θ, ϕ (cid:105) , | π − θ, π + ϕ (cid:105)} basis:0 | H (cid:105) = a | θ, ϕ (cid:105) + b | π − θ, π + ϕ (cid:105) (C2) | V (cid:105) = c | θ, ϕ (cid:105) + d | π − θ, π + ϕ (cid:105) (C3) a = 1 √ (cid:0) sin θ e − i ϕ + cos θ e i ϕ (cid:1) ,b = − i √ (cid:0) cos θ e − i ϕ − sin θ e i ϕ (cid:1) ,c = i √ (cid:0) sin θ e − i ϕ − cos θ e i ϕ (cid:1) ,d = 1 √ (cid:0) cos θ e − i ϕ + sin θ e i ϕ (cid:1) . Expressing | π − θ, ϕ (cid:105) and | θ, π − ϕ (cid:105) in the | H (cid:105) , | V (cid:105) basis: | π − θ, ϕ (cid:105) = a | H (cid:105) + c | V (cid:105)| θ, π − ϕ (cid:105) = b | H (cid:105) + d | V (cid:105) U {| H (cid:105) , | V (cid:105)} = (cid:34) e iψ a + e iχ b e iψ ac + e iχ bde iψ ac + e iχ bd e iψ c + e iχ d (cid:35) (C4)From this the determinant of the U {| H (cid:105) , | V (cid:105)} matrix is e i ( ψ + χ ) and this unitary matrix can be converted intoa SU(2) matrix W Γ ( α ) by multiplying U {| H (cid:105) , | V (cid:105)} withphase element as: W Γ ( α ) = e − i ( ψ + χ )2 · U {| H (cid:105) , | V (cid:105)} = e − i ( ψ + χ )2 (cid:34) e iψ a + e iχ b e iψ ac + e iχ bde iψ ac + e iχ bd e iψ c + e iχ d (cid:35) From this, the required retardance
Γ ( r, φ ) and fast-axis α ( r, φ ) of this d-plate can be extracted through Eqs.(7 and 8). [1] Shivaramakrishnan Pancharatnam. Generalized theoryof interference and its applications. In Proceedings of theIndian Academy of Sciences-Section A , volume 44, pages398–417. Springer, 1956.[2] Michael V Berry. The adiabatic phase and pancharat-nam’s phase for polarized light.
Journal of Modern Op-tics , 34(11):1401–1407, 1987.[3] Qiwen Zhan. Cylindrical vector beams: from mathemat-ical concepts to applications.
Advances in Optics andPhotonics , 1(1):1–57, 2009.[4] Zhan Qiwen.
Vectorial optical fields: Fundamentals andapplications . World scientific, 2013.[5] Lorenzo Marrucci, C Manzo, and D Paparo. Opticalspin-to-orbital angular momentum conversion in inho-mogeneous anisotropic media.
Physical review letters ,96(16):163905, 2006.[6] David L Andrews.
Structured light and its applications:An introduction to phase-structured beams and nanoscaleoptical forces . Academic press, 2011.[7] Halina Rubinsztein-Dunlop, Andrew Forbes, Michael VBerry, Mark R Dennis, David L Andrews, MasudMansuripur, Cornelia Denz, Christina Alpmann, PeterBanzer, Thomas Bauer, et al. Roadmap on structuredlight.
Journal of Optics , 19(1):013001, 2016.[8] Carmelo Rosales-Guzmán, Bienvenu Ndagano, and An-drew Forbes. A review of complex vector light fieldsand their applications.
Journal of Optics , 20(12):123001,2018.[9] Ralf Dorn, Susanne Quabis, and Gerd Leuchs. Sharperfocus for a radially polarized light beam.
Physical reviewletters , 91(23):233901, 2003.[10] Les Allen, Marco W Beijersbergen, RJC Spreeuw, andJP Woerdman. Orbital angular momentum of lightand the transformation of laguerre-gaussian laser modes.
Physical Review A , 45(11):8185, 1992.[11] Yijie Shen, Xuejiao Wang, Zhenwei Xie, Changjun Min, Xing Fu, Qiang Liu, Mali Gong, and Xiaocong Yuan. Op-tical vortices 30 years on: Oam manipulation from topo-logical charge to multiple singularities.
Light: Science &Applications , 8(1):1–29, 2019.[12] K Yu Bliokh, FJ Rodríguez-Fortuño, Franco Nori, andAnatoly V Zayats. Spin–orbit interactions of light.
Na-ture Photonics , 9(12):796, 2015.[13] Yachao Liu, Yougang Ke, Hailu Luo, and ShuangchunWen. Photonic spin hall effect in metasurfaces: a briefreview.
Nanophotonics , 6(1):51–70, 2017.[14] Nir Shitrit, Igor Yulevich, Elhanan Maguid, Dror Ozeri,Dekel Veksler, Vladimir Kleiner, and Erez Hasman. Spin-optical metamaterial route to spin-controlled photonics.
Science , 340(6133):724–726, 2013.[15] Andrea Aiello, Peter Banzer, Martin Neugebauer, andGerd Leuchs. From transverse angular momentum tophotonic wheels.
Nature Photonics , 9(12):789–795, 2015.[16] Yuichi Kozawa and Shunichi Sato. Optical trapping ofmicrometer-sized dielectric particles by cylindrical vectorbeams.
Optics Express , 18(10):10828–10833, 2010.[17] Brian J Roxworthy and Kimani C Toussaint Jr. Opti-cal trapping with π -phase cylindrical vector beams. NewJournal of Physics , 12(7):073012, 2010.[18] Michael A Taylor, Muhammad Waleed, Alexander B Stil-goe, Halina Rubinsztein-Dunlop, and Warwick P Bowen.Enhanced optical trapping via structured scattering.
Na-ture Photonics , 9(10):669–673, 2015.[19] Matthias Meier, Valerio Romano, and Thomas Feurer.Material processing with pulsed radially and azimuthallypolarized laser radiation.
Applied Physics A , 86(3):329–334, 2007.[20] Manuel Erhard, Robert Fickler, Mario Krenn, and An-ton Zeilinger. Twisted photons: new quantum perspec-tives in high dimensions.
Light: Science & Applications ,7(3):17146, 2018.[21] V.G. Chigrinov, V.M. Kozenkov, and H.S. Kwok.
Pho- toalignment of Liquid Crystalline Materials: Physics andApplications . Wiley Series in Display Technology. Wiley,2008.[22] Jihwan Kim, Yanming Li, Matthew N Miskiewicz, Chul-woo Oh, Michael W Kudenov, and Michael J Escuti.Fabrication of ideal geometric-phase holograms with ar-bitrary wavefronts. Optica , 2(11):958–964, 2015.[23] Wei Ji, Chun-Hong Lee, Peng Chen, Wei Hu, Yang Ming,Lijian Zhang, Tsung-Hsien Lin, Vladimir Chigrinov, andYan-Qing Lu. Meta-q-plate for complex beam shaping.
Scientific reports , 6:25528, 2016.[24] Dianmin Lin, Pengyu Fan, Erez Hasman, and Mark LBrongersma. Dielectric gradient metasurface optical ele-ments. science , 345(6194):298–302, 2014.[25] Mohammadreza Khorasaninejad, Wei Ting Chen,Robert C Devlin, Jaewon Oh, Alexander Y Zhu, andFederico Capasso. Metalenses at visible wavelengths:Diffraction-limited focusing and subwavelength resolu-tion imaging.
Science , 352(6290):1190–1194, 2016.[26] Jacob Scheuer. Optical metasurfaces are coming of age:Short-and long-term opportunities for commercial appli-cations.
ACS Photonics , 2020.[27] Xiaohui Ling, Xinxing Zhou, Xunong Yi, Weixing Shu,Yachao Liu, Shizhen Chen, Hailu Luo, Shuangchun Wen,and Dianyuan Fan. Giant photonic spin hall effect inmomentum space in a structured metamaterial with spa-tially varying birefringence.
Light: Science & Applica-tions , 4(5):e290–e290, 2015.[28] Mandira Pal, Chitram Banerjee, Shubham Chandel,Ankan Bag, Shovan K Majumder, and Nirmalya Ghosh.Tunable spin dependent beam shift by simultaneouslytailoring geometric and dynamical phases of light ininhomogeneous anisotropic medium.
Scientific reports ,6:39582, 2016.[29] Robert C Devlin, Antonio Ambrosio, Noah A Rubin,JP Balthasar Mueller, and Federico Capasso. Arbitraryspin-to–orbital angular momentum conversion of light.
Science , 358(6365):896–901, 2017.[30] Mushegh Rafayelyan and Etienne Brasselet. Laguerre–gaussian modal q-plates.
Optics letters , 42(10):1966–1969, 2017.[31] Noah A Rubin, Gabriele D’Aversa, Paul Chevalier, Zhu-jun Shi, Wei Ting Chen, and Federico Capasso. Matrixfourier optics enables a compact full-stokes polarizationcamera.
Science , 365(6448):eaax1839, 2019.[32] Amir Arbabi, Yu Horie, Mahmood Bagheri, and AndreiFaraon. Dielectric metasurfaces for complete control ofphase and polarization with subwavelength spatial res-olution and high transmission.
Nature nanotechnology ,10(11):937–943, 2015.[33] JP Balthasar Mueller, Noah A Rubin, Robert C Devlin,Benedikt Groever, and Federico Capasso. Metasurfacepolarization optics: independent phase control of arbi-trary orthogonal states of polarization.
Physical reviewletters , 118(11):113901, 2017.[34] Bruno Piccirillo, Vincenzo D Ambrosio, Sergei Slus-sarenko, Lorenzo Marrucci, and Enrico Santamato. Pho-ton spin-to-orbital angular momentum conversion viaan electrically tunable q-plate.
Applied Physics Letters ,97(24):241104, 2010.[35] Sergei Slussarenko, Anatoli Murauski, Tao Du, VladimirChigrinov, Lorenzo Marrucci, and Enrico Santamato.Tunable liquid crystal q-plates with arbitrary topologi-cal charge.
Opt. Express , 19(5):4085–4090, Feb 2011. [36] Ebrahim Karimi, Bruno Piccirillo, Eleonora Nagali,Lorenzo Marrucci, and Enrico Santamato. Efficient gen-eration and sorting of orbital angular momentum eigen-modes of light by thermally tuned q-plates.
AppliedPhysics Letters , 94(23):231124, 2009.[37] Shivaramakrishnan Pancharatnam. Achromatic combi-nations of birefringent plates part-i. In
Proceedings ofthe Indian Academy of Sciences-Section A , volume 41,pages 130–136. Springer, 1955.[38] Shivaramakrishnan Pancharatnam. Achromatic combi-nations of birefringent plates part-ii. In
Proceedings ofthe Indian Academy of Sciences-Section A , volume 41,pages 137–144. Springer, 1955.[39] Thorlabs super achromatic waveplates.[40] Xunong Yi, Ying Li, Xiaohui Ling, Yachao Liu, YougangKe, and Dianyuan Fan. Addition and subtraction oper-ation of optical orbital angular momentum with dielec-tric metasurfaces.
Optics Communications , 356:456–462,2015.[41] Sam Delaney, María M Sánchez-López, Ignacio Moreno,and Jeffrey A Davis. Arithmetic with q-plates.
Appliedoptics , 56(3):596–600, 2017.[42] B Radhakrishna, Gururaj Kadiri, and G Raghavan.Wavelength-adaptable effective q-plates with passivelytunable retardance.
Scientific reports , 9(1):1–9, 2019.[43] Olga Korotkova.
Random light beams: theory and appli-cations . CRC press, 2013.[44] Enrique J Galvez. Complex light beams.
Deep Imagingin Tissue and Biomedical Materials: Using Linear andNonlinear Optical Methods , page 31, 2017.[45] Zdeněk Bouchal. Nondiffracting optical beams: physicalproperties, experiments, and applications.
Czechoslovakjournal of physics , 53(7):537–578, 2003.[46] Michael Mazilu, D James Stevenson, Frank Gunn-Moore,and Kishan Dholakia. Light beats the spread:’non-diffracting’ beams.
Laser & Photonics Reviews , 4(4):529–547, 2010.[47] Nikolaos K Efremidis, Zhigang Chen, Mordechai Segev,and Demetrios N Christodoulides. Airy beams and accel-erating waves: an overview of recent advances.
Optica ,6(5):686–701, 2019.[48] Massimo Granata, Christelle Buy, Robert Ward, andMatteo Barsuglia. Higher-order laguerre-gauss mode gen-eration and interferometry for gravitational wave detec-tors.
Physical review letters , 105(23):231102, 2010.[49] Liu Tao, Anna Green, and Paul Fulda. Higher-orderhermite-gauss modes as a robust flat beam in inter-ferometric gravitational wave detectors. arXiv preprintarXiv:2010.04338 , 2020.[50] Stefan W Hell and Jan Wichmann. Breaking the diffrac-tion resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy.
Optics let-ters , 19(11):780–782, 1994.[51] Wentao Yu, Ziheng Ji, Dashan Dong, Xusan Yang, Yun-feng Xiao, Qihuang Gong, Peng Xi, and Kebin Shi.Super-resolution deep imaging with hollow bessel beamsted microscopy.
Laser & Photonics Reviews , 10(1):147–152, 2016.[52] NB Simpson, L Allen, and MJ Padgett. Optical tweez-ers and optical spanners with laguerre–gaussian modes.
Journal of modern optics , 43(12):2485–2491, 1996.[53] NB Simpson, K Dholakia, L Allen, and MJ Padgett. Me-chanical equivalence of spin and orbital angular momen-tum of light: an optical spanner.
Optics letters , 22(1):52–
54, 1997.[54] EM Wright, J Arlt, and K Dholakia. Toroidal op-tical dipole traps for atomic bose-einstein condensatesusing laguerre-gaussian beams.
Physical Review A ,63(1):013608, 2000.[55] Mushegh Rafayelyan, Titas Gertus, and Etienne Bras-selet. Laguerre-gaussian quasi-modal q-plates fromnanostructured glasses.
Applied Physics Letters ,110(26):261108, 2017.[56] Jay N Damask.
Polarization optics in telecommunica-tions , volume 101. Springer Science & Business Media,2004.[57] Radhakrishna Bettegowda. Prescription for transformingpolarization states of light using two-quarter waveplates.
Optical Engineering , 56(3):034110, 2017.[58] Karol Salazar-Ariza and Rafael Torres. Trajectories onthe poincaré sphere of polarization states of a beam pass-ing through a rotating linear retarder.
JOSA A , 35(1):65–72, 2018.[59] R Simon and N Mukunda. Universal su (2) gadget forpolarization optics.
Physics Letters A , 138(9):474–480,1989.[60] Rajendra Bhandari. Polarization of light and topologicalphases.
Physics Reports , 281(1):1–64, 1997.[61] MW Beijersbergen, RPC Coerwinkel, M Kristensen, andJP Woerdman. Helical-wavefront laser beams producedwith a spiral phaseplate.