Customization and Information Encoding of Longitudinally Polarized Light Fields

The tightly focused structured light field displays a non‐negligible component of the electric field in the direction of propagation, i.e., the longitudinal light field. The longitudinal light field results in various novel optical phenomena, such as transverse spin angular momentum and 3D topological light fields. However, it remains a challenge to generate customized longitudinal light fields. Here, based on inverse design theory, the longitudinal light field can be customized on demand under the tightly focusing condition, which allows to generate the longitudinal light field with controllable structured distribution for propagating waves or manipulate the complex structures for all the three components of the electric field. In experiment, an experimental system is built integrating the generation, customization, and detection of structured longitudinal light field, and the customization of the longitudinal light fields with various patterns is experimentally realized. The customized longitudinal light field structures may find applications such as information encoding, long‐distance light manipulation, and novel light–matter interactions. As an example, information encoding is realized by the structured longitudinal light fields.


Introduction
As is well known, the electric field of an infinitely extended plane electromagnetic wave is perpendicular to the direction of its propagation.However, it is not the case for tightly focused structured light field because there is a nonzero component of the longitudinal polarization (i.e., the electric field component parallel to the propagation direction). [1,2]Under the paraxial approximation, the longitudinal component is much weaker than the transverse component and can be neglected.When the light field is focused with a high numerical aperture (NA) objective lens, the longitudinal component becomes nonnegligible. [3,4][9] In addition, the longitudinal field has attracted numerous research interests in light-matter interactions.The longitudinal field can enable the z-polarization sensitive detection in micro-Raman spectroscopy, [10] impart the energy gain into a particle and accelerate it, [11] and interact with a single ion. [12]The longitudinal component of the tightly focused femtosecond pulses can ablate a hole beyond the diffraction limit on various materials. [13,14]21][22] Over the past two decades, many measures have been taken to enhance the longitudinal field, for instance, using annular apertures [13,23] and diffractive optical elements, [24] combined with tightly focusing radially polarized field.In addition, by manipulating the distributions of the amplitude, phase, and polarization state of the incident light, special longitudinal fields, e.g., longitudinal optical needle [24][25][26] and longitudinal polarization vortex structures, [27,28] have been generated.However, these approaches typically determine the longitudinal field by the transverse components of a given light field.In other words, if the incident light field is given, the longitudinal field is determined.Therefore, it is still a huge challenge to customize the longitudinal field structures on demand until now.Although it has been theoretically shown that the shaping of 3D vectorial structured field is possible to generate the customized complex longitudinal field structure recently, [29] it is yet to be realized experimentally.The reverse engineering approach can be used to customize the focusing field under tight focusing conditions, [30,31] but it is mainly for the total intensity and is not applicable to the longitudinal light field.
[34] While for the longitudinal field, the direct measurement is difficult, which can be implemented indirectly by single molecules probing, [35] nanotomography, [36,37] scanning near-field optical microscopy, [38,39] recording of the focal pattern in photoresist, [40] and so on.However, these detection methods depend on the point-by-point scanning and complex light-matter interaction, which is slow and not applicable to fast optical field acquisition.To solve this problem, a polarization-conversion microscopy (PCM) method has been proposed. [41]Based on the PCM method, the longitudinal and transverse fields can be measured simultaneously.In addition, when a circularly polarized light is tightly focused, a strong longitudinal field with the vortex structure appears in the vicinity of the focus, which is described by the spin-to-orbital angular momentum conversion. [22,42]The related demonstration and characterization of the topological charge of the longitudinal field vortex are just qualitative, and the real-time and accurate measurement is absent.
Here, we show that it is possible to customize the longitudinal field.We build an optical system to both customize and measure the longitudinal field structures by using the PCM method.To illustrate our approach in detail, we first generate the longitudinal field lattices and achieve the information encoding by the longitudinal fields.We then generate the customized diffractionfree longitudinal needle and tube fields.Based on the PCM method, the spin-to-orbit conversion under the tightly focusing condition is visualized intuitively.Our approach facilitates an intuitive understanding of the tightly focused fields and provides a promising way to manipulate the complex 3D topological vector light fields and explore novel light-matter interaction involving longitudinal fields.

Theory of Customizing Longitudinal Field
To customize the longitudinal field structures, we introduce the theoretical description of the relationship between the tightly focused field and the corresponding incident field based on the inverse Fourier algorithm, which can be briefly described as follows (see Section S1, Supporting Information, for details).For the incident field E i (x,y) and the focal field E f (x,y,z), the spatial frequency spectra are Ẽi ðk x , k y Þ and Ẽf ðk x , k y ; zÞ, respectively, which have the following correlation Equation (1) shows that once the focal field is determined, the corresponding incident field is uniquely determined.Where f is the focal length of the objective lens and k = 2πλ is the wave number in the free space, as shown in Figure 1a.k x and k y are the spatial frequencies (components of wave vector) in the (see Section S1, Supporting Information), and Ẽi ðk x , k y Þ ¼ ℱfE i ðx, yÞg and Ẽf ðk x , k y ; zÞ ¼ ℱfE f ðx, y, zÞg (ℱf⋅g is the Fourier transform), respectively.Equation ( 1) means that for any target focal field E f , we can find the corresponding incident field E i which is the key to realize the customization of the target field.
Without loss of generality, we take the creation of the longitudinal field vortex with a topological charge of þ1 as a simulation example to test the customization of the longitudinal field.As shown in Figure 1b, we first give a preset longitudinal field E f pz (hereafter, the subscript "p" means the preset field), which can be defined with preset amplitude and phase structures.However, the longitudinal field never exists alone, but must be accompanied with the related transverse field.According to the correlation between the longitudinal and transverse fields (see Section S2, Supporting Information), the corresponding transverse field can be calculated by the transformation (named as the process "ⓐ") In Equation ( 2), the preset field Ẽf pz can be chosen freely in any plane z behind the focusing lens, without loss of generality, we choose the focal plane, i.e., Ẽf ∂y êy is the transverse differential operator (ê x and êy are the unit vectors in the x and y directions, respectively), ℱ À1 f⋅g is the inverse Fourier transform, and the transverse field is E f p⊥ ¼ ðE f px , E f py Þ, thus we obtain the 3D components of the preset field.Second, we need to take the preset field as the tightly focused field.At the same time, the tightly focused field is also a propagating field because the z-component of the wave vector is real.For a propagating field, there must be k z > 0, i.e., the preset field E f p (E f px , E f py , E f pz ) must obey certain constraints and only the field meeting the constraints can be set as the target focal field to be customized.The constraint is carried out by the spatial filtering in spatial-frequency domain.As shown in Figure 1c, the propagating fields along z-direction is restricted by the condition k 2 ⊥ < ðNA ⋅ kÞ 2 , i.e., the transverse wave vector are restricted in a circular region that is the pupil of the objective lens.After the spatial filtering (named as the process "ⓑ") shown in Figure 1c, the really existed target focal field E f can be obtained by The intensities and phases of the x, y, and z components of the real target focal field are shown in Figure 1d.Finally, when the tightly focused field is known, it is easy to obtain the corresponding incident field based on the transformation relationship between the incident field and the focal field, and substituting the target focal field described in Equation ( 3) into (1), we can calculate the corresponding incident field E i (named as the process "ⓒ") shown in Figure 1e.This incident field is usually complex, however, it can always be written as the superposition of two orthogonal polarization components, e.g., left-and right-handed circularly polarized (LCP E i l and RCP E i r ) components, horizontal (E i x ) and vertical (E i y ) components, and radial (E i ρ ) and azimuthal (E i ϕ ) components, where the relationship between the cylindrical coordinates and the Cartesian coordinates is For the incident field shown in Figure 1e, E i ϕ ¼ 0. In turn, the incident field can be focused by an objective lens; thus, the z-component of its focal field is the customized longitudinal field structure on demand.

Experimental Details
The incident field E i obtained from Equation ( 1)-(3) usually has the spatially varying amplitude, phase, and polarization.A vector optical field (VOF) generator is used to generate E i , as shown in Figure 2a.The VOF generator consists of a reflective phase-only spatial light modulator, a 4f system (composed of lenses, L3 and L4), a pair of quarter-wave plates, and a Rochi grating.The fundamental Gaussian beam from a continuous-wave laser at a wavelength of 532 nm is expanded by a pair of lenses (L1 and L2) to shape into a nearly uniform intensity, and is then incident into the VOF generator through a beam splitter.Finally, the generated incident field E f can be written as where E l (x,y) and E r (x,y) are the amplitude distributions of LCP and RCP components, respectively, Φ l (x,y) and Φ r (x,y) are the corresponding phase distributions, and êl and êr are the unit vectors of the LCP and RCP, respectively.To realize the complex amplitude manipulation for the LCP and RCP components in Equation ( 4), a 2D holographic grating [43] combined with a double-phase technique is adopted, [44,45] as shown in Figure 2b, where ,y) and E r (x,y).M 12 (x,y) = M 1 (x,y) À M 2 (x,y), M 1 (x,y) and M 2 (x,y) are binary periodic functions giving by 0 or þ1, and In the special case that the LCP and RCP components have uniform and equal amplitudes (E l = E r = E 0 ), we have E 0 e jβðx,yÞ ½cos αðx, yÞê x þ sin αðx, yÞê y , and it is clear that β(x,y) and α(x,y) dominate the spatially varying phase and polarization of the incident field respectively, where β(x,y) = [Φ l (x,y) þ Φ r (x,y)]/2, α(x,y) = [Φ l (x,y) À Φ r (x,y)]/2, ψ 1 ðx, yÞ ¼ Φ l ðx, yÞ, and ψ 2 ðx, yÞ ¼ Φ r ðx, yÞ.
For the general case that the amplitude distributions of LCP and RCP components are nonuniform [E l (x,y) and E r (x,y) and unequal [E l (x,y) 6 ¼ E r (x,y)], the amplitude can also be manipulated.The generated incident field E i can be tightly focused by a high NA lens to form a customized target field, which can be measured by the PCM method (see Section S3 and S4, Supporting Information, for details).

Results and Discussions
Based on the aforementioned theory and experimental setup, we measure the tightly focused fields of four typical light fields (including linearly polarized light in Figure S2, Supporting Information, linearly polarized vortex in Figure S3, Supporting Information, circularly polarized light in Figure S4, Supporting Information, and radially polarized light in Figure S5, Supporting Information) by the PCM method (see Section S5, Supporting Information).These results confirm our idea to be valid.Subsequently, we will explore the customized lattices composed of longitudinal field spots with Gaussian intensity profile.For this purpose, we define the preset lattice of the longitudinal field spots as where N is the number of spots, (x n , y n ) indicates the coordinates of the nth spot, and a 0 is the radius of each Gaussian spot (here a 0 = 0.2λ and λ is the laser wavelength), respectively.Substituting Equation ( 5) into ( 2) and ( 3), the expression of the customized tightly focused field can be obtained.Based on this and the generation method mentioned earlier, as examples, Figure 3a-c shows three customized results corresponding to N = 3, N = 4, and N = 5, respectively, in which the distance between two adjacent spots is 1.8λ.It can be seen that the longitudinal fields jE f z j 2 exhibit the expected lattice structures composed of Gaussian spots, and the corresponding transverse fields (jE f ⊥ j 2 ¼ jE f x j 2 þ jE f y j 2 ) appear as the same lattices formed by donut-shaped spots.
According to Equation ( 5), there is no doubt that more complex longitudinal field lattice can be constructed.In addition, the longitudinal component of the focal field cannot be acquired by direct observation using conventional optical equipment.For this reason, encoding information in the longitudinal field E z of a tightly focused field is possible.Inspired by this, a special encoding concept, i.e., encoding information based on the longitudinal field, is proposed and realized here.First, as examples, the Arabic numerals (from "0" to "9") are set as the plaintexts, we construct the preset longitudinal field lattices composed of Gaussianprofile spots in the same plane along the z-direction (i.e., focal plane with z = 0), whose arrangements correspond to the geometries of numbers from "0" to "9" based on Equation (5), respectively.Second, based on Equation (2), the corresponding transverse field can be obtained.As mentioned earlier, the corresponding transverse field has the same lattice, but it is composed of donut-shaped spots (see Figure S6 and Section S6, Supporting Information).To hide the information of the plaintext in the transverse field, some extra transversely polarized donut-shaped spots need to be added and make the transverse fields of the plaintext (Arabic numerals from "0" to "9") be the same.After determining all the components of the preset field, according to Equation ( 1)-( 3), the incident field can be obtained, and the incident field and the focal field satisfy the Fourier transform relationship.As the plaintext information of focal field cannot be seen at all in the incident field, we termed the incident field as the ciphertext.Finally, the sender transmits the ciphertext carrying the information to the user (or receiver) by switching different incident fields, and the receiver recovers the plaintext information through the longitudinal field measurement device, which is equivalent to information decryption.The decrypted information is the obtained longitudinal field spots for images of the numbers from "0" to "9" (see the simulated results in top row of Figure S7, Supporting Information, and the experimental results in top row of Figure 3d), while all the corresponding transverse field spots hide the information (see the simulated results in bottom row of Figure S7, Supporting Information, and the experimental results in bottom row of Figure 3d).Clearly, the information of plaintext is presented as the Gaussian-like spot in longitudinal field for images of the numbers from "0" to "9" while the corresponding transverse fields are the same, meaning that we can encode information in the longitudinal field, especially, the longitudinal field can be customized independently at will.
Figure 4 presents the experimental results for the customized diffraction-free longitudinal optical needle.To generate it, the preset longitudinal field in the focal plane is set to be E f pz ¼ J m ðk r rÞ expðjmϕÞ, where J m (•) is the mth-order Bessel function of the first kind and k r is the radial wave vector.Substituting the preset field into Equation ( 1)-( 3), the customized diffractionfree longitudinal optical field can be obtained.To generate the longitudinal optical needle and optical tube, we need to set m = 0 and m = 1, respectively.Figure 4 illustrates only the generation of the longitudinal needle with m = 0 (the optical tube results of m = 1 can be found in Figure S9 and S10 of Section S7, Supporting Information).By scanning along the z direction, the 3D intensity distribution of the transverse and longitudinal components can be acquired.Figure 4a shows the intensity evolution of the longitudinal field, obviously it is indeed an ultra-long diffraction-free optical needle along the propagating direction, i.e., it propagates without divergence within a distance of 16λ (it is about 4λ in ref. [24]).The intensity distributions in the xy plane at the focal plane of z = 0 and the xz plane are presented in Figure 4b,c, respectively.Figure 4d reflects the intensity profiles along the x axis in the xy plane.Specifically, for the customized diffraction-free longitudinal optical needle, its corresponding transverse field is like an optical tube (see Figure S9 and Section S8, Supporting Information).
In addition, when a circularly polarized light is tightly focused, a spin-dependent optical vortex can be generated, where the spin angular momentum is transformed into the orbital angular momentum giving rise to a helical phase for the longitudinal field.Based on the setup shown in Figure 2a, the topological charge of the longitudinal field vortex is measured (see Figure S12 and Section S8, Supporting Information).Our method transforms the unmeasurable longitudinal field into measurable transverse field and provides a promising tool for easy, fast characterization for the topological charge of the longitudinal field experimentally.

Conclusion
We have demonstrated how to manipulate and customize the longitudinally polarized field on demand by constructing the lattices arranging the longitudinal field spots, encoding the information based on the longitudinal field spots, and generating the longitudinal optical needle and tube.Furthermore, we have realized the accurate measurement of the topological charge of the longitudinal field vortex.Our experimental results are in good agreement with the simulated results, which provide us with some benefits in the foreseeable future.First, it offers an easier and feasible way to customize the required longitudinally polarized fields with the predefined intensity and phase profiles with arbitrary structures, which allows us to manipulate the complex 3D structured light fields, including complex topologies.Second, it provides a good feasibility and solution for those experiments of the light-matter interaction that requires complex structural light.Hence, our work provides a useful tool for effectively studying and exploring novel physical phenomena involving the longitudinal light field.

Figure 1 .
Figure 1.Illustrative diagram for generating the customized longitudinal field structures.a) Geometry for vector diffraction of non-paraxial tightly focusing system.b) The preset longitudinal field (E f pz ) and the corresponding transverse fields (E f px and E f py ).c) The spatial frequency filtering.d) The intensity (jE fx,y,z j 2 ) and phase (argðE f x,y,z Þ) distributions for the x, y, and z components of the target focal field behind the filter.e) The calculated incident field for realizing the customized target field in experiment.

Figure 3 .
Figure 3.The customized tightly focused fields.a-c) The simulated and experimental results of the longitudinal field lattices (jE f z j 2 ) with three, four, and five spots and the corresponding transverse fields jE f ⊥ j 2 , respectively.d) The experimental results of information encoding by the longitudinal field spots jE f z j 2 (decrypted information) and the corresponding transverse field jE f ⊥ j 2 .The images in (a-c) have a size of 7λ Â 7λ and the images in (d) have a size of 17λ Â 17λ.