Retrieving Jones matrix from an imperfect metasurface polarizer

Abstract. Optical metasurfaces, which consist of subwavelength scale meta-atoms, represent a novel platform to manipulate the polarization and phase of light. The optical performance of metasurfaces heavily relies on the quality of nanofabrication. Retrieving the Jones matrix of an imperfect metasurface optical element is highly desirable. We show that this can be realized by decomposing the generalized Jones matrix of a meta-atom into two parallel ones, which correspond to the ideal matrix and a phase retardation. To experimentally verify this concept, we designed and fabricated metasurface polarizers, which consist of geometric phase-controlled dielectric meta-atoms. By scanning the polarization states of the incident and transmitted light, we are able to extract the coefficients of the two parallel matrices of a metasurface polarizer. Based on the results of the Jones matrix decomposition, we also demonstrated polarization image encryption and spin-selective optical holography. The proposed Jones matrix retrieval protocol may have important applications in computational imaging, optical computing, optical communications, and so on.


Introduction
As one of the most important intrinsic properties of light, polarization builds the link between orthogonal electric vectors involving amplitude and phase.][10] Traditional polarizers are usually bulky and have many limitations for optical integration.The emergence of metasurfaces, which consist of subwavelength scale metaatoms, provides a new route to address this issue.8][19][20][21][22] The metasurface optical element, which has the ability to manipulate the electric field in different directions, can be described mathematically using the Jones matrix. 23,24By engineering the materials and geometrical parameters of a meta-atom or meta-molecule, it is possible to construct the corresponding Jones matrix. 25,26][29][30][31] Despite the rapid development of nanofabrication technology, the imperfection of metasurface polarizers hinders their practical applications.To partially circumvent this constraint, we propose to experimentally retrieve the Jones matrix of an imperfect metasurface polarizer.This approach allows us to accurately determine each component of the metasurface polarizers ' Jones matrix at specific wavelengths.As a proof of concept, we utilized the retrieved Jones matrix to experimentally verify the outgoing PS.Moreover, by arranging the dielectric meta-atoms in a judicious manner, we demonstrated the polarization image encryption and chiroptical holography using the metasurface polarizers.The proposed methodology in this work may greatly release the strict manufacturing requirements of the metasurface optical elements and thus benefits a wide variety of applications of polarization optics.

Jones Matrix Decomposition
In conventional optics, in order to generate the arbitrary PS of light, some specific assembly of a linear polarizer and various wave plates are usually used in the light path.Recent studies show that a single metasurface which consists of subwavelength scale meta-atoms can be utilized to mimic the multiple optical elements in traditional optics. 25,27As shown in Fig. 1, the combination of two birefringent dielectric meta-atoms can be used to design the metasurface polarizers. 27The Jones matrix J of the metasurface polarizer can be decomposed to the superposition of that of the two birefringent meta-atoms, which is shown as 1(e) and 1(f).Among various factors, the imperfect etching depth and angle of the meta-atoms are the two common issues that will affect the performance of the metasurface polarizers.For the birefringent dielectric metaatoms, which are designed to introduce proper phase retardation between two orthogonal electric field components, the planar geometrical parameters determine the introduced phase difference and the height affects the accumulated phase retardation.
The etching angle and depth determine both the planar geometry and height of the meta-atoms, therefore, influence their optical functionalities and conversion efficiency.In this work, we assume that the etching process will not break the symmetry of the dielectric meta-atoms.These kinds of imperfections can be regarded as planar plates and usually exert little effects on the polarization of the incident light.The corresponding Jones matrix can be rewritten as Rðψ − 45 degÞ where the α and β are the coefficients of the target Jones matrix and imperfection part, respectively.When β equals zero, the metasurface will become a perfect polarizer.Both the amplitude and phase of the two coefficients are associated with the wavelength of incident light and the imperfections of the meta-atoms.First, we designed and fabricated the linear metasurface polarizer which consists of silicon nitride (SiN x ) meta-atoms sitting on a fused silica substrate.Through the Lumerical FDTD simulations at the wavelength of 660 nm, we built the phase retardation library of the birefringent meta-atoms with a height of 1400 nm [Fig.2(a)].The meta-atoms with length and width varying from 100 to 380 nm are placed in the square lattices with periods of 400 and 450 nm in Figs.2(b) and 2(c), respectively.According to the phase retardation library, we selected two types of meta-atoms to compose the meta-molecule.First, the first meta-atom as a half-wave plate was selected, corresponding to the latter part in Eq. ( 1), whose phase retardations of the fast and slow axes are ϕ 0 and ϕ 0 þ π, where ϕ 0 is an arbitrary real number.Then the second meta-atom corresponding to the former part in Eq. (1) was selected, in which phase retardations of the fast and slow axes are ϕ 0 − ϕ and ϕ 0 þ ϕ, where ϕ ¼ 2χ is determined by the target function of this polarizer.Based on the concept described in Fig. 1(c), different combinations of the meta-atoms were utilized to design three typical polarizers, including horizontal linear polarizer (LP-Pol-H, l 1 ¼ 210 nm, w 1 ¼ 210 nm, l 2 ¼ 160 nm, w 2 ¼ 370 nm, and p ¼ 400 nm), left circular polarizer (LCP-Pol, l 1 ¼ 380 nm, w 1 ¼ 110 nm, l 2 ¼ 370 nm, w 2 ¼ 150 nm, and p ¼ 450 nm), and right circular polarizer (RCP-Pol, l 1 ¼ 110 nm, w 1 ¼ 380 nm, l 2 ¼ 370 nm, w 2 ¼ 150 nm, and p ¼ 450 nm).To verify the performance of the metasurface polarizers, the p = 400 nm @660 nm p p l w h polarization-dependent transmission spectra of the three metasurface polarizers were numerically calculated in the wavelength range from 500 to 1000 nm in Figs.2(d)-2(f).For the linear metasurface polarizer with an optical axis along a horizontal (H) direction, it has a high polarization extinction ratio in the wavelength range between 650 and 700 nm.
From the calculated results, we can see that the bandwidths of these three kinds of polarizers are quite narrow, which will definitely limit the practical applications.To partially circumvent this issue, we experimentally retrieve the target Jones matrices from the imperfect metasurface polarizers.As shown in Fig. S1(a) (Sec. 1 in the Supplementary Material), we carried out the following optical measurements.The output from a supercontinuum laser is normally incident on the metasurface polarizers from the substrate side.The polarization ellipse of the transmitted light can be sketched by rotating the linearly polarized analyzer and recording the transmitted power (Fig. 3 and Sec. 1 in the Supplementary Material).To retrieve the four complex terms in the Jones matrix of a linear metasurface polarizer, six kinds of PSs of the incident light were used, including linear polarizations along H, V, 45 deg, 135 deg directions, left-and right-circular polarizations (LCP and RCP).By fitting the six measured ellipses (dot orange lines), the Jones matrix of an imperfect linear polarizers can be determined at each wavelength (Secs. 2 and 3 in the Supplementary Material).Using the measured Jones matrix of the metasurface polarizer (LP-Pol-H) and the six kinds of input PSs, we can calculate the corresponding polarization ellipses and their rotating directions (solid blue lines).In addition, it is shown that the above methodology can be applied to different wavelengths of the incident light.For example, at the wavelengths of 690 and 760 nm, where the Jones matrices are close to or deviate from the ideal ones, we find that the calculated polarization ellipses agree well with the experiment ones.By scanning the wavelengths of the incident light from 640 to 840 nm, we can experimentally extract the coefficients α and β of the target Jones matrix and imperfection part, respectively.The wavelength-dependent ratio of jα∕βj is shown in Sec. 4 in the Supplementary Material, from which we can evaluate the level of imperfection of the metasurface polarizers.As shown in Sec. 5 in the Supplementary Material, the optical properties of the circular metasurface polarizers (LCP-Pol and RCP-Pol) were characterized using the same methods as that in Fig. 3.Both the left-and right-circular metasurface polarizers exhibit good polarization effect between 600 and 700 nm.Notably, the circular metasurface polarizers act as both a half-wave plate and a polarization filter, which is distinguished from their conventional counterparts.

Polarization Image Encryption with Linear Metasurface Polarizers
We show that the linear metasurface polarizers can be used for polarization image encryption by delicately arranging the rotation angles of the meta-atoms in real space.The center of the rotation of each polarizer is the center of the unit cell with an area size of 800 nm × 800 nm.Each unit contains two meta-atoms, this is to ensure that there are no additional coupling effects due to the rotations [Fig.4 wavelengths of 690 and 760 nm, respectively.In the experiment, a circularly polarized Gaussian beam from the supercontinuum laser is normally incident onto the META image, and the normalized intensity of each letter was measured after passing through a linear analyzer with a fast axis along the horizontal direction.For the incident light at the wavelength of 690 nm [Fig.The calculated intensity distributions agree well with the ones captured by the CCD camera.Furthermore, by extracting the complex coefficients α and β from Eq. ( 2), the intensity ratio of the target polarization components and the residual one is obtained.The coefficients of the metasurface polarizer's Jones matrices here act as the keys to revealing the useful information in the transmitted light.

Optical Holography with Circular Metasurface Polarizers
Based on the concept of geometric phase, we also demonstrated the spin-selective optical holography [32][33][34][35] by engineering the meta-atoms' rotation angles, as shown in Figs.5(a) and 5(b).
For an ideal circular metasurface polarizer, when the meta-atom is rotated by an angle of φ, the transmitted light will carry a geometric phase of 2σφ, where σ ¼ AE1 correspond to the leftand right-circular PSs of the incident light.The mechanisms can be explained by applying the Jones matrices of the left-and right-circular metasurface polarizers on the Jones vectors of the incident light with arbitrary PS, which are shown as where E x and E y are the electric fields of the incident light, and ½1; i T and ½1; −i T represent the left-and right-circular PSs of the transmitted light, respectively.It is worth noting that in order to reduce the coupling effects of the adjacent meta-atoms, the period of the unit cell is set to be 1.0 μm along both the x and y directions.Meanwhile, the geometrical parameters of the meta-atoms remain the same as those of the uniform metasurfaces in Fig. 2. We designed and fabricated two kinds of phase-type metasurface holograms, based on the RCP-Pol [Figs.(d) and 5(l)] of the holographic images ("R" for the RCP-Pol, "L" for the LCP-Pol) were calculated using the commercial software VirtualLab Fusion.As shown in Figs.5(e) and 5(m), the holographic images were reconstructed for a Gaussian beam (waist radius: 140 μm and λ ¼ 700 nm), which matches the experimental conditions.The SEM images of the circular metasurface polarizers are in Figs.5(f) and 5(n).In the holography experiment [Fig.S2(c) in the Supplementary Material], the incident light from a supercontinuum laser is normally incident on the metasurfaces (see Sec. 4.1 for more details).By changing the circular PSs of the incident light, the spin-selective holographic images of the RCP-Pol and LCP-Pol devices are shown in Figs.5(g)-5(j) and Figs.5(o)-5(r), respectively.As expected, the holographic images can only be observed in the LCP-RCP and RCP-LCP measurement schemes for the RCP-Pol and LCP-Pol devices, respectively.Benefitting from this spin selective transmission property, there is no twin image generation, which exists in the typical geometric phase-based metasurface holography.By introducing the phase gradient into the metasurface holograms, the converted polarized light is separated from the residual light.This also allows us to easily separate the holographic images and the residual light, which correspond to the target Jones matrix and residual part.The optical performance of the metasurface holograms was also carried out at 680 and 760 nm, and similar phenomena were observed (see Sec. 7 in the Supplementary Material).

Discussion and Conclusions
In summary, we have proposed a new strategy for retrieving the Jones matrix of an imperfect metasurface polarizer.Through  a series of polarization measurements, we can comprehensively characterize the optical properties of the metasurface polarizers.Moving from theory to practice, we fabricated and measured the real Jones matrices of three metasurface polarizers at different wavelengths.Utilizing the linear metasurface polarizers, we demonstrated the polarization image encryption in real space, where the real Jones matrix plays a critical role in separating the target polarization component and the residual.By introducing the concept of geometric phase into the design of circular metasurface polarizers, we also presented the spin selective optical holography phenomenon.In addition, there is a one-to-one correspondence between the incident PSs and the transmitted ones, which may enable us to achieve a full Stokes polarimetry by a single metasurface optical chip.The proposed Jones matrix retrieval approach opens a new route for extracting useful information from imperfect metasurface optical elements and may have important applications in polarization imaging, 36,37 advanced light source, 38 information multiplexing, 39,40 and so on.It should be noted that the traditional polarization optical elements are still playing important roles due to their high efficiency, extinction ratio, and wide bandwidth.Under specific application scenarios that requires small volume, light weight, and high integration level, the metasurface could be a promising platform.Thanks to the rapid development of nanofabrication, the large-scale fabrication of metasurfaces is compatible with the mature CMOS technology.By combining electron beam lithography, DUV projection lithography, and nanoimprinting technologies, 41,42 the mass production costs can be greatly reduced in the future.

Device fabrication
All the metasurface devices in this work were fabricated using the electron beam lithography and inductively coupled plasma etching processes.The procedures are as follows: a fused silica substrate with a 1400 nm thick silicon nitride layer, which was grown by plasma-enhanced chemical vapor deposition, was spin-coated with the positive tone electron beam resist and the charge dissipation solution, respectively.After baking the resist at a temperature of 180°C and the charge dissipation layer at a temperature of 90°C, the designed meta-atom patterns were written by exposing the resist with the electron beam lithography then developed in the MIBK/IPA 1:3 developer and washed in DI water.Afterward, a 20 nm thick chromium layer was deposited via the electron beam evaporation to fill the developed patterns, and the samples were immersed into the acetone for the lift-off process.The chromium patterns act as the hard mask in the inductively coupled plasma etching process (gas: CHF 3 and O 2 ), protecting the covered area from exposure to the plasma, and the silicon nitride meta-atoms were formed.Finally, the chromium residual was removed from the top of the meta-atoms using the chromium etchant.

Optical experiment
There are three optical setups for characterizing the metasurface polarizers' Jones matrix (Fig. S1 in the Supplementary Material).The incident light source is a supercontinuum laser (NKT), and the output wavelength ranging from 640 to 1100 nm is controlled by an acousto-optic modulator.All the polarization states are defined when the incident light is viewed against the propagation direction of light.An assembly of a linear polarizer and a quarter-wave plate is used to obtain the LCP and RCP states of incident light.In the case of linear polarization measurement, one linear polarizer and one linear analyzer were used.The incident light is focused onto the metasurface polarizers from the substrate side by a lens with a focal length of f ¼ 100 mm and the transmitted light is collected by a 10× objective lens.The power of the transmitted light was measured by a Thorlabs power meter.For the polarization image encryption measurement, the transmitted signals were captured by a CCD camera.For the optical holography experiment, the focal length of lens 1 is 250 nm which was used to generate a Gaussian beam with a waist radius of ∼140 μm.
where χ and ψ are the ellipticity angle and the main axes angle of the PS of light, respectively.As shown in Figs.1(a) and 1(b), the phase retardations of the fast and slow axes of the first metaatom are −2χ and 2χ, respectively.The second meta-atom acts as a half-wave plate with phase retardations of 0 and π.The rotation angle of the first and second meta-atoms are ψ − 45 deg and ψ, respectively [Fig.1(c)].Therefore, by controlling the birefringence of the two meta-atoms, one can convert the incident light with arbitrary PS to any position ðψ; χÞ on the Poincaré sphere [Fig.1(d)].However, once the phase retardations of a meta-atom deviate from the ideal values, the performance of the entire metasurface polarizer will deteriorate, as shown in Figs.

Fig. 1
Fig. 1 Design concept of the metasurface polarizers.(a) A 45 deg counterclockwise rotated metaatom with birefringence in a square lattice of period p.The phase retardation in different axes depends on the output PS's ellipticity angle χ.(b) A meta-atom, which is equivalent to a half-wave plate in a square lattice of period p. (c) The unit cell of a metasurface polarizer consists of two above-mentioned meta-atoms, which has a 2p period.The two meta-atoms fill the diagonal position and the rotation angle of these combinations depends on the output PS's main axes angle ψ.(d) Arbitrary PS on the Poincaré sphere can be described by the two parameters χ and ψ.(e) The schematic illustration of an ideal metasurface polarizer.Under the pumping of arbitrary PS, the metasurface polarizer can completely convert it into the target PS ðψ; χÞ.(f) The schematic illustration of the imperfect metasurface polarizer.Only a certain percentage (α) of the transmitted light has the target PS ðψ; χÞ, and the coefficient β corresponds to the residual due to the imperfections of the meta-atoms.

Fig. 2
Fig. 2 Selection of the unit cell of metasurface polarizers.(a) Sketch of a SiN x dielectric metaatom, which is sitting on a fused silica substrate.φ x and φ y indicate the phase retardation that the transmitted light carries in the x and y axes, respectively.Structure parameters, h, l, and w , correspond to the height, length, and width of the meta-atom, respectively, and p is the period of the square lattice.(b), (c) Phase retardation library at the wavelength of 660 nm.The height h of the meta-atoms is kept at 1400 nm, and l and w are varied from 100 to 380 nm.The periods of the unit cell in panels (b) and (c) are 400 and 450 nm, respectively.(d)-(f) Simulation transmission coefficients of three kinds of metasurface polarizers, including horizontal linear polarizer (LP-Pol-H, l 1 ¼ 210 nm, w 1 ¼ 210 nm, l 2 ¼ 160 nm, w 2 ¼ 370 nm, and p ¼ 400 nm), left circular polarizer (LCP-Pol, l 1 ¼ 380 nm, w 1 ¼ 110 nm, l 2 ¼ 370 nm, w 2 ¼ 150 nm, and p ¼ 450 nm), and right circular polarizer (RCP-Pol, l 1 ¼ 110 nm, w 1 ¼ 380 nm, l 2 ¼ 370 nm, w 2 ¼ 150 nm, and p ¼ 450 nm).For (d) LP-Pol-H, four combinations of polarization measurement schemes (horizontal polarization H and vertical polarization V) are calculated.H-H means that both the incident light and transmitted light are linearly polarized.Panels (e) and (f) show optical properties of the LCP-Pol and RCP-Pol.LCP and RCP: left-and right-circular polarizations.

Fig. 3
Fig. 3 Experimental retrieval of the Jones matrix of the linear metasurface polarizer.The PSs of the incident light include linear polarizations along H, V , 45 deg, 135 deg directions, and left-and right-circular polarizations (LCP and RCP).After passing through the LP-Pol-H metasurface, the polarization ellipses directions of the transmitted light at wavelengths of 690 and 760 nm were measured.The dot and solid lines correspond to the measured and calculated data, respectively.From the calculated results, we can obtain the spin directions of the electric fields, which are indicated by the arrows.Each element of the Jones matrix is shown by the bar charts, in which the height and color are associated with the amplitude and phase.
4(e)], the contrast between the two perpendicular polarizer regions (letters: T and A) is distinct, indicating the high-polarization extinction ratio of the linear metasurface polarizers.As the fast axis of the designed linear metasurface polarizers in M and E has the projection angle of φ ¼ 45 deg; 135 deg with respect to the horizontal axis, the intensity of these two letters is basically the same as each other according to the Malus's Law.When the wavelength of incident light is switched to 760 nm, the intensities of the four letters deviate from the theoretical values due to the existence of the residual light [Fig.4(g)].In Figs.4(f) and 4(h), we show that the intensity distribution of the META image can be calculated through the measured Jones matrix of the linear metasurface polarizer (see Sec. 6 in the Supplementary Material).

Fig. 4
Fig. 4 Polarization image encryption with the linear metasurface polarizers.(a) A schematic drawing of a tilted linear metasurface polarizer, which is rotated by an angle φ.(b) The layout of the linear metasurface polarizers.For letters M, E, T, and A, the corresponding rotation angles are φ ¼ 45 deg; 135 deg; 0 deg, and 90 deg, respectively.The red pattern represents the incident Gaussian beam.(c), (d) The scanning electron microscope (SEM) images of the metasurfaces.The scale bars of the SEM images are (c) 20 and (d) 1 μm, respectively.(e), (g) For incident light with LCP state, the intensity distribution of the four letters is measured at the wavelengths of 690 and 760 nm using a horizontal linear analyzer.(f), (h) The calculated intensity distribution of the four letters is based on the measured Jones matrix.The results in (e)-(h) are normalized to the maximum intensity values.

Fig. 5
Fig. 5 Optical holography with circular metasurface polarizers.(a), (b) Schematics of the Fourier space holography based on RCP-Pol and LCP-Pol.The working wavelength is 700 nm.For the incident light with arbitrary PS, the circularly polarized output light carries a phase factor of 2φ, where φ is the rotation angle of the metasurface polarizer.(c), (k) The target intensity distribution of the holographic images.(d), (i) The calculated phase distributions of metasurface holograms.(e), (m) The reconstructed holographic images using the conditions in the experiment.(f), (n) The SEM images of RCP-Pol and LCP-Pol hologram devices (scale bar: 500 nm).The holographic images of (g)-(j) RCP-Pol and (o)-(r) LCP-Pol hologram devices were obtained under the measurements of four polarizer and analyzer (LCP and RCP) combinations.