Infrared full-Stokes polarimetry by parametric up-conversionZhanghang Zhu†,1 Di Zhang†,1 Fei Xie,1 Jiaxin Chen,1 Shengchao Gong,1 Wei Wu,1 Wei Cai,1 Xinzheng Zhang,1

Mengxin Ren,1, 2, a) and Jingjun Xu1, b)1)The Key Laboratory of Weak-Light Nonlinear Photonics, Ministry of Education,School of Physics and TEDA Applied Physics Institute, Nankai University, Tianjin, 300071,P.R. China2)Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006,P.R. China

Polarimetry aims to measure polarization information of an optical field, providing a new freedom to enhanceperformance of optical metrology. Furthermore, the polarimetry in infrared (IR) range holds a promise for awide range of academic and industrial applications because the IR light relates to unique spectral signaturesof various complex chemicals. However, a primary challenge in IR applications remains lacking efficientdetectors. Motivated by such a constraint, we present in this paper a nonlinear up-conversion full-StokesIR Retrieval Polarimetry (IRP) that is able to decipher the IR polarization using a well-commercial andhigh-performance visible light detector. Assisted by a nonlinear sum-frequency generation (SFG) processin a lithium niobate thin film, the polarization states of the IR light are encoded into the visible SF wave.Based on a Stokes-Mueller formalism developed here, the IR polarization is successfully retrieved from SFlight with high precision, and polarization imaging over the targets with either uniform or non-uniformpolarization distributions are demonstrated. Our results form a fresh perspective for the design of noveladvanced up-conversion polarimeter for a wide range of IR metrological applications.

Infrared (IR) light is electromagnetic radiation withwavelength longer than that of visible (VIS) light.1 Al-though invisible to human naked eyes, the IR light holdssignificant usefulness in various cutting-edge applicationsincluding biological sensing,2 astronomical observation,3

and environmental science,4 owing to its capacity forunique chemical specificity. Until now, the IR relatedapplications are mainly challenged by the lack of effi-cient detectors.5 For example, becuase the IR sensitivesemiconductors have small band gap energy, their per-formance is easily disturbed by ambient thermal fluctu-ation in terms of dark current and noise. Furthermore,the available pixel number of the arrayed IR detectorsis limited. This renders the IR cameras a poor res-olution compared with their visible counterparts, suchas complementary metal-oxide semiconductors (CMOS)and charge-coupled devices (CCD).6

An appealing solution to above constraints is para-metric frequency up-conversion, such as sum-frequencygeneration (SFG).7 In such architecture, instead of di-rectly detecting the IR fundamental frequency (FF) pho-tons, the IR signal (at frequency ω1) is firstly trans-ferred to the visible range with the help of another pumplight (ω2) via a second order nonlinear (χ(2)) material.8

Then the generated SF photons (ω3 = ω1 + ω2, follow-ing the energy conservation law) are detected by high-performance visible detectors. The first study of suchparametric up-conversion of IR light could be traced backto 1960s, when Midwinter and Warner demonstrated aconversion of 1.7 µm radiation to 493 nm by a lithiumniobate (LN) crystal.9 One year later, such up-conversion

a)Electronic mail: ren mengxin@nankai.edu.cnb)Electronic mail: jjxu@nankai.edu.cn

technique was further applied to realize IR up-conversionimaging.10 Over the past decades, the parametric up-conversion detection has received great interests, andseveral technical improvements were made in various as-pects, such as finding new nonlinear materials for higherup-conversion efficiency,11 ameliorating sensitivity of thevisible detectors,12 manufacturing high power IR sourcesat various wavelengths for target illumination,13,14 op-timizing system designs for better imaging field-of-viewand resolution.15,16

However, the current IR up-conversion techniques arestill designed to acquire the intensity and spectral infor-mation of the IR light, while the polarization informa-tion is simply dropped. Complementary to the intensityand spectrum, the polarization is another fundamentalcharacteristic that describes how the light’s electric fieldvector oscillates. While the intensity and spectrum tellus about the geometries and material compositions, po-larization interrogation can bring numerous additionalrichness from the targets. Specifically, by analyzing thelight polarization changes by scattering or transmittingthrough the objects, people can infer birefringence orstress inside material volumes, and roughness or textureof a surface, moreover, enhance contrast for objects thatare difficult to distinguish otherwise.17,18 In this context,polarimetry, as a technique invented to measure the po-larization state of light, has become an emerging tech-nique to enhance many fields of optical metrology, opticalsensing, and so on.19

Enlightened by the respective merits of the two tech-niques of the nonlinear IR up-conversion and the po-larimetry, it would be a good idea to further combinethem together. Such incorporation could inherit advan-tages of both techniques, i.e. detecting IR polarizationinformation while using the VIS detectors. This would

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IR Retrieval Polarimetry

χ(2)

(a) (b)

ϕ

ξ1S

2S

3S

21 (2) , M SGT HWP QWP Mask L1 L3

LN

L4

QWP GT

L5 CCDGT L2

Pump

Probe

PSG

Delay Line

PSAx(o)

y(o)

FIG. 1. A schematic of IRP system. (a) Operation principle of the IRP system. FF signal (ω1) and pump light (ω2) areup-converted into SF wave (ω3) through a second order nonlinear material. The nonlinear SFG process can be presented by a

Mueller matrix M(χ(2),Sω2

), via which the Stokes vectors of Sω1 can be calculated reversibly from Sω3 . (b) A sketch of the

IRP setup. An optical pump-probe system was built. The path difference between the pump and probe pulses is compensatedby a mechanical delay line. The pump polarization is vertically polarized by a Glan-Taylor (GT) prism, and the polarizationof the probe light is prepared by a polarization state generator (PSG) consisting of a GT polarizer, a half-wave plate (HWP)and a quarter-wave plate (QWP). The optical beams are focused onto the LN film respectively afterwards by lens pairs withfocal lengths of 500 mm (L1, L2) and 100 mm (L3). The generated SF light is collected by a lens (L4, 75 mm), followed by apolarization state analyzer (PSA) comprising a rotating QWP, and a GT prism, and finally imaged onto a visible CCD cameraafter a lens (L5, 200 mm). Physical orientation and the shape of the polarization ellipse are defined by angles of azimuth (φ)and ellipticity (ξ), respectively. Positive values of φ and ξ correspond to the clockwise rotation of the polarization azimuth anda right-handed polarization ellipse, as observed against the propagation direction. Intensity images by the signal light for ‘L’,‘N’ and ‘house’ masks are given in the top-right corner.

consequently benefit many fields of IR related applica-tions. But to the best of our knowledge, such combina-tion still remains unexplored. In this paper, we experi-mentally demonstrate an IR Retrieval Polarimetry (IRP)that manages to decipher the polarization information ofthe IR signal from the generated SF light. We develop aStokes-Mueller formalism based on nonlinear optics thatcorrelates the input IR light polarization states with theSF light. In such algorithmic framework, the incomingIR signals and outgoing SF radiations are representedby 4×1 Stokes vectors, and the nonlinear SFG process isrepresented by a 4×4 Mueller matrix. We further experi-mentally achieve the IR polarization imaging of differenttargets with uniform or non-uniform polarization distri-butions using our IRP system, which reconstructs the po-larization of the IR signal light with high precision. SuchIRP technique would pave the way to the design of novelIR polarimetry for advanced remote sensing, astronomyand industrial inspection.

I. STOKES-MUELLER FORMALISM FOR IRP

First, we develop a Stokes-Mueller algebra for retriev-ing the polarization of the IR signal light based on thatof the SF wave, which essentially forms the mathemati-cal basis for the operation of our IRP. We adopted here afour-element Stokes vector S = [s0, s1, s2, s3]T (T denotesmatrix transpose) to describe the polarization state of

light.20 The Stokes description of light has a chief advan-tage that it operates in the form of intensity, which is realvalued and is easily measurable using photo-detectors inexperiments. Furthermore, in contrast to other repre-sentations, such as the Jones vector that is limited to thecase of fully polarized light,21 the Stokes vector also spansthe space of unpolarized and partially polarized light.Each Stokes element is defined as a sum or differencebetween intensities of different orthogonal polarizationbases. Specifically, s0 is the total intensity of the light,s1 is the intensity difference between linear polarizationcomponents along 0◦ and 90◦, s2 is the intensity differ-ence between +45◦ and −45◦ linear polarizations, and s3is the intensity difference between right- and left-handedcircular polarization components. In such representation,the polarization dependent interaction between the lightand matter is denoted by a 4×4 Mueller matrix (M), andthe incident polarization state S relates to the outputstate S′ via S′ = M · S. Such Stokes-Mueller formalismacts as a basis for the traditional linear optical polarime-try to characterize the signal light polarization.21 Herewe further develop a new nonlinear Stokes-Mueller for-malism for the IRP technique, which aims to reconstructthe IR polarization from that of the SF light, rather thanmeasuring that of the IR light directly, as illustrated byFig. 1(a).

According to the nonlinear optics, the generated SFwaves are closely related to the second order nonlinearpolarizations P(2) excited inside the materials. Specifi-

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s1

s3

s2

(b)

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s1

s3

s2

(a)

(c) (d)

ϕ (°) ϕ (°)

ξ(°)

ξ(°)

Meas. Retrv.

Signal sphere 1( )S SF sphere 3( )S

FIG. 2. Calibration of IRP. (a) Poincare sphere of FF signal light (at ω1). 72 different signal polarization states (reddots) that uniformly sample the sphere were chosen as input polarizations for calibrating IRP. (b) Poincare sphere of generatedSF light. (c) The three-dimensional FF Poincare sphere is expanded into a planar map in a coordinate of azimuth–ellipticityangles (φ− ξ). The originally input FF polarization states are presented by the dots in color, and the reconstructed results aregiven by small gray dots. The well overlap between them demonstrates the high precision of our IRP in reconstructing the FFpolarizations. (d) The φ− ξ map of the SF Poincare sphere. The SF dots generated by the FF light are shown using the samecolor as those in (c).

cally, the P(2) for the SFG process follows22

P(2)i (ω3) = 2

∑j,k

ε0χ(2)ijkE

ω1j Eω2

k , (1)

in which footnotes i, j, k stand for the unit vector alongthe Cartesian x-, y- or z-axes, and Eω1

j and Eω2

k arethe electric fields of the signal and pump light, respec-

tively. χ(2)ijk is the element of the nonlinear susceptibility

tensor, which generates P(2) vector polarized along thei-direction under the combined interaction of the fun-damental electric fields along j- and k-directions. Such

P(2)i acts as a secondary source to radiate the nonlinear

SF waves Eω3 polarized along the i-direction, and mag-nitude of the Eω3 is proportional to P(2). By furtherconsidering the mathematical relationship between Sω3

and Eω3 , we could derive

Sω3 = M(χ(2),Sω2

)· Sω1 , (2)

in which M(χ(2),Sω2

)is a 4×4 Mueller matrix describ-

ing the SFG process. Equation (2) has a similar form as

the linear Stokes-Mueller formalism. However, differentfrom the Mueller matrix in linear optics that only relatesto medium’s properties such as birefringence or opticalchirality,21 the SFG matrix M here is a function of notonly properties of the material (i.e. nonlinear χ(2)), butalso the pump polarization Sω2 . Once the SFG matrixM and Stokes parameters of the SF light (Sω3) are de-termined, the polarization state of the signal light couldbe calculated via

Sω1 = M−1(χ(2),Sω2

)· Sω3 . (3)

II. SETUP OF IRP

An x-cut LN thin film was used as the SFG materialin our experiment. The thickness of the LN film is about200 nm, which resides on a 500 µm thick silicon dioxide(SiO2) substrate (NANOLN, Jinan Jingzheng Electron-ics Co., Ltd). LN is a typical optical material show-ing ultra-broadband transparency and strong second or-

4

der nonlinearity covering the visible to mid-IR spectralrange, which renders it a feasible material platform forwide photonic and optoelectronic applications. The LNcrystal belongs to a point group of 3m. Resulting fromthe symmetry restrictions, χ(2) tensor of the LN has 11non-vanishing components, in which only 4 elements are

independent: χ(2)eee, χ

(2)eoo, χ

(2)ooo, χ

(2)ooe.23 The experimen-

tal axes were defined to overlap with the LN principalcrystallographic coordinate. Optical axis of the LN isnominated as z(e)-axis, and the light propagated along−x-direction, as shown in Fig. 1(b).

A schematic of the experimental setup is shown inFig. 1(b). An optical pump-probe system was built.The near-infrared (NIR) laser pulses at a wavelengthof 808 nm were generated by a Kerr lens mode-lockedTi:sapphire laser oscillator (Spectra Physics, Maitai,pulse width of 230 fs, repetition rate of 80 MHz). Thepulse train was divided into two paths using a beamsplitter. A delay line, consisting of a linear translationstage in the pump beam path, was used to compensatethe length difference between the pump and probe pathsand ensured the fine control of the pulse synchronization.The pulses were adjusted to be well overlapped tempo-rally to guarantee the strongest SFG. The optical beamswere focused onto the LN film afterwards by lens pairswith focal lengths of 500 mm and 100 mm, forming fo-cal points with the diameter of 135 µm. The intensityof the pump and probe beams were about 142 MW/cm2

and 130 MW/cm2 on the LN surface, respectively. Thepolarization of the pump beam was fixed along the ordi-nary (o-) axis of the LN crystal. The generated SF lightat 404 nm was collected in the forward direction by a lenswith the focal length of 75 mm. Short-pass filters (BG40,not shown) were inserted in the SF emission path toblock the remaining 808 nm light. The SF light then en-tered a home-built full-Stokes polarization state analyzer(PSA) comprising a rotating quarter-wave plate (QWP),a Glan-Taylor (GT) calcite polarizer, and afterwards wasdetected by a monochrome electron-multiplying charge-coupled device (EMCCD, iXon Ultra 888). 91 framesof SF signal were recorded while the QWP was rotatedfrom 0 to 180◦ with a discrete step of 2◦. Through Fouriertransformation of the recorded SF intensity as a functionof the QWP rotation angle, the Stokes parameters of theSF wave could be obtained.24

III. CALIBRATION OF IRP

In order to accurately reconstruct Sω1 of the incidentIR signal light via the measured SF Stokes parameters ac-cording to Eq. (3), the first objective is to determine theMueller matrix M

(χ(2),Sω2

)of the IRP system. Such

process could be referred as calibration of the IRP, whichcould be accomplished by resorting to multiple sets ofknown input and output polarization pairs.

The polarization states of the signal beam incidenton the LN were prepared by a polarization state gen-

erator (PSG) consisting of a GT polarizer, a half-waveplate (HWP) and a QWP. For the calibration end, 72different signal polarization states that uniformly sam-ple the Poincare sphere were chosen as input signal po-larizations, whose Sω1 parameters were further experi-mentally characterized using the PSA [as illustrated byred dots in Fig. 2(a)]. The corresponding generated Sω3

were measured as well and labeled by blue dots on theSF Poincare sphere in Fig. 2(b). The 16 elements of Mlinking Sω1 to Sω3 can be numerically determined usinga least squares method. Afterwards, the FF polarizationstates can be retrieved through Eq. (3). For the clearercomparison between the original and the retrieved IR sig-nal polarizations, the three-dimensional Poincare spheresare expanded into planar maps, as shown in Fig. 2(c,d),in forms of angles of azimuth [φ = 1

2 arctan( s2s1

)] and

ellipticity [ξ = 12 arctan( s2√

s21+s22)], which correspond to

the physical orientation and the shape of the polariza-tion ellipse, respectively. The well overlap between thereconstructed polarization states (small gray dots) andthe directly measured ones (large color dots) in Fig. 2(c)demonstrates the excellent precision of our IRP in re-constructing FF signal polarization states from the SFpolarizations.

IV. STOKES POLARIMETRIC IMAGING

We now apply our IRP in full-Stokes polarization imag-ing. Such polarization imaging technique aims to mapthe polarization states across a scene of interest. In thetraditional IR imaginary polarimetry, the polarizationdistribution over the cross-section of the signal beam isdirectly characterized by the PSA equipped by a IR cam-era. However, here we would show the retrieval of thespatial polarization information contained in the IR sig-nal beam via analyzing that of the SF wave. To demon-strate such application, binary masks with transparentletters of ‘L’ and ‘N’ were used as the targets, which werefabricated by focused ion beam milling through a 200 nmthick opaque gold film supported by a SiO2 substrate [theintensity images of the masks are given in Fig. 1(b)]. Themasks were inserted into the signal beam, and were pro-jected onto the LN film plane. The polarization states ofthe signal light illuminating the mask were adjusted bythe PSG in front. In this configuration, the polarizationstates across the signal beam were uniformly distributed.After interacting with the pump beam, both the morpho-logical and polarization information of the masks con-tained in the signal beam were successfully transferredinto the generated SF light and recorded by the PSA.The SF Stokes parameters were analyzed at each pixel,and the polarization images at the SF wavelength wereconstructed afterwards. Figure 3 presents the results ofthe masks by our IRP system. In the literatures of polar-ization imaging, different sets of polarization parametersare commonly used. The first one is the Stokes param-

5

s2

s3

s1

(a)ϕ

(°)

ξ(°

)

SF Meas. Signal Retrv. Signal Meas.

(b)

FIG. 3. Polarization images of ‘L’ and ‘N’. (a) Images of Stokes parameters s1, s2, and s3. The first, second and thirdcolumns give the experimentally measured results of SF waves, the retrieved FF signal, and the directly measured FF results,respectively. Outlines of the letters are sketched by dashed lines. Scale bar of 100 µm is given in bottom-left conner of the firstpicture. (b) Images are given in forms of azimuth (φ) and ellipticity (ξ) of the polarization ellipse. It is clear that the retrievedFF polarization parameters reproduce the directly measured FF results very well, confirming the precision of our technique.

eters (s). In contrast to the traditional intensity pho-tography that gives the same images despite the signallight polarization varies, the s1, s2 and s3 present a goodcapability in distinguishing different signal polarizations.The Stokes parameters are uniformly distributed insidethe letter area, while the background is noisy because noreliable SF signal was detected resulting from the opacityoutside the letter areas. For each mask, the Stokes im-ages at the SF wavelength are given in the first column.The retrieved FF images following Eq. (3) are given in thesecond column, and the directly measured FF results aregiven in the third column. It is clear that the retrievedFF polarizations reproduce the directly measured FF val-ues very well, indicating the fidelity and precision of ourtechnique in reconstructing the FF polarization images.Images of φ and ξ are further plotted in Fig. 3(b) to intu-itively present the geometric features of the polarizationellipse, which also confirm the good consistence betweenthe retrieved and the directly measured FF signal values.

We further implement our IRP technique to the tar-get that contains an inhomogeneous birefringent struc-ture, which causes multiple polarization components inthe signal light. Such scenario holds a high applicationsignificance in the fields of remote sensing for mineralexploration, as well as microscopy for characterizing the

anisotropic biological tissues. To demonstrate such func-tionality, a commercial depolarizer was adopted as thetarget. A ‘house’ shaped mask was adopted [as shown bythe intensity image in Fig. 1(b)] and put on the surfaceof the depolarizer. Such depolarizer is a patterned mi-croretarder array, which consists of a thin film of birefrin-gent liquid crystal polymer sandwiched between two glassplates. The fast-axis orientation and phase retardation ofthe liquid crystal were designed to vary with a periodicalpattern inside the depolarizer. Thus, after transmittingthrough such depolarizer, a spatially varied polarizationdistribution would be produced over the cross section ofthe signal light. It is clear that the birefringence featureof the depolarizer is not discernible by the intensity im-age in Fig. 1(b), however, it is very well resolved in thepolarization images in Fig. 4. The pseudo-color plot es-sentially traces the contour of the spatial birefringencevariation of the liquid crystal, and each band of colorin the polarization images represents a region with thesame polarization state. Similar as Fig. 3, the retrievedFF polarization images here (the second column of Fig. 4)reproduce the directly measured FF values (the third col-umn) very well, confirming the precision of our techniqueagain.

6

s2

s3

s1

SignalRetrv.

SFMeas.

SignalMeas.

ϕ(°)

ξ(°)

FIG. 4. Polarization images of a ‘house’ mask witha non-uniform birefringence distribution. The imagesare given in forms of Stokes parameters (s), azimuth (φ) andellipticity (ξ). The retrieved FF signal polarization images(the second column) reproduce the measured FF values (thethird column) very well.

V. CONCLUSION

In conclusion, we build a nonlinear up-conversion full-Stokes IRP that manages to retrieve the polarization in-formation of the incident IR probe light from that ofthe up-converted SF wave. The IR polarization im-ages of targets with either uniform or non-uniform po-larization distributions are successfully reconstructed us-ing our IRP system with high precision. Despite wefocus here on the degenerate SFG process that boththe pump and probe light have the same wavelengthat NIR, it is expected that our method also validatesfor the non-degenerate SFG process where the signalphoton can be in mid- or long-infrared range and up-converted by visible pump light.25,26 Thus it is applica-ble to vibrational spectroscopy for birefringent moleculeidentification, and thermal imaging for night vision de-tection. Furthermore, utilizing the current state-of-the-art scheme of nonlinear enhancement by nanostructuressuch as metasurfaces27–30 or intracavity up-conversionconfiguration,31,32 the efficiency of the SFG process fromthe LN film could be furthermore effectively improved.We could foresee a direct impact of our results on a

variety of IR polarimetric applications, such as opticalcrystallography, chemical sensing, disease diagnosis, andexplosive detection, etc.

ACKNOWLEDGMENTS

Z.Z. and D.Z. contribute equally to this work. Thiswork was supported by National Key R&D Pro-gram of China (2017YFA0305100, 2017YFA0303800,2019YFA0705000); National Natural Science Founda-tion of China (92050114, 91750204, 61775106, 11904182,12074200, 11774185); Guangdong Major Project of Ba-sic and Applied Basic Research (2020B0301030009); 111Project (B07013); PCSIRT (IRT0149); Open ResearchProgram of Key Laboratory of 3D Micro/Nano Fabrica-tion and Characterization of Zhejiang Province; Funda-mental Research Funds for the Central Universities (010-63201003, 010-63201008, 010-63201009, 010-63211001);Tianjin Youth Talent Support Program. We thankNanofabrication Platform of Nankai University for fabri-cating samples.

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