Spectrally resolving single-shot polarimeterSebastian Knitter,* Tim Hellwig, Michael Kues, and Carsten Fallnich

Institut für Angewandte Physik, Universität Münster, Corrensstraβe 2, 48149 Münster, Germany*Corresponding author: sebastian.knitter@uni‑muenster.de

Received May 31, 2011; revised July 5, 2011; accepted July 6, 2011;posted July 8, 2011 (Doc. ID 148456); published August 5, 2011

We demonstrate a spectrally resolving single-shot polarimeter. The system consists of a commercial imagingspectrograph, modified by a birefringent wedge and a segmented polarizer. The physical operating principleand limitations of the apparatus as well as preliminary polarimetric measurements on the emission of randomlasers are reported. © 2011 Optical Society of AmericaOCIS codes: 120.2130, 260.2130.

Light emitted from luminescent samples (i.e., in organicchemistry or biology) is most often characterized by theintensity spectrum. In some cases, the polarization ofemitted or reflected light (e.g., in thin film chemistry, lifescience, or ultrafast optics) is also a quantity of interestthat yields additional information about the system underinvestigation [1–3]. We present an instrument, that char-acterizes the sample’s emission by the complete Stokesvector ~SðλÞ ¼ ðS0ðλÞ; S1ðλÞ; S2ðλÞ; S3ðλÞÞ for a set of hun-dreds of wavelengths λ at a time. Although there arenotions of other spectral domain polarimeters in the lit-erature [4,5], the presented scheme allows turning anyavailable imaging spectrograph with a two-dimensionalarray detector into a single-shot spectropolarimeter withdiffraction-limited resolution. In this way, one mightmake use of cooled detectors, which are specifically de-signed for use in spectroscopic applications and featureexcellent performance in quantum yield, readout noise,and linearity.The technique involves dispersing the light spectrally

along the horizontal axis of a two-dimensional sensor bymeans of a grating spectrograph, and intensity codingthe polarization in the vertical direction by a division-of-amplitude-type mechanism [6]. The components used forthe latter task are a birefringent wedge prism (BW) and asegmented (mosaic) polarizer (MP).Light passing the BW experiences a linear gradient

of birefringence along the vertical direction and thus achange in its polarization state. The polarizer, acting onthese states, maps them to distinct intensities. A Czerny–Turner monochromator with toroidal mirrors, as part ofthe spectrograph, images this intensity modulation fromthe entrance plane of the instrument into the detectionplane, where the camera’s pixels constitute sensors atthe end of a detection channel. Because of the linear nat-ure of the detection system, the intensity pattern alongeach individual wavelength can be utilized to reconstructthe polarization states by inversion of the system’s trans-fer matrix. Two important considerations must be madeat this point: first, the BW and the MPmust be close to theentrance slit of the spectrograph to prevent the mixing ofintensity channels due to diffraction. Second, the wedgeangle must be chosen so small that the polarization statedoes not change over the size of one pixel by more thanthe required polarimetric instrument accuracy. In prac-tice, for a pixel size of 20 μm and an overall height of100 pixels, a wedge–polarizer and polarizer–slit distance

of a few millimeters and a wedge angle of 1° allow foran overall accuracy of 10% or better.

A calibration procedure [7] yields the wavelength-dependent two-dimensional transfer matrix AðλÞ forthe detection system (lenses, mirrors, grating, and detec-tor response), which connects an arbitrary input Stokesvector ~SðλÞ with the intensity readings ~IðλÞ along aparticular column of the detector ~IðλÞ ¼ AðλÞ~SðλÞ. To re-construct any polarization state ~S, the Moore–Penrosepseudoinverse of A (i.e., A−1) will be calculated and ap-plied to the intensity vector ~I [8].

Considering the simplest case, in which only foursensors (i ¼ 1…4) are used, A ¼ Aij is a 4 × 4 instrument

matrix. The ith rowof A itself is the first rowof theMuellermatrix Mi modeling the polarimetric evolution of

polarized light, before impinging on sensor i: ~S0i ¼ Mi

~S.The instrument matrix, for a set of four beams of equal

initial polarization ~S, passing through an inhomogen-eously birefringent and polarizing medium is, therefore,proportional to

Ai1 ¼ 1;

Ai2 ¼�

cos 2αiðcos22βi þ sin22βi cos δiÞþ sin 2αi cos 2βi sin 2βið1 − cos δiÞ

�;

Ai3 ¼�

sin 2αiðsin22βi þ cos22βi cos δiÞþ cos 2αi cos 2βi sin 2βið1 − cos δiÞ

�;

Ai4 ¼�

cos 2αi sin 2βi sin δi− sin 2αi cos 2βi sin δi

�;

where αi is the orientation of the polarizers, βi is theorientation of the fast axis of the birefringent material,and δi is the maximum retardation in the ith channel.For the inverse to exist, detðAÞ ≠ 0must be satisfied. In thespecial case of a single-crystal BW retarder, all angles βiare equal. Even though it involves a fair amount of algebra,it can easily be shown that detðAÞ ¼ 0, if not at least one ofthe polarizers is oriented differently, from all others. Forthis reason the polarimeter must include an MP, in orderto unambiguously reconstruct arbitrary polarizationstates.

The experimental setup is displayed in Fig. 1. Lightleaving the sample (S), was collimated using an asphericlens (L, f ¼ 15mm). A cylindrical lens (CL) was used to

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create a vertical focal line in the plane of the monochro-mator’s entrance slit. Between the cylindrical lens andthe slit, the BW (material: crystaline quartz) and the MPwere placed. The slow and fast axes of the quartz crystallay perpendicular to the direction of incidence and thethickness of the crystal varied linearly in the vertical di-rection by 20 μm=mm. The MP (custom product, CodixxAG) was made up of one nonpolarizing segment (op-tional, for reference measurements) and two differentlyoriented polarizers with a relative offset angle of 45 deg.The spectral dispersion in the horizontal plane was rea-lized by a commercial grating spectrograph (Acton-Research SP-2500i, f ¼ 0:5m) with a diffraction-limitedresolution of 0:1nm.A halogen lamp, in combination with additional polar-

izers as well as half- and quarter-wave plates, was usedfor the calibration of the instrument. This incandescentlight source was chosen for its temporally stable andessentially flat emission spectrum in the desired wave-length range of 550–570 nm, covering the emission band-width of Rhodamine 6G dye. The calibration procedureitself [7] was in part automated by means of motorizedrotation stages. Figure 2 shows spectropolarigrams asread from the sensor array (size 1340 × 100 × 16 bits) fordifferent input polarizations and their corresponding re-constructions. The calculation of the inverse transfermatrix AðλÞ−1 (1340 × 100 × 4 elements) required about30 min on an eight-processor Xeon computer. However,once AðλÞ−1 was known, the full spectral and polarimetricinformation could be recovered and displayed in realtime on an office-grade Pentium computer. The framerate is momentarily limited to 10Hz by the readout timeof the array sensor (Princeton Instruments, Pixis 100BR-excelon, 20 μm × 20 μm pixels). By using the internalbinning functions of the camera, the frame rate can beincreased at the expense of lowered spectral resolutionand/or polarimetric accuracy. The covered spectralrange can be altered by choosing a grating with a differ-ent grating constant for the monochromator.A figure-of-eight test was performed by turning a

broadband quarter-wave plate by 360 deg in front of thepolarized emission from a halogen lamp. 95% of all recon-structed states over all wavelengths lay in the proximity

of 3:8 deg on the surface of the Poincaré sphere and�3:5% in magnitude of the theoretical values, computedwith Mueller calculus [9]. Figure 3 shows how the relativeuncertainty dropped with increasing number of pixelrows being used. The relative error decreased ratherrapidly toward 50 pixels and saturated below 10% forthe maximum number of 100 pixels. Also, the accuracydecreased dramatically when the average signal-to-noiseratio was below 100∶1. Generally, the inherent detectornoise, the retardation accuracy (λ=100) of the broad-band wave plates being used for calibration, and thediffraction-induced blurring around the polarizer seg-ments accounted for most of the residual error.

The system reported in this Letter is excellently suitedfor investigations on fluorescent media in liquids, forexample, on random lasing from nanoparticles in a dyesolution. In such media, narrow emission lines appear

Fig. 1. Experimental setup in (a) top and (b) side views:S, sample; L, lens; λ=4 and λ=2, quarter- and half-wave plates;CL, cylindrical lens; BW, birefringent wedge; MP, mosaicpolarizer; CCD,CCD camera.

Fig. 2. Sensor-array readings for (a) vertically linear and(c) right-handed circular polarization and their correspondingreconstructions [(b) and (d), dotted, S0; dashed, S1; dashed–dotted, S2; solid, S3]. The raw images show a slightly inclinedhorizon, which was attributed to an imperfect mounting flangebetween monochromator and camera. The tilt is included in thecalibration and does not corrupt the measurement itself.

Fig. 3. Relative error of the polarization measurement de-creased rapidly with larger numbers of incorporated sensorrows during reconstruction and saturated below 10%.

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randomly atop the usual emission spectrum, due to ascattering-induced coherent feedback mechanism [10].Because of the inherent mobility of the particles, randomlasing emission spectra change with every pulse from thepump laser and so does the wavelength-dependent polar-ization. Using the setup described here, the emissionfrom random lasers could be characterized not only bythe spectral intensity S0ðλÞ, but also by three additionalparameters: S1ðλÞ; S2ðλÞ; S3ðλÞ.The sample contained TiO2 particles (Sachtleben-

Chemie, ϕ ¼ 300 nm, rutile-type) at a concentration of 5 ·109 cm−3 immersed in a solution of dye in ethanol (Rho-damine 6G, 5mM). It was pumped by the frequency-doubled output of an Nd:YAG laser with a pulse durationof 9 ns at a repetition rate of 10Hz, focused to a spot ofapproximately 100 μm in diameter with a pulse energy of12 μJ. A quarter-wave plate was employed to circularizethe pump laser polarization, to prevent a prepolarizedfluorescent emission caused by optically induced align-ment of dye molecules [11].Figure 4 shows the reconstructed polarization states.

Great care must be taken in interpreting the actualpolarization states, since they were not normalized. Themeasurement showed a degree of polarization DoP ¼ðS2

1 þ S22 þ S2

3Þ1=2=S0 of less than 0.4 for the fluorescentbackground, which is still unexpectedly high, since,e.g., an incandescent lamp with principally unpolarizedoutput would yield a DoP of less than 0.1. This behaviorcannot yet be explained conclusively, but it might beattributed to a not-negligible retardation error (λ=100)

of the quarter-wave plate, used to circularize the pumppolarization. It must also be noted that the signal-to-noiseratio dropped due to the small emission cross section ofRhodamine dye and the absence of random lasing modesbelow 555 nm and above 565 nm. This leads to an in-creased measurement error in these regions and a see-mingly increasing DoP. However, the increased DoP atthe position of most random lasing modes was signifi-cant. Also, the state of polarization showed distinctfeatures at these particular wavelengths. Even at a firstglance, spectral emissions from random lasing samplesshow an intricate polarization structure, which will besubject to further investigation.

In conclusion, we presented a technique to upgradeany imaging grating spectrograph equipped with a two-dimensional sensor array in the detection plane, to beused as a spectrally sensitive polarimeter. The spectralresolution of a modified unit is still limited by diffrac-tion as well as the size of the detector pixels, while theacquisition frame rate is only limited by the sensitivity ofthe detector and, in the end, by its maximum data trans-fer rate. We believe that this technique might assist re-searchers far beyond the obvious communities with anoninvasive, reliable, and fast technique to characterizepolychromatic light.

This work was supported by the German FederalMinistry of Education and Research (BMBF) withinthe compound project PEARLS under grant number13N10154.

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Fig. 4. Spectral polarization measurement of random lasingemission. Vertical lines indicate the spectral positions of ran-dom laser modes.

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