Improved measurement of polarization state in terahertz polarization spectroscopy M. Neshat∗ and N. P. Armitage Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA ∗Corresponding author: [email protected] CompiledJanuary24,2012 A calibration scheme is presented for improved polarization state measurement of terahertz pulses. In this scheme the polarization response of a two-contact terahertz photoconductive detector is accurately measured, andisusedtocorrectfortheimpactofthenon-idealitiesofthedetector.Experimentalresultsshowexcellent 2 sub-degree angular accuracy and at least 60% error reduction with this scheme. (cid:13)c 2012 Optical Society of 1 America 0 OCIS codes: (040.2235)Farinfraredorterahertz;(120.3940)Metrology;(120.5410)Polarimetry. 2 n a Terahertzpolarimetryisaquicklygrowingcharacteriza- We model the polarization response of the detector J tiontoolforthestudyofeffectssuchasbirefringenceand with a polarization vector P =[p p ] which is gener- d x y 3 themagneto-opticKerreffect[1]-[2].Recentlyanumber ally frequency dependent. In the frequency domain, the 2 of polarization sensitive methods for terahertz pulse ra- detected signal can be expressed by an inner product as diation have been proposed [3]– [7]. Most of these meth- ] s ods are based on photoconductive antenna detectors, in I(ω,φ)=P (ω)·E (ω,φ), (1) d d c which it is assumed that the detector has an ideal linear i where I is the Fourier transform of the detected signal, t polarization response over the entire frequency range. p E istheelectricfieldvectorimpingingonthedetector,ω d o However, it has been shown recently that this assump- istheangularfrequencyandφistheangularorientation . tion is generally not valid [8] and that in addition to the s of the analyzing polarizer. The polarization vector can c geometrical structure, the polarization response of the beexpressedintheformofanormalizedJonesvectoras i s photoconductive detector depends strongly on the opti- y cal and terahertz alignments [8,9]. Therefore, it is clear (cid:2) (cid:113) (cid:113) (cid:3) P = 1+s 1−sexp(jδ) , (2) h that calibration schemes are necessary to compensate d 2 2 p for a non-ideal polarization response of the detector in where the polarization parameters −1 ≤ s(ω) ≤ 1 [ order to get the highest accuracy. In this letter, we pro- andδ(ω)aredeterminedthroughthecalibrationscheme 1 pose a calibration scheme for widely used two-contact over the desire frequency range. In Fig. 1, E represents t v photoconductive detectors for accurate measurement of the transmitted polarization state, and is related to E 5 d the polarization state of pulsed THz radiation. through the Jones matrix of the rotated analyzing po- 5 7 Fig. 1 illustrates our terahertz time-domain polariza- larizer as 4 tionspectroscopysetup.Itusesan8f confocalgeometry (cid:104) E (ω,φ) (cid:105) (cid:104) cos2φ cosφsinφ (cid:105) 1. with THz TPX lenses, which are less prone to misalign- Ex(ω,φ) = cosφsinφ sin2φ × y d 0 ments and polarization distortion as compared to off- 2 axis parabolic mirrors. Two identical photoconductive (cid:104) Ex(ω) (cid:105) . (3) 1 E (ω) antennaswithsubstratelensareusedasemitterandde- y t : v tectors. A rotatable analyzing polarizer is placed in the In the proposed calibration scheme, the sample is re- i collimated beam immediately before the detector, and a X movedandthepolarizerattheemittersideisadjustedto fixed polarizer is placed immediately after the emitter. r yieldaknownpolarizationstatesuchthatintheabsence a Polarizerswerewiregridwithwirediameterandspacing of the sample E = E = [1 0]. Once the polarization i t of 10 µm and 25 µm, respectively, and field extinction stateofE isknown,P isdeterminedbysolving(1)and t d ratio of ∼40:1 at 1 THz. The space with terahertz wave (3) for two scans corresponding to two different angular propagation is enclosed and purged with dry air during orientation of the analyzing polarizer. For φ = ±45◦, s measurements. The laser source is a 800 nm Ti:sapphire and δ are readily calculated as femtosecond laser with pulse duration of < 20 fs and 85 MHz repetition rate, which is divided into pump and 1−|r|2 s= , δ =arg(r), probe beams. The temporal THz pulse is recorded by 1+|r|2 scanning the retro-reflector and varying the time delay I(ω,+45◦)−I(ω,−45◦) r = . (4) between terahertz pulse and the sampling probe laser. I(ω,+45◦)+I(ω,−45◦) Thetemporalsignalisthentakenintothefrequencydo- Oncethedetectorpolarizationvectorischaracterized, main through a Fourier transform. asamplecanbeputinplace,thatchangesE .Thenthe t 1 X ZZ Polarizer Sample Analyzer FL2 FL1 M7 M5 Ei Et Ed THz  THz emitter TL1 TL2 TL3 TL4 detector RR Pump M1 M2 M4 Probe M6 fspulse laser M3 BS (M:mirror, RR: retro‐reflector, BS: optical beam splitter, FL: optical focusing lens, TL: THz lens) Fig. 1. Experimental THz-TDS setup for polarization state measurement (M:mirror, RR: retro-reflector, BS: optical beam splitter, FL: optical focusing lens, TL: THz lens). new polarization state E is obtained from (1) and (3) outthedetectorresponse,awiregridpolarizersimilarto t for two angles φ=±45◦ as theanalyzingpolarizerwasusedasasample.Inthiscase, the polarization state after the sample polarizer is well (cid:104) E (ω) (cid:105) 1 (cid:104) p −p p +p (cid:105) x = x y x y × known and can be compared with that from measure- Ey(ω) t p2x−p2y px−py −(px+py) ment. The sample polarizer was installed in a precision (cid:104) I(ω,+45) (cid:105) rotation stage with 0.08◦ accuracy. The polarizer axis . (5) I(ω,−45) wasdeterminedaccuratelybythediffractedpatternofa red laser passing through the wire grid. The sample po- In principle any two orientation of the analyzing po- larizer was rotated from 0◦ (polarizer axis along x-axis) larizer can be chosen for this analysis as long as they to70◦with10◦increment.Polarizationstatesweremeas- satisfy the following conditions; φ1, φ2 (cid:54)= ±90◦ and ured for each position of the sample polarizer by using |φ1−φ2| (cid:54)= 0◦,±180◦, however, it has been shown that (5). Fig. 3(a) shows the extracted angle from the meas- φ1 =+45◦andφ2 =−45◦providesthebestaccuracy[6]. uredpolarizationstate.Anexcellentagreementbetween Two commercially available photoconductive dipole set and extracted rotation angle is shown in Fig. 3(a). antennas (TERA8-1) with 20 µm length and 5 µm gap Figs.3(b)-(d)comparetheerroroftheextractedrotation sizes from MenloSystems were used. Fig. 2 shows the anglebetweencalibratedanduncalibratedmeasurement measured (s,δ) parameters over the frequency range for for displayed frequencies. Based on this comparison the one of these dipoles. For an ideal detector with linear root-mean-square (RMS) error is reduced considerably polarization along the x-axis (s,δ) would equal (1,0) by at least 60% after applying the calibration. (Pd = [1 0]) for all frequencies. It is clear that errors As a proof-of-principle for this method of obtaining introduced from performing polarimetry without a cali- polarizationstates,westudiedtheeffectoftheproposed bration step, while small, are not negligible. calibration scheme on measuring birefringence through polarimetry.A50mm-diametersapphirewaferwith0.47 1.002 50 mm thickness and C axis in the plane of the wafer was 1.001 35 used as sample. For a uniaxial crystal with its optical axisinthex-yplane,thephaseretardation(∆)between 1 20 the optical axis and its perpendicular direction and the 0.999 5 angular orientation (θ) of the optical axis itself can be er0.998 −10 obtained simultaneously from its Jones matrix as paramet0.997 −25 δ (deg.) θ = 1tan−1(cid:20) 1 (cid:21), ∆=−2tan−1(cid:20) 1 (cid:21), s 0.996 −40 2 (cid:60)(R) sin(2θ)(cid:61)(R) 0.995 −55 R= Ext(ω), (6) E (ω) 0.994 −70 yt 0.993 −85 where (cid:60)(.) and (cid:61)(.) denote the real and imaginary op- erators, respectively, and E and E are the x- and y- 0.99220 0 400 600 800 1000 1200 140−0100 xt yt componentofthetransmittedpolarizationstate,respec- Frequency (GHz) tively, when E = [1,0]. Fig. 4 compares the measured i Fig. 2. Measured polarization parameters (s,δ) of a 20 birefringencemagnitudeofthesapphirewafer,extracted µm dipole with 5 µm gap size. from the phase retardation (∆) in (6), with and with- out applying the calibration scheme. For this measure- Inordertoevaluatetheimprovementfromcalibrating ment,theangularorientationθwasobtainedfrom(6)as 2 80 2.4 2.2 376 GHz 70 2.0 60 1.8 CUanlcibarliabtreadted Rotation angle (deg.)345000 Error (deg.)11110.....62408 20 0.6 0.4 10 0.2 200 0 400 600 800 1000 1200 1400 0.00 10 20 30 40 50 60 70 Frequency (GHz) Angular orientation (deg.) (a) (b) 2.4 2.4 2.2 902 GHz 2.2 1203 GHz 2.0 2.0 1.8 Calibrated 1.8 Calibrated Uncalibrated Uncalibrated 1.6 1.6 Error (deg.)111...024 Error (deg.)111...024 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.00 10 20 30 40 50 60 70 0.00 10 20 30 40 50 60 70 Angular orientation (deg.) Angular orientation (deg.) (c) (d) Fig.3.(a)Extractedrotationangleofthesamplepolarizer,(b)-(d)comparisonoftheerroroftheextractedrotation angle between calibrated and uncalibrated measurements for displayed frequencies. θ = 43.5◦ by averaging (cid:60)(R) over the frequency range Conventional method where it behaves smoothly. Also shown in Fig. 4 is the 0.365 Pol. Mea. w/o calib. birefringence calculated from the difference between the Pol. Mea. w/ calib. 0.363 indices of refraction for ordinary and extraordinary rays (∆n = n −n ) using a more conventional method of 0.361 e o aligning the THz electric field along crystalline axes in 0.359 seperatemeasurements.AsseeninFig.4,thecalibration 0.357 has considerably improved the birefringence extracted n ∆ from polarization measurement when compared to the 0.355 uncalibratedmeasurement.Itshouldbeemphasizedthat 0.353 in the polarization measurement, unlike the more con- 0.351 ventionalmethod,itisnotnecessarytoknowθ apriori. 0.349 ThisworkwasmadepossiblebysupportfromDARPA 0.347 YFA N66001-10-1-4017 and the Gordon and Betty Moore Foundation. 0.345 0.6 0.8 1 1.2 1.4 1.6 Frequency (THz) Fig. 4. Measured birefringence of a sapphire wafer ob- References tained from polarimetry with and without calibration, and from conventional method. 1. E. 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