Polarization constraints in reciprocal unitary backscattering
Qiaozhou Xiong, Nanshuo Wang, Xinyu Liu, Si Chen, Cilwyn S. Braganza, Brett E. Bouma, Linbo Liu, Martin Villiger
PPolarization constraints in reciprocal unitary backscattering
Qiaozhou Xiong, † Nanshuo Wang, † Xinyu Liu, Si Chen, CilwynS. Braganza, Brett E. Bouma,
3, 4
Linbo Liu,
1, 2, ∗ and Martin Villiger ∗ School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore, 639798, Singapore. School of Chemical and Biomedical Engineering,Nanyang Technological University, Singapore, 637459, Singapore. Harvard Medical School and Massachusetts General Hospital,Wellman Center for Photomedicine, Boston, Massachusetts 02114, USA. Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: March 5, 2019)We observed that the polarization state of light after round-trip propagation through a bire-fringent medium frequently aligns with the employed input polarization state “mirrored” by thehorizontal plane of the Poincar´e sphere. In this letter we explore the predisposition for this mirrorstate and demonstrate how it constrains the evolution of polarization states as a function of theround-trip depth into weakly scattering birefringent samples, as measured with polarization-sensitiveoptical coherence tomography (PS-OCT). The constraint enables measurements of depth-resolvedsample birefringence with PS-OCT using only a single input polarization state, which offers a criticalsimplification compared to the use of multiple input states.
PACS numbers: 42.30.Wb, 42.25.Ja
Polarization offers access to unique, distinguishing sig-natures of samples for diverse applications from remotesensing [1 and 2] to biomedical optics [3–5]. Convention-ally, multiple input polarization states are required inaddition to polarization-diverse detection to fully charac-terize the polarization properties of a sample, promptingcomplex instrumentation. Alleviating these hardware re-quirements would enable more widespread exploration ofthis compelling contrast mechanism.In previous experiments, we observed that the polar-ization state of backscattered or reflected light, whenmeasured through identical illumination and detectionpaths, frequently evolved through the employed inputpolarization state but with reversed handedness, corre-sponding to the input state mirrored by the horizontalplane of the Poincar´e sphere [6]. Earlier investigationsof the polarization properties of single mode fibers re-ported on aspects of the polarization mirror state [7–9],yet without elucidating its manifestation. To examinethe polarization mirror state, we measured the round-tripsignal through a 1.5-m-long single-mode optical fiber. In-stead of using a conventional polarimeter, we employedinterferometric measurements for the later coherence gat-ing experiments, as depicted in Fig. 1(a). Light from asuper-continuum source was linearly polarized, preparedwith an achromatic quarter-wave plate (QWP) to dif-ferent input polarization states, and split into referenceand sample arms. A linear polarizer oriented at 45 ◦ in the reference arm defined the reference polarizationstate independent of the input polarization. For polar-ization diverse detection, the sample and reference lightwas combined in a polarization-maintaining fiber to thendirect each of the fibers two linear eigenstates towards agrating-based spectrometer (760–920 nm). The recorded fringe signals reveal the amplitude and relative phase ofthe two orthogonal electromagnetic field components inthe sample arm and hence the polarization state of thesample light.Employing a polarization controller to alter the bire-fringence of the fiber, we measured the time-varying po-larization state resulting from randomly moving the pad-dle positions of the controller. Visualized in Fig. 1(b) asthe normalized Stokes vectors of the center wavelength(840 nm) in the Q , U , and V -coordinates of the Poincar´esphere, the polarization mirror state manifests by therepeated crossing of the polarization state evolution in aspecific state m , highlighted by the red arrow in Fig. 1(b).Repeated with different launching polarization states s (indicated by the blue arrows in Figs. 1(c,d)), we recog-nize that m = D · s , where D = diag(1 , , − m cor-responds to the input state mirrored by the horizontal QU -plane, explaining its designation as the polarizationmirror state. The input states were determined by re-flecting the light to the detector in free space, withoutthe fiber in place. All measurements were performed inthe fixed coordinates of the receiver and are independentof the orientation of the coordinates in the illuminationpath.To appreciate the mirror state phenomenon, we con-sider a general retarder R ( x ) with its retardation varyingas a function of x , e.g. the polarization controller’s pad-dle positions. The retarder may be preceded by a staticelement P . The combined system, illustrated in Fig. 2(a),transforms the input polarization state s into the outputstate t : t = D · P (cid:62) · R (cid:62) ( x ) · D · R ( x ) · P · s = T ( x ) · s (1) a r X i v : . [ phy s i c s . op ti c s ] M a r FIG. 1. Demonstration of the polarization mirror state. (a)Schematic drawing of the optical system employed throughoutall experiments. QWP: quarter-wave plate; BS: beamsplitter.(b)-(d) Polarization state evolution on the Poincar´e sphere asa result of moving the three paddles of the polarization con-troller when using circularly, elliptically and linearly polarizedinput states, respectively, indicated by the blue arrows.
Here, T = D · P (cid:62) · R (cid:62) · D · R · P , where (cid:62) denotestranspose, and all vectors and matrices are in the rotationgroup SO(3). We chose to follow the convention of main-taining the orientation of the spatial xy -coordinates irre-spective of the lights propagation direction [10 and 11].In reciprocal media, the reverse transmission through ele-ment R is described by D · R (cid:62) · D [12 and 13] (see supple-mentary material section 1 [14]). It is important to notethat the round-trip transmission T is D -transpose sym-metric T = D · T (cid:62) · D , which makes T a linear retarder.The round-trip effectively cancels any optical activity orcircular retardation and relates to the weak localizationof light [15]. The effect of T on the input state can bedescribed by a rotation vector τ ( x ) lying in the QU -planeof the Poincar´e sphere, with its direction indicating therotation axis, and its length defining the amount of ro-tation. Considering their 2 π -ambiguity, the rotation vec-tors of all possible linear retarders are confined to a cir-cle with a radius of π within the QU -plane (Fig. 2(b)).When moving the polarization controller paddles, τ ( x )traces out an intricate path in the QU -plane, as shown FIG. 2. Theoretical explanation of the polarization mir-ror state. (a) Model of the round-trip propagation througha reciprocal sample, comprising R ( x ) with varying retar-dation and a static element P . (b) The rotation vec-tors of all linear retarders localize within a circle of radius π within the QU -plane of the Poincar´e sphere. The or-ange curve represents the end points of the rotation vec-tors mapping the randomly chosen input polarization state[cos( − π/ · cos( π/ , cos( − π/ · sin( π/ , sin( − π/ (cid:62) ex-actly to its mirror point. The green trace represents the sim-ulated rotation vector evolution for a synchronous movementof the polarization controller paddles. The purple curves rep-resent the rotation vectors of D · P (cid:62) · R (cid:62) ( x ) · D · R ( x ) · P ,where R ( x ) is a linearly increasing linear retarder, for threerepresentative sets of distinct P and R ( x ). by the green line in Fig. 2(b) for simulating a synchronousmovement of the three paddles.There exists only a single rotation vector within the QU -plane that rotates a given input state s onto an ar-bitrary output state t . This rotation vector is defined bythe intersection of the QU -plane and the plane bisecting s and t . In order for s to pass through t , τ ( x ) has toevolve through this specific point within the π -circle ofthe QU -plane. The only exception, there exists a con-tinuum of rotation vectors that map s onto its mirrorstate m = D · s , because the QU -plane coincides withthe bisecting plane in this case. These rotation vectorsare located on a curve τ m within the QU -plane (orangecurve in Fig. 2(b), and supplementary material section2 [14]). Every intersection of τ ( x ) with τ m correspondsto s evolving through the mirror state m , explaining itsfrequent realization.Importantly, the presence of diattenuation that in-duces polarization-dependent loss would skew the mea-sured polarization states and frustrate the repeated evo-lution through the mirror state. The mirror state onlymanifests in systems that can be accurately modeled withunitary transmission matrices.We next used PS-OCT to measure the polarizationstate of light backscattered within a scattering sample asa function of its round-trip depth [16 and 17]. At the scaleof the axial resolution of OCT, tissue can be modeled asa sequence of homogeneous linearly birefringent layerswith distinct optic axis orientations. R ( x ) describes inthis case a linear retarder with a retardance that linearlyincreases with depth x , resulting in D · R (cid:62) · D = R . P contains the combined effect of system componentsand preceding tissue layers. The resulting rotation vec-tors τ ( x ) form regular curves across the π -circle (pur-ple curves in Fig. 2(b)). All possible traces intersect thecurve τ m precisely once, ensuring periodic crossing of m .To inspect in more detail the evolution of t , we take itsderivative with respect to x , and substitute s = T (cid:62) ( x ) · t : ∂ t ∂x = ∂ T ( x ) ∂x · T (cid:62) ( x ) · t = β ( x ) × t (2)Because T (cid:62) · T is the identity matrix, ( ∂ T /∂x ) · T (cid:62) = − T · (cid:0) ∂ T (cid:62) /∂x (cid:1) is skew-symmetric and can be expressedas the cross-product operator β × , which is constant fora retardance that linearly increases with x (see supple-mentary material section 3 [14]). Accordingly, within asingle sample layer, t evolves on the Poincar´e sphere withconstant speed rotating around the apparent optic axis β on a circle constrained to pass through m .For experimental validation, we prepared a scatter-ing phantom consisting of three linearly birefringent lay-ers with distinct optic axis orientations [18] (Fig. 3(a)).Without the fiber segment in the sample arm, we focusedthe light with a 30 mm focal length lens into the sam-ple, achieving a full-width at half maximum (FWHM)spot diameter of ∼ µ m, and scanned with galvanomet-ric mirrors in the lateral direction. The spectrometer’sbandwidth offers an axial resolution of ∼ . µ m. At eachscanning location, using PS-OCT, we constructed theStokes vector as a function of depth in the sample. Toremove speckle and improve the signal, we spatially fil-tered the original Stokes vectors with a two-dimensionalGaussian kernel of 20 µ m 1 /e width in the axial directionand 80 µ m in the lateral direction. Finally, we computedthe normalized three-component Stokes vector r ( z ) as afunction of depth, shown in Figs. 3(b-d) for three dis-tinct input polarization states at one lateral sample lo-cation. We then fitted circles to the polarization stateevolution within each layer. The circles (in purple color)demonstrate a close match with the measured polariza-tion states and all circles evolve through the polarizationmirror state m (indicated by red arrows), as expected.Using a single input polarization state for PS-OCT, itis straightforward to compute the cumulative retardationthat propagation through the sample to a given depthand back imparts on the input polarization state [19].Yet, cumulative retardation can be difficult to interpretin samples with a layered architecture, and it is more in-sightful to compute local retardation, i.e. the derivativeof the retardance of T ( x ) with depth, which is given bythe norm of β and is proportional to the sample birefrin-gence [20–22] at that depth location. Following Eq. (2)we have FIG. 3. Evolution of coherence-gated polarization statesin a three-layer birefringence phantom. (a) Schematic sketchof the phantom consisting of three layers with distinct opticaxis orientations. (b)-(d) Polarization state evolution (color-coded corresponding to the layers in (a)) for a circularly po-larized (b), elliptically polarized (c), and linearly polarizedinput state (d). (cid:12)(cid:12)(cid:12)(cid:12) ∂ t ∂x (cid:12)(cid:12)(cid:12)(cid:12) = | β | sin θ (3)where we used | t | = 1. θ is the angle between the rota-tion vector β and the polarization state t and is needed todeduce local retardation. With only a single input statethis angle is generally unknown. Using, instead, two in-put polarization states oriented at 90 ◦ to each other onthe Poincar´e sphere reveals the orientation of the appar-ent optic axis. However, recognizing that the evolution of t is constrained to go through m , it is possible to recoverthe orientation and magnitude of β from measurementswith only a single input polarization state. Owing to thisconstraint, both ∂ t /∂x and ( t − m ) lie within the sameplane orthogonal to β . Hence, the direction of β can beobtained by the cross-product β = ( ∂ t /∂x ) × ( m − t ),and sin θ = | β × t | / | β | , allowing to calculate, aftersome algebraic manipulations: β = ∂ t ∂x × ( m − t )1 − t (cid:62) · m (4)To validate the ability of the polarization mirror stateto reconstruct local retardation, we imaged a tissue-like FIG. 4. Local retardation imaging of birefringence phantom using the polarization mirror state constraint. (a,b) Schematicdrawings of the two-layer phantom in either orientation. (c,d) Corresponding cumulative retardation and (e,f) local retardationimages, respectively. Scale bars in (c) measure 100 µ m (vertical) and 400 µ m (horizontal). phantom consisting of a long birefringent band followedby four parallel elements with distinct birefringence levelsand an optic axis orientation different from the long band[18] (Fig. 4(a) and 4(b)) and the gap was filled with non-birefringent matrix.For reconstruction of local retardation, we em-ployed the pre-calibrated polarization mirror state m ,and implemented Eq. (4) by approximating t =( r [ p + 1] + r [ p ]) / ∂ t /∂x = ( r [ p + 1] − r [ p ]) / ∆ z ,where p is the pixel index along depth z = p · ∆ z , and ∆ z is the axial sampling distance. To avoid high-frequencynoise introduced by taking the difference between adja-cent points, we axially averaged the reconstructed rota-tion vector β ( z ) with a Gaussian window of the sameaxial size as used to filter the Stokes vectors. The normof β , scaled to degrees of retardation per depth ( ◦ /µ m),reveals the sample’s local retardation, imaged with eitherside of the sample facing up (Figs. 4(e) and 4(f)). Forcomparison, the cumulative retardation of T ( x ) was com-puted by evaluating the angle between r ( z ) at each depthand r ( z surf ), where z surf is the axial location of the samplesurface within each depth profile (Figs. 4(c) and 4(d)).Whereas cumulative retardation is difficult to interpret,the local retardation clearly reveals the individual sam-ple segments with their distinct levels of birefringenceand is recovered irrespective of the sample orientation[22 and 23]. To demonstrate local retardation imaging inbiological tissue, we measured ex vivo swine retina (Sup-plementary material section 4 [14]).Previous strategies to reconstruct local birefringencefrom single-input-state PS-OCT rely on the intrinsicsymmetry of the imaging system [21 and 24] and assumethat the optical elements in the illumination and detec-tion paths have no impact on the polarization states.Most OCT instruments for clinical applications, however,use fiber-based optical components with distinct illumi-nation and detection paths, which breaks the intrinsic D -transpose symmetry [25]. Crucially, the evolution of t through the mirror state persists also in systems withdistinct retardation in the illumination and detection op-tics. This is equivalent to left-multiplying Eq. (1) with an additional matrix B . Although the apparent cumulativeretarder that maps the input state onto the measuredoutput state is no longer a linear retarder in this case, B simply alters the location of the circular evolution of thepolarization states on the Poincar´e sphere to go throughthe actual mirror state to B · m .A remaining challenge manifests whenever t alignswith m , which impairs the reconstruction of local re-tardation (cyan arrows in Fig. 4(f)). This correspondsto the effective polarization state in the target layer toorient along one of that layers optic axes, and even pre-vents the cumulative retardation from accumulating re-tardance. Using circularly polarized input light requiresa half wave of retardation to realize this alignment, whichis uncommon in many biological samples. Yet, some tis-sues feature substantial birefringence and controlling theinput state is not necessarily possible. The resulting ar-tifact can be avoided by introducing a modest amountof polarization mode dispersion (PMD) into the systemand using spectral binning for reconstruction [26]. Be-cause PMD disperses the input polarization state acrossthe spectral bins, simultaneous alignment of t with m inall bins is very unlikely.Coupling the sample light through the 1.5-m-long sin-gle mode fiber twisted around the polarization con-troller paddles provided sufficient PMD for our broad-bandwidth source. For spectral binning, we multipliedthe spectral fringe signals with Hanning windows h ( k, n )of width ∆ k/N centred on n · ∆ k/ (2 N ) within the avail-able k -support, ∆ k , n ∈ [1 , N − N = 5, resultingin 9 spectral bins, to compute the binned Stokes vec-tors r ( z, n ). We also evaluated the degree of polariza-tion DOP = (cid:68)(cid:0) Q + U + V (cid:1) / /I (cid:69) , where (cid:104)(cid:105) indicatesaveraging over the spectral bins, and Q , U , V and I are the spatially filtered Stokes components before nor-malization. Following the identical processing for localretardation for each bin as described above, we obtainedthe rotation vectors β ( z, n ). Fig. 5(a-d) illustrates thelocal retardation of bins 1 and 9, together with a map w ( z, l ) = | t − m | expressing the reliability of the givenStokes vector by the distance from its mirror state, for a FIG. 5. Inaccurate estimation of local retardation can beavoided with a small amount of polarization mode dispersion(PMD) in combination with spectral binning. (a)-(f) Cross-sectional images of a two-layer phantom. (a) Local retarda-tion reconstructed using only the 1st spectral bin. (b) Lo-cal retardation reconstructed using only the 9th spectral bin.(c,d) Reliability metric maps of the 1st and 9th spectral bin,respectively. (e) Local retardation reconstructed using theentire spectrum without spectral binning. (i) Local retarda-tion image reconstructed with spectral binning combining allbins. Scale bars in (f) measure 100 µ m (vertical) and 400 µ m(horizontal). tissue-like birefringence phantom. Bin 9 results in highlocal retardation values but with little reliability, unlikebin 1, which indicates more modest local retardation yetwith higher reliability. The β ( z, n ) with high reliabilityof all bins describe the same sample retardation but maybe offset in their relative orientation due to system PMD.The required rotation G ( n ) to align the vectors of eachbin to the central bin N in the least-square sense is givenby:max G ( n ) Tr G ( n ) · (cid:88) z,l β ( z, l, N ) · β (cid:62) ( z, l, N ) · w ( z, l ) (5)where z and l are point indices in the axial and lateraldirections, respectively, G ( n ) is assumed constant withinan entire B-scan, and the sum is taken over all points withsufficient DOP > . > × (cid:80) β · β (cid:62) · w = U · W · V † , where † denotes conjugate transpose, the solution to Eq. (5) isobtained by G = V · U † . Lastly, the aligned rotationvectors are averaged among the spectral bins consideringtheir weights w ( z, l ), and then axially filtered, as pre-viously, to obtain the final local retardation image, freefrom artifacts, as demonstrated in Fig. 5(f).In conclusion, we demonstrated the peculiar proper-ties of the mirror polarization state that manifest whenmeasuring backscattered light along identical illumina-tion and detection paths free of polarization-dependentloss. In PS-OCT, the mirror state constrains the evolu- tion of the depth-dependent polarization states and en-ables local retardation imaging, which previously has notbeen available to PS-OCT without substantially morecomplex measurements using multiple input states.Support is acknowledged from a National ResearchFoundation Singapore (NRF-CRP13-2014-05), Ministryof Education Singapore (MOE2013-T2-2-107 & RG83/18 (2018-T1-001-144)), and NTU-AIT-MUV programin advanced biomedical imaging (NAM/15005), and partby the National Institutes of Health grants P41EB-015903, R03EB-024803. † These authors contributed equally ∗ These authors contributed equally.Corresponding authors: [email protected]@mgh.harvard.edu[1] K. Sassen, Bulletin of the American Meteorological Soci-ety , 1848 (1991).[2] J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A.Shaw, Applied optics , 5453 (2006).[3] N. Ghosh and A. I. Vitkin, Journal of biomedical optics , 110801 (2011).[4] V. V. 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