Real-time Phase Conjugation of Vector Vortex Beams
A. G. de Oliveira, M. F. Z. Arruda, W. C. Soares, S. P. Walborn, R. M. Gomes, R. Medeiros de Araújo, P. H. Souto Ribeiro
RReal-time Phase Conjugation of Vector Vortex Beams
A. G. de Oliveira, M. F. Z. Arruda,
1, 2
W. C. Soares, S. P. Walborn, R. M. Gomes, R. Medeiros de Ara´ujo, and P. H. Souto Ribeiro Departamento de F´ısica, Universidade Federal de Santa Catarina, Florian´opolis, SC, 88040-900, Brazil Instituto Federal do Mato Grosso, Sorriso, MT, 78890-000, Brazil Universidade Federal de Alagoas, Campus Arapiraca, Arapiraca, AL, 57309-005, Brazil Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brazil Instituto de F´ısica, Universidade Federal de Goi´as, Goiˆania, GO, 74690-900, Brazil (Dated: October 17, 2019)Vector vortex beams have played a fundamental role in the better understanding of coherenceand polarization. They are described by spatially inhomogeneous polarization states, which presenta rich optical mode structure that has attracted much attention for applications in optical com-munications, imaging, spectroscopy and metrology. However, this complex mode structure can bequite detrimental when propagation effects such as turbulence and birefringence perturb the beam.Optical phase conjugation has been proposed as a method to recover an optical beam from per-turbations. Here we demonstrate full phase conjugation of vector vortex beams using three-wavemixing. Our scheme exploits a fast non-linear process that can be conveniently controlled via thepump beam. Our results pave the way for sophisticated, practical applications of vector beams.
PACS numbers:
Vector vortex beams are at the heart of many ap-plications in optics, due to their more complex opti-cal mode structure, described by anisotropic polarizationstates that are not spatially homogeneous in the trans-verse plane. They can provide crucial benefits in impor-tant applications, such as increased transmission ratesin classical and quantum optical communications in freespace [1–8], as well as in optical fibers [9, 10]. Uniqueapproaches for enhanced-resolution imaging [11, 12] andremote optical sensing [13–15] have been demonstratedwith vector beams. The radial vector beam has tighterfocusing properties that have been used in nanoscale opti-cal imaging [16, 17] and laser cutting [18], while it’s radialpolarization boosts efficiency in tip-enhanced near-fieldspectroscopy and imaging [19–21]. While vector beamsshow much promise for interesting applications, the spa-tial mode structure of these beams can make them quiteprone to distortions caused by turbulence or diffusive pro-cesses, while the polarization properties are degraded bydynamical optical birefringence, which can appear duringpropagation in optical fibers, for example.As is well known, optical wavefronts suffer distortionduring propagation in anisotropic media or multiple lightscattering [22–24], preventing focusing and imaging. Onemethod to correct these perturbations is through phaseconjugation, as was first reported in 1970 by Zeldovichet al. [25]. They observed that light emerging in reversefrom a distorted glass displayed a spatial profile thatwas free from the distortions of the diffusive medium,after back scattering on a carbon disulfide (CS ) gas cell,which functioned as a phase conjugating mirror (PCM).The PCM essentially realized the time reversal of the op-tical field, which allowed the distortions to be correctedwhen the field back-propagated through the medium.Wavefront correction by optical phase conjugation has been used to improve high-resolution imaging [26] andto construct laser oscillators [27, 28]. More recently, itwas used in biological tissue to control waves in spaceand time [29, 30], and is viewed as an essential compo-nent for long distance optical fiber communications withultra-high bit-rate [31].Various techniques have been explored to produce aphase-conjugate beam, including degenerate four-wavemixing [24, 32–34], backward stimulated Brillouin, Ra-man, and Rayleigh-wing or Kerr scattering [25], as wellas single- or multiphoton-pumped backward stimulatedemission (lasing) [35]. Though most experiments havefocused on spatial properties of the field, some haveconcerned phase conjugation of isotropic polarizationstates[24]. Phase conjugation of anisotropic vector fieldswas realized in Refs. [36, 37] in photorefractive media,however the required exposure time of 250s is typicallymuch too long a time scale to correct wavefront distor-tions accrued during propagation.Here we demonstrate phase conjugation of vectorbeams using Stimulated Parametric Down-Conversion(StimPDC), a three-wave mixing process [38, 39]. Stim-PDC has previously been used to realize phase conju-gation in the transverse spatial profile [40, 41], includ-ing light beams possessing orbital angular momentum[42, 43]. We employ an arrangement of non-linear crys-tals that produces an idler beam that is the phase con-jugate of the seed beam, in both the polarization andspatial degrees of freedom. StimPDC presents a numberof advantages as a phase conjugation process. First, theprocess is fast, occurring essentially in real time, as theemission is stimulated and therefore follows the seed fre-quency. This renders StimPDC as a promising techniquefor correcting dynamic distortions of optical beams ingeneral. Second, we show that the polarization state of a r X i v : . [ phy s i c s . op ti c s ] O c t R LR R
WAVEFRONT POLARIZATION VECTOR BEAM R E G U L A R M I RR O R C O N J U G A T I O N M I RR O R FIG. 1: Comparison between an ordinary mirror and a phaseconjugation mirror in terms of wavefront, polarization andvector beam. Black arrows indicate direction and sense ofpropagation. the pump beam provides convenient control parameters,allowing one to rotate the polarization states affected byphase conjugation and switch conjugation on and off.
Phase Conjugation
Let us recall the differences be-tween an ordinary mirror and a ideal PCM. While theordinary mirror reverses only the normal component ofthe wavevector, the ideal PCM not only reverses the en-tire wavevector (cid:126)k , but also conjugates the field amplitude(phase profile: φ ( (cid:126)r ⊥ ) → − φ ( (cid:126)r ⊥ ); polarization: (cid:126)ε → (cid:126)ε ∗ ).Fig. 1 a) shows a diverging beam incident on an ordi-nary mirror. After reflection, it continues diverging. Onthe other hand, the same beam incident on a PCM con-verges backwards towards the source, as illustrated inFig.1 d). This is equivalent to time reversal of the field.Thus, reflection upon a PCM is not specular, meaningthat the angle of incidence is not equal to the angle ofreflection. In terms of the polarization of the light beam,reflection of a circularly polarized field from an ordinarymirror and a PCM is illustrated in Fig. 1 b) and Fig.1 e), respectively. The reflected beam changes its circu-lar polarization state from R ( L ) to L ( R ) when reflectedfrom the ordinary mirror, but remains unchanged whenreflected from the PCM, since the polarization is con-jugated and also the wave vector is reversed. We canuse this picture to infer what happens to a vector beam,as shown in Figs. 1 c) and f). There is no difference be-tween specular and PCM reflection for linear polarizationcomponents. However, for circular components they areopposed. The PCM reflects an arbitrary and unknowninput state conjugating all polarization states present inthe transverse spatial profile. This cannot be achievedusing specular devices without the previous knowledgeof the input state. In this sense, phase conjugation canbe used to restore arbitrary vector beams perturbed bypropagation through some anisotropic and/or birefrin- PumpPump S e e d Crystals21Crystals21
FIG. 2: a) Two-crystal type-I parametric down-conversionsource for the generation of polarization Bell-states. b) Stim-ulated parametric down-conversion for two-crystal source con-figuration. gent medium.StimPDC differs from the ideal PCM considered in Fig.1 in that it is a forward process, in which the pump, seedand idler all propagate in the forward direction. Thegenerated idler beam displays the conjugate of the trans-verse phase profile of the seed laser beam (signal). Inthis process, the wavevector (cid:126)k is not reversed but ratherreflected in the transverse plane.
Spontaneous three-wave mixing in two crys-tals
We first demonstrate a polarization-conjugationdevice using StimPDC based on a two-crystal geome-try, inspired by a spontaneous parametric down con-version (SPDC) source of polarization-entangled pho-tons [44] as illustrated in Fig. 2 a). The optical axesof the crystals are crossed, so that one of them emitsvertically-polarized pairs of photons and the other emitshorizontally-polarized ones. Here we focus on the po-larization degree of freedom, so that pump, signal andidler fields will be treated as ideal monochromatic planewaves, which can be achieved using spatial and spectralfilters.Let us consider that the pump beam is described bythe general polarization state | ϑ p , ϕ p (cid:105) = cos ϑ p | H (cid:105) + e i ( ϕ p − Φ) sin ϑ p | V (cid:105) , (1)where the extra phase − Φ is added to offset the phaseaccrued between the SPDC crystals. If light emitted bythe two crystals is spectrally and spatially indistinguish-able, the polarization state of the SPDC pair will be com-pletely determined by the pump beam: | ψ (cid:105) s,i = cos ϑ p | V (cid:105) s | V (cid:105) i + e iϕ p sin ϑ p | H (cid:105) s | H (cid:105) i . (2)It will be convenient to define the orthonormal states |± ϑ s , ϕ s (cid:105) = cos ϑ s | H (cid:105) ± e iϕ s sin ϑ s | V (cid:105) , (3)so that the SPDC state (2) can be rewritten as | ψ (cid:105) s,i = | + ϑ s , ϕ s (cid:105) | α (cid:105) + |− ϑ s , ϕ s (cid:105) | β (cid:105) , (4)where | α (cid:105) = sin ϑ p ϑ s | H (cid:105) + e − i ( ϕ p + ϕ s ) cos ϑ p ϑ s | V (cid:105) , | β (cid:105) = sin ϑ p ϑ s | H (cid:105) − e − i ( ϕ p + ϕ s ) cos ϑ p ϑ s | V (cid:105) . (5)The state (4) represents the possible polarization-stateoutputs for spontaneous emission from the two-crystalsource. Phase conjugation in stimulated emission
In Stim-PDC, a seed beam stimulates the emission–say–in thesignal mode, also stimulating emission in the idler mode,since the down-converted photons are produced in pairs.In order to maximize efficiency, the optical mode of theseed laser must match the desired signal mode.
FIG. 3: Poincar´e sphere with N/S poles given by a) H/V po-larization, b) R/L polarization and c) Signal and idler polar-izations are mirror images through the δ -rotated conjugationplane in StimPDC when ϑ p = π/ Let us consider the two-crystal configuration shown inFig. 2 b), where the seed beam is prepared with po-larization state | ϑ s , ϕ s (cid:105) , which will stimulate signal pho-tons also in polarization | ϑ s , ϕ s (cid:105) . As a consequence ofthe polarization correlations described in Eq. (4), whichare imposed by the two-crystal configuration and phasematching conditions, this enhances the emission of idlerphotons with polarization | α (cid:105) given in Eq. (5). If thestimulated emission rate in the idler beam is high enough,the spontaneous emission, which is always present, canbe neglected, and the signal and idler fields are approxi-mately described by coherent states.It is illustrative to describe the polarization states us-ing the Poincar´e sphere, which has the circular polar-izations at the poles and is a parametrization of thelast three Stokes’ parameters in spherical coordinates( ρ, θ, φ ), as illustrated in Fig. 3 b). The polar and az-imuthal angles θ and φ are used to describe polariza-tion in the circular {| R (cid:105) , | L (cid:105)} basis, similarly to how ϑ and ϕ describe polarization in the {| H (cid:105) , | V (cid:105)} basis (seeequations (1) and (3) and Fig. 3 a)). While the latteris a privileged basis for describing the coupling of lightwith the each crystal, the former has been convention-ally used in the Poincar´e representation and is also con-venient for the geometrical interpretation of our results,as introduced below. Henceforth, we will refer to theStokes vector (cid:126)S = ( S , S , S ) T , omitting dependence on the intensity parameter S , which is not relevant to ouranalysis. On the Poincar´e spheres, the polarization state | ϑ µ , ϕ µ (cid:105) has Stokes vector (cid:126)S µ = cos ϑ µ sin ϑ µ cos ϕ µ sin ϑ µ sin ϕ µ = sin θ µ cos φ µ sin θ µ sin φ µ cos θ µ , (6)where µ = p, s, i may represent pump, signal/seed oridler.In two-crystal StimPDC, the idler polarization state | α (cid:105) is completely defined by the Stokes parameters of thepump and seed beams. Indeed, from equation (5), onecan show that (cid:126)S i = 12 S s, − S p, S p, S s, − S p, S s, − S s, S p, − S p, S s, . (7)The above equation is our first theoretical result andwe will use it to analyze the polarization (vector) con-jugation of the signal beam. Polarization conjugation interms of Stokes vectors is defined as (cid:126)S ∗ = ( S , S , − S ) T [24]. Comparing this definition with Eq. (7), we seethat, if the pump beam polarization is linear diagonal( D ) (cid:126)S p = (0 , , T , the idler is given by (cid:126)S i = ( S s, , S s, , − S s, ) T = (cid:126)S s ∗ , (8)equal to the conjugate of the polarization of the seedbeam. As the conjugation flips only the sign of the thirdStokes vector component, we can interpret this result ge-ometrically in the following way: on the Poincar´e sphereof polarization, signal and idler are mirror images of eachother by reflection on the equatorial plane.Note that choosing the pump beam with linear anti-diagonal ( A ) polarization (cid:126)S p = (0 , − , T gives (cid:126)S i = ( S s, , − S s, , S s, ) T , (9)so that signal and idler are mirror images of each otherby reflection on the vertical plane S = 0. In fact, thereare many interesting intermediate polarization transfor-mations that can be implemented by choosing differentpump beam polarization states. The particular case ofthe pump beam containing equal amounts of H and V polarizations in a coherent superposition ( ϑ p = π/ S p, = 0) is especially interesting, since thecoupling to both crystals is the same. In this case, wemay rewrite equation (5) as | α (cid:105) = cos ϑ s | H (cid:105) + e i (2 δ − ϕ s ) sin ϑ s | V (cid:105) , (10)where δ = − ϕ p /
2. The relative-phase 2 δ − ϕ s shows that,in the Poincar´e sphere of polarization, the signal and idlerbeams are mirror images of each other upon reflection ona plane resulting from rotation of the equatorial planearound the H/V axis by an angle δ , as illustrated in Fig.3 c). Pump
Crystals HWP QWPHWP QWP HWP PBSQWP Lens
SPCMCCD
HWPQWP PBS
FIG. 4: Experimental setup. A diode laser oscillating at thewavelength 405 nm is used to pump two identical type-I BBO( β barium borate) crystals in sequence. The optical axis ofthe crystals are rotated by 90 degrees from each other, so thatcrystal 1 produces vertically polarized photon pairs and crys-tal 2 generates horizontally polarized photons. This arrange-ment has been employed to generate polarization-entangledphoton pairs [44], though here we will perform polarization-dependent stimulated emission. A seed diode laser at 780 nmis aligned to the desired signal direction and stimulates theconversion to the wavelengths 780 nm (signal-direct stimula-tion) and 840 nm (idler indirect stimulation), in non collinearphase-matching, so that signal and idler directions of propa-gation form an angle of about 4 ◦ . The polarization states ofthe pump and seed beams are prepared using a half-wave plate(HWP) and a quarter-wave plate (QWP) for each beam. Thisallows us to pump and seed the parametric down-conversionprocess with arbitrary and controlled polarization states. Thepolarization state of the idler field, which is indirectly stimu-lated, is analyzed by measuring its Stokes parameters. Thismeasurement is realized with an adjustable polarization ana-lyzer, consisting of a QWP, a HWP and a polarizing beamsplitter (PBS). The idler is detected with a single-photoncounting module(SPCM), in front of which there is 10 nmbandwidth interference filter centered at 840 nm. There isalso a polarization analysis scheme for the seed, which is de-tected with a CCD camera. Experiment - isotropic polarization states
Werealized phase conjugation using the experimental setupsketched in Fig. 4. We prepared the seed beam in six dif-ferent polarization states and measured the correspond-ing idler beam polarization states, for both A - and D -polarized pump beams. Full polarization tomography ofthe seed/signal and idler beams was performed and theStokes parameters were extracted, and plotted on thePoincar´e sphere.Results for isotropic polarization states are illustratedusing the Poincar´e sphere, shown in Fig. 5 a) for a D -polarized pump beam. We observe that seed (signal) andidler polarization states are located at opposite hemi-spheres, corresponding to phase conjugation. The con-jugate of a given polarization state on the sphere is ob-tained by changing the sign of the latitude while keepingthe longitude ( θ −→ π − θ ). Our results clearly illustratethis effect for both circular and elliptical states.For a A -polarized pump beam, however, polarizationconjugation does not occur, in the usual sense. Rather,the idler is close to the mirror image of the seed through FIG. 5: Poincar´e spheres when pump beam polarization islinear a) diagonal and b) antidiagonal. Solid (open) circlescorrespond to the seed (idler). In both cases, the seed beam isprepared in six different polarization states: R (right-circular;white), L (left-circular; black), E , E , E and E (elliptical;red, blue, yellow and green, respectively), represented by filledcircles. Blue and red discs seem outside the sphere surface dueto uncertainties that are no represented in this picture. SeeFig. 6 for the error bars.FIG. 6: Spherical-coordinate angles: experimental data foridler’s θ i and φ i versus signal’s θ s and φ s ) for a)-b) diagonalpump and c)-d) antidiagonal pump. The solid straight linesrepresent the theoretical predictions. the vertical plane S S , as shown in Fig. 5 b). The abil-ity to control phase conjugation is an interesting feature,in contrast to the conjugation of the spatial degrees offreedom, which always occurs.For a more quantitative view, we plot idler versusseed/signal angular coordinates in the Poincar´e sphere. Diagonal Pump
Seed R E1 H D E2 L E3 V A E4Idler (theory) L E2 H D E1 R E4 V A E3 MeanFidelity (%) 86.4 81.3 92.7 84.1 84.0 86.6 83.0 93.7 87.6 83.4 86.3
Antidiagonal Pump
Seed R E1 H D E2 L E3 V A E4Idler (theory) R E4 H A E3 L E2 V D E1 MeanFidelity (%) 85.4 85.1 93.4 87.6 88.5 87.2 81.6 91.9 80.3 80.9 86.2
TABLE I: Fidelities between theoretical and measured idlerpolarization states pumping with diagonally and antidiago-nally polarized beams for several seed preparations.FIG. 7: Vector vortex beams. Images were obtained exper-imentally. Arrows correspond to theoretical polarization di-rections. The radial vector beam a) is first transformed to theanisotropic beam b), used as the seed beam. The resultingidler phase conjugated c) and not conjugated d).
The results (points with error bars) are shown in Fig. 6along with theory (solid line). To compare polar ( θ ) andazimuthal ( φ ) coordinates, we assigned colors to each ex-perimental point, with labels denoting polarization of theseed beam.In addition, in Table I, we list the fidelity [45, 46] ofthe measured idler’s polarization state with respect tothe theoretical prediction. All calculated fidelities lie be-tween 80% and 94% with average of 86 .
3% for diagonalpump and 86 .
2% for antidiagonal pump. Our experi-mental results clearly demonstrate phase conjugation ofisotropic polarization states. Discrepancies with theorycome mainly from depolarization of the measured light,which we attribute to scattering on the crystals surfacesas well as spatial walk-off, which could be reduced byusing thinner crystals.
Experiment - anisotropic polarization states
We now show that this setup realizes phase conjuga-tion of vector beams. A vortex half-wave plate (ThorlabsWPV10L-780) is placed in the path of the seed beam in
FIG. 8: Measurement results showing phase conjugation. Po-larization projections onto H/V, A/D, and R/L bases. 1a)–6a) seed beam, 1b)–6b) idler beam conjugated, 1c)–6c) idlerbeam not conjugated.
Fig. 4, which produces a radial vector beam containingonly linear polarization, as shown in Fig. 7a). The profilewas measured and the arrows indicate the polarizationstates in the profile. To better illustrate phase conjuga-tion of the vector beam, we apply a QWP to the radialbeam so that a vortex beam containing both linear andcircular polarization states are generated, as indicated inFig. 7b). This beam is used as the seed in StimPDC.Figs. 7c) and d) show images of the doughnut-shapedintensity profile of the idler beam obtained in StimPDCwhen the pump beam is D - and A - polarized respectively.A signature of phase conjugation is the inversion of senseof rotation of the circularly polarized components (recallthat both beams propagate forward).The intensity profiles alone are not sufficient to showthat phase conjugation occurs. To do so, we performpolarization measurements on the seed and idler beams,and examine the images of the resulting beams using aCCD camera. Fig. 8 shows images of the seed (columna) and idler (column b for D-polarized pump and columnc for A-polarized pump) beams, upon projections on thelinear polarization bases H/V (rows 1 and 2) and A/D(rows 3 and 4) and circular polarization basis R/L (rows5 and 6). Phase conjugation of the vector beam can beobserved more clearly by comparing the images from pro-jection in the circular polarization basis. Figs. 8 5b) and6b) present a diagonal(anti-diagonal) Hermite-Gaussianshape contrasting with the anti-diagonal(diagonal) shapeof the seed in Figs. 8 5a) and 6a) respectively, showingphase conjugation, while Figs. 8 5c) and 6c) show anti-diagonal(diagonal) Hermite-Gaussian shape, indicatingno phase conjugation (again, recall that all beams prop-agate forward). At the same time, there is no differencebetween conjugation and no-conjugation, when the pro-jections onto linear polarization bases are performed. Conclusion
Optical phase conjugation has con-cerned the preparation of a light beam that is the timereversal of another one in terms of its angular spec-trum. Similarly, vector beam phase conjugation denotestime reversal that also includes the polarization degree offreedom. We demonstrate theoretically and experimen-tally the process of vector beam phase conjugation usingthree-wave mixing in a two-crystal source. The schemeis fast and can be conveniently controlled by manipu-lating the pump beam, and thus can be used in appli-cations in which real-time corrections must be made tovector beams propagating in anisotropic and/or birefrin-gent media. We expect our results to broaden the rangeof possibilities for the use of vector beams in real worldapplications.The authors would like to thank the Brazilian AgenciesCNPq, FAPESC, FAPERJ, FAPEG and the BrazilianNational Institute of Science and Technology of QuantumInformation (INCT/IQ). This study was funded in partby the Coordena¸c˜ao de Aperfei¸coamento de Pessoal deN´ıvel Superior - Brasil (CAPES) - Finance Code 001. [1] V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita,S. Slussarenko, L. Marrucci, and F. Sciarrino, NatureCommunications , 961 (2012).[2] Y. Zhao and J. Wang, Opt. Lett. , 4843 (2015).[3] O. J. F. as, V. D’Ambrosio, C. Taballione, F. Bisesto,S. Slussarenko, L. Aolita, L. Marrucci, S. P. Walborn,and F. Sciarrino, Sci. Rep. , 8424 (2015).[4] G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, andR. R. Alfano, Opt. Lett. , 4887 (2015).[5] G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie,T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R.Alfano, et al., Opt. Lett. , 1980 (2015).[6] J. Zhang, F. Li, J. Li, Y. Feng, and Z. Li, IEEE Photon.J. , 7907008 (2016).[7] P. Li, B. Wang, and X. Zhang, Opt. Express , 15143(2016).[8] B. Ndagano, I. Nape, M. A. Cos, C. Rosales-Guzman, and A. Forbes, J. Lightwave Tech. , 292 (2018).[9] P. Gregg, M. Mirhosseini, A. Rubano, L. Marrucci,E. Karimi, R. W. Boyd, and S. Ramachandran, Opt.Lett. , 1729 (2015).[10] B. Ndagano, R. Br¨uning, M. McLaren, M. Duparr´e, andA. Forbes, Opt. Express , 17330 (2015).[11] D. P. Biss, K. S. Youngworth, and T. G. Brown, Appl.Opt. , 470 (2006).[12] M. Yoshida, Y. Kozawa, and S. Sato, Opt. Lett. , 883(2019).[13] V. D’Ambrosio, N. Spagnolo, L. D. Re, S. Slussarenko,Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita,and F. Sciarrino, Nature Communications , 2432 (2013).[14] F. Tppel, A. Aiello, C. Marquardt, E. Giacobino, andG. Leuchs, New Journal of Physics , 073019 (2014).[15] S. Berg-Johansen, F. T¨oppel, B. Stiller, P. Banzer,M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, andC. Marquardt, Optica , 864 (2015).[16] Q. Zhan, Adv. Opt. Photon. , 1 (2009).[17] R. Chen, K. Agarwal, C. Sheppard, and X. Chen, OpticsLetters , 3111 (2013).[18] V. G. Niziev and A. V. Nesterov, Journal of Physics D:Applied Physics , 1455 (1999).[19] Z. D. Schultz, S. J. Stranick, and I. W. Levin, Anal Chem. , 9657 (2009).[20] N. Kazemi-Zanjani, S. Vedraine, and F. Lagugn´e-Labarthet, Opt. Express , 25271 (2013).[21] F. Lu, T.-X. Huang, H. Lei, H. Su, H. Wang, M. Liu,W. Zhang, X. Wang, and T. Mei, Sensors , 3841(2018).[22] R. A. Fisher, Optical phase conjugation (Academic Press,2012).[23] K. R. MacDonald, W. R. Tompkin, and R. W. Boyd,Optics letters , 485 (1988).[24] R. W. Boyd, K. R. MacDonald, and M. S. Malcuit, in Laser Wavefront Control (International Society for Op-tics and Photonics, 1989), vol. 1000, pp. 69–81.[25] B. Y. Zel’Dovich, V. Popovichev, V. Ragul’Skii, andF. Faizullov, in
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