Polarization resolved radiation angular patterns of orientationally ordered nanorods
PPolarization resolved radiation angular patterns of orientationallyordered nanorods
Alexei D. Kiselev ∗ Saint Petersburg National Research University of Information Technologies,Mechanics and Optics (ITMO University),Kronverskyi Prospekt 49, 197101 Saint Petersburg, Russia (Dated: September 12, 2019)We employ the transfer matrix approach combined with the Green’s functionmethod to theoretically study polarization resolved far-field angular distributionsof photoluminescence from quantum nanorods (NRs) embedded in an anisotropicpolymer film. The emission and excitation properties of NRs are described by theemission and excitation anisotropy tensors. These tensors and the solution of theemission problem expressed in terms of the evolution operators are used to derive theorientationally averaged coherency matrix of the emitted wavefield. For the case ofin-plane alignment and unpolarized excitation, we estimate the emission anisotropyparameter and compute the angular profiles for the photoluminescence polarizationparameter such as the degree of linear polarization, the Stokes parameter s , the el-lipticity and the polarization azimuth. We show that the alignment order parameterhas a profound effect on the angular profiles. I. INTRODUCTION
Over the last two decades quantum nanorods (NRs) have been the subject of intensestudies as semiconductor nanoheterostructures that possess a unique combination of geom-etry and size dependent emission and excitation properties [1–8] (see also a review [9]). Inaddition to quantum confinement effects coming into play at length scales comparable to thebulk exciton Bohr radius, these structures feature linearly polarized photoluminescence [1]and excitation (absorption) anisotropy [10–12].The linear polarization of emission is governed by the fine structure of the ground excitonstate. It is determined by a number of factors such as the fine structure splittings, theselection rules and the exciton oscillator strengths [2–6, 9, 13]. In particular, both excitationand emission of single cadmium selenide (CdSe) quantum rods are found to exhibit strongpolarization dependence, indicating that dipole moment exists along the long axis of therods, e.g., the unique c -axis of the wurtzite structure [14].There is a variety of applications utilizing the linear polarized emission from the NRsthat are used as efficient light emitters for lasing [15], biological labeling [16] and generationof nonclassical light [17]. In liquid crystal display devices, it was found that using NRs asbacklight source may significantly enhance the optical efficiency of the backlighting system [8,18, 19].Since the emission and excitation properties of NRs crucially depend on their orientation,it is of paramount importance for any application utilizing the polarized emission from NRs ∗ Email address: [email protected] a r X i v : . [ phy s i c s . op ti c s ] S e p to control and determine their alignment in a film. There are several methods to achieveunidirectional alignment of NRs that have been discussed in the past few years [7, 8, 18, 20].One of the most promising approaches uses the photoalignment technique to align NRs inthe liquid crystal polymer (LCP) matrix brought in contact with the photoaligning azo-dyelayer through the precise control over the orientation of photosensitive dye molecules [21, 22].There are several techniques developed for determination of the three-dimensional (3D)orientation of the transition dipoles of single molecules. These include polarization-sensitivedetection of fluorescence through a high-N.A. objective originally proposed in [23] and themethods based on different versions of emission pattern imaging [24, 25].It was demonstrated that far-field polarization microscopy can yield the 3D orientationof CdSe quantum dots [26]. In Ref. [27] it was shown that the 3D orientation of a singlefluorescent nanoemitter can be determined by polarization analysis of the emitted light usingthe model based on the theoretical results obtained in Refs. [28–31]. Results of polarimetricmeasurements performed on core/shell cadmium selenide/cadmium sulfide (CdSe/CdS) dot-in-rods [27, 32] turned out to be consistent with the hypothesis of a linear dipole tilted withrespect to the rod axis. Orientation of gold and rare-earth-doped nanorods was also recentlystudied in Refs. [33, 34].Angular radiation patterns of nanoemitters such as NRs strongly depend on its orienta-tion. In Ref. [35], orientation of the emissive dipole moments was deduced from measure-ments of the far-field polarized angular radiation patterns of organic light-emitting diodes(OLED)s in electrical operation. Angular distributions of polarized light from multilayerLED structures studied in [36–40] are found to be important for optimization of light ex-traction efficiency and performance of LED devices.The radiation patterns of light emitted by NRs, however, have received little attentionand are much less studied as compared to the LED systems. In this work, we adapt asystematic approach and theoretically study polarization resolved angular distributions ofphotoluminescence from NRs embedded in the liquid crystal polymer (LCP) film and alignedby the azo-dye photoaligning layer using the photoalignment technique. This geometry waspreviously described by Tao Du at al. in Ref. [21].One of the key features of such a multilayer system is that both the LCP film and theazo-dye layer are optically anisotropic. As it was demonstrated for emission from hyper-bolic metamaterials [41], such an anisotropic environment will profoundly influence angularradiation patterns.Our theoretical approach to the emission problem developed to analyze the combinedeffects of NR alignment and optical anisotropy of surrounding media on the angular radiationdistributions is based a suitably modified version of the transfer matrix method. We showthat this method can be used in combination with the Green’s function technique to obtainour key analytical result giving the orientationally averaged coherency matrix of NR emissionexpressed in terms of the evolution operators and the orientational averages. The importantpoint is that the coherency matrix also depends on the emission and excitation anisotropyparameters determined by the transition dipole moments, the level populations and the localfield screening factors. Our goal is to examine how the angular dependence of the polarizationstate of emitted light is affected by the orientational ordering, optical anisotropy and theemission/excitation anisotropy parameters.The paper is organized as follows.In Sec. II we present our theoretical approach. After introducing the emission and exci-tation anisotropy tensors in Sec. II A, we briefly discuss the angular spectrum representationand the evolution operators in Sec. II B. Necessary details on the transfer matrix methodare provided in Sec. II C. In Sec. II D, we compute the dyadic Green’s function and solvethe single-emitter problem. It is found that the far-field eigenwave amplitudes of the emit-ted wavefield can be expressed in terms of the evolution operators. The expression for theorientationally averaged coherency matrix of the emitted light is obtained in Sec. II E. Theanalytical results are employed to perform numerical analysis of the angular profiles for thepolarization characteristics of the emitted light in Sec. III. Finally, in Sec. IV, we drawthe results together and make some concluding remarks. Technical details are relegated toAppendix A. II. THEORYA. Emission and excitation anisotropy tensors
Semiconductor core/shell nanoheterostructures representing the nanorods (NRs) studiedin Ref. [21] are also known as the dot-in-rods where a spherical core is surrounded by arod-like shell. For the CdSe(cadmium selenide)/CdS (cadmium sulfide) dot-in-rods withadjusted geometry of the hexagonal crystal structures, the wurtzite c -axis of both the coreand the shell is along the long axis of the rod shell, because the growth process of the shellalong the c -axis is determined by the crystal anisotropy of the core.The band-edge exciton fine structure of CdSe nanocrystals consists of eight states withthe total angular momentum projection on the c -axis F ∈ { , ± , ± } [5, 6, 42]: |± L (cid:105) , |± L,U (cid:105) and | L,U (cid:105) , where the superscripts L and U indicate lower and upper sublevels,respectively. There are five bright (dipole allowed) exciton states: |± L,U (cid:105) and | U (cid:105) . Forthe state | U (cid:105) , the transition dipole moment (cid:104) | p | U (cid:105) , where p is the momentum operator,is directed along the c -axis (1D dipole), whereas the transition dipole moments (cid:104) | p |± L,U (cid:105) lie in the plane normal to ˆ c (2D dipole), where ˆ c is the unit vector parallel to the c -axis.For emitted lightwave linearly polarized along the polarization unit vector ˆ e , the emissionprobabilities for the bright exciton states are proportional to both the magnitudes of theprojections of the transition dipoles on the polarization vector and the populations of theexciton states. So, the polarization dependent factors of the probabilities can be written inthe form: N |(cid:104) | (cid:0) ˆ e · p (cid:1) | U (cid:105)| = D (cid:107) (cid:0) ˆ e · ˆ c (cid:1) , N ± |(cid:104) | (cid:0) ˆ e · p (cid:1) |± L,U (cid:105)| = D ( L,U ) ⊥ (cid:104) − (cid:0) ˆ e · ˆ c (cid:1) (cid:105) , (1)where | (cid:105) stands for the ground state; N and N ± are the level populations. Formula (1)describes the uniaxial transition anisotropy that leads to the linearly polarized emission.This anisotropy is governed by a number of factors such as the fine structure splittings,the selection rules and the exciton oscillator strengths. Sensitivity of these factors to thesize and shape of the nanostructures provides a way to control the optical properties ofnanorods [2, 3, 5, 6].For nanorods embedded in surrounding dielectric media, both the emission and absorptionproperties of NRs are additionally influenced by the effects of dielectric confinement throughthe modification of the interactions between charge carriers and the local field effect [43].The latter is the effect of dielectric screening arising from the difference between the external(outside) electric field and the internal local field inside the nanostructures. A systematictheoretical treatment of such screening generally requires using the methods of the effectivemedium theory (details can be found, e.g., in the monographs [44, 45]). This theory hasbeen applied to interpret optical properties of nanostructures [10–12, 46–49], liquid crystalsystems [50–52] and hyperbolic metamaterials [53–57].In the simplest case of cylindrically symmetric and ellipsoidally shaped NRs surroundedby an isotropic dielectric medium, the local field effect can be described in terms of twodepolarization factors, N (cid:107) and N ⊥ , related to the screening factors, L (cid:107) = (1 + N (cid:107) ( (cid:15) em /(cid:15) m − − and L ⊥ = (1 + N ⊥ ( (cid:15) em /(cid:15) m − − , where (cid:15) m ( (cid:15) em ) is the dielectric constant of thesurrounding medium (NR emitter), for the electric field components directed along andnormal to the c -axis, respectively. For prolate NRs with sufficiently large aspect ratio, thecomponent along the c -axis is characterized by the smallest depolarization factor is N (cid:107) and,in contrast to the normal components, is almost insensitive to the screening effect.The local field screening effect can be taken into account by using Eq. (1) with therenormalized transition coefficients: D ( LF ) (cid:107) = D (cid:107) | L (cid:107) | and D ( LF ) ⊥ = D ⊥ | L ⊥ | , where D ⊥ = D ( L ) ⊥ + D ( U ) ⊥ . Clearly, this effect introduces additional the effective transition anisotropyleading to enhancement of the degree of linear polarization.In a phenomenological approach, a quantum nanoemitter (NR) is regarded as a pointoscillating dipole located at r em with the current density J em ( r ) = J em δ ( r − r em ). In thisapproach, the emission probability and the above transition anisotropy renormalized bythe local field effect can be taken into account by replacing the product of the currentdensity amplitudes with the emission dipole tensor averaged over the quantum state of thenanoemitter.From Eq. (1), this emission anisotropy tensor is uniaxially anisotropic and can be writtenin the following general form: (cid:104) J (em) α [ J (em) β ] ∗ (cid:105) q ≡ J (em) αβ = J (em) ⊥ δ αβ + ( J (em) (cid:107) − J (em) ⊥ ) c α c β ≡ J em [ δ αβ + u em c α c β ] , (2)where α, β ∈ { x, y, z } , ˆ c = ( c x , c y , c z ), δ αβ is the Kronecker symbol; an asterisk indicatescomplex conjugation and u em = ( J (em) (cid:107) − J (em) ⊥ ) /J (em) ⊥ is the emission anisotropy parameter .The tensor coefficients, J (em) (cid:107) and J (em) ⊥ , that enter relation (2) are assumed to be proportionalto the corresponding renormalized transition coefficients D ( LF ) (cid:107) and D ( LF ) ⊥ .The geometry of our system is schematically depicted in Fig. 1. This system consists ofthree layers: (a) the film containing the nanorods (emitters) and liquid crystal monomers;(b) the azo-dye photoaligning layer and (c) the glass substrate.In order to describe light emission from an orientationally ordered ensemble of nanorodswe begin with the single-emitter problem. Solution of this problem will yield the expressionsfor the far-field eigenwave amplitudes E (em) p and E (em) s . Then the intensities of the s- andp-polarized waves registered at the detection point are given by I p,s = J exc (ˆ c ) (cid:104)| E (em) p,s | (cid:105) q (3)where J exc (ˆ c ) is the excitation rate which is proportional to the probability of absorptionand depends on the polarization of exciting light. Similar to the emission probability, thisdependence is determined by the excitation (absorption) anisotropy tensor J (exc) αβ = J (exc) ⊥ δ αβ + ( J (exc) (cid:107) − J (exc) ⊥ ) c α c β ≡ J exc [ δ αβ + u exc c α c β ] . (4)Figure 1: Schematic of emitters embedded in the top layer of the multi-layer structure.So, for the exciting light linear polarized along the unit vector ˆ e exc , the excitation rate isgiven by J exc (ˆ c ) = J exc (cid:104) u exc (cid:0) ˆ c · ˆ e exc (cid:1) (cid:105) , (5)where u exc = ( J (exc) (cid:107) − J (exc) ⊥ ) /J (exc) ⊥ is the excitation anisotropy parameter . B. Operator of evolution
In our subsequent calculations, we shall use the transfer matrix approach which can beregarded as a modified version of the well-known transfer matrix method [58, 59] and waspreviously applied to study the polarization-resolved conoscopic patterns of nematic liquidcrystal cells [60–62]. This approach has also been extended to the case of polarizationgratings and used to deduce the general expression for the effective dielectric tensor ofdeformed helix ferroelectric liquid crystal cells [63, 64].In this approach, we deal with a harmonic electromagnetic field characterized by thefree-space wave number k vac = ω/c , where ω is the frequency (time-dependent factor isexp {− iωt } ), and consider the multi-layer slab geometry shown in Fig. 1. In this geometry,each optically anisotropic layer is sandwiched between the bounding surfaces (substrates)normal to the z axis and is characterized by the dielectric tensor (cid:15) ij (in what follows, themagnetic permittivities are equal to unity).Further, we restrict ourselves to the case of stratified media and use the angular spectrumrepresentation [65–67] of the electromagnetic fields taken in the following form: { E ( k P , z ) , H ( k P , z ) } = (cid:90) { E ( r ) , H ( r ) } exp( − i k P · r P )d r P (6)where r = z ˆ z + r P and the vector k P /k vac = q P = ( q ( P ) x , q ( P ) y ,
0) = q P (cos φ P , sin φ P ,
0) (7)represents the lateral component of the wave vector. Then we write down the representationfor the electric and magnetic fields, E and H , E = E z ˆ z + E P , H = H z ˆ z + ˆ z × H P , (8)where the components directed along the normal to the bounding surface (the z axis) areseparated from the tangential (lateral) ones. In this representation, the vectors E P = E x ˆ x + E y ˆ y ≡ (cid:18) E x E y (cid:19) and H P = H × ˆ z ≡ (cid:18) H y − H x (cid:19) are parallel to the substrates and give thelateral components of the electromagnetic field.We can now substitute the relations (8) into the Maxwell equations ∇ × E = iµk vac H , (9a) ∇ × H = − ik vac D + 4 πc J em , (9b)where D = ε · E is the electric displacement field; k vac = ω/c is the free-space wave number and J em ( r ) = − iω µ em δ ( r − r em ) is the current density of the dipole emitter located at r = r em ,and eliminate the z components of the electric and magnetic fields to obtain equations forthe tangential components of the electromagnetic field that can be written in the following4 × − i∂ τ F = M · F + F J ≡ (cid:18) M M M M (cid:19) (cid:18) E P H P (cid:19) + F J ( k P ) δ ( τ − τ em ) , τ ≡ k vac z, (10)where M is the differential propagation matrix and its 2 × M ij are givenby M (11) αβ = − (cid:15) − zz q ( P ) α (cid:15) zβ , M (22) αβ = − (cid:15) − zz (cid:15) αz q ( P ) β , (11a) M (12) αβ = µδ αβ − q ( P ) α q ( P ) β (cid:15) zz , (11b) M (21) αβ = (cid:15) αβ − (cid:15) αz (cid:15) zβ (cid:15) zz − µ − p ( P ) α p ( P ) β , p P = ˆ z × q P . (11c)The last term on the right hand side of Eq. (10) F J ( k P ) = (cid:18) − q P (cid:15) − zz J z ( k P ) J P ( k P ) − (cid:15) (cid:48) z (cid:15) − zz J z ( k P ) (cid:19) (12)is expressed in terms of the Fourier amplitude of the emitter current density J ( k P , τ ) = 4 πc (cid:90) J em ( r P , z )e − i k P · r P d r P = J ( k P ) δ ( τ − τ em ) , τ em ≡ k vac z em , (13)where J ( k P ) = − πi µ em k vac e − i k P · r em = J z ( k P )ˆ z + J P ( k P ) . (14)At F J = , general solution of the homogeneous system (10) F ( τ ) = U ( τ, τ ) · F ( τ ) (15)can be conveniently expressed in terms of the evolution operator which is also known as the propagator and is defined as the matrix solution of the initial value problem − i∂ τ U ( τ, τ ) = M ( τ ) · U ( τ, τ ) , (16a) U ( τ , τ ) = I , (16b)where I n is the n × n identity matrix. Basic properties of the evolution operator are discussedin Appendix A of Ref. [64].For uniformly anisotropic planar structures with the dielectric tensor of the form: (cid:15) ij = (cid:15) z δ ij + ( (cid:15) (cid:107) − (cid:15) z ) m i m j + ( (cid:15) ⊥ − (cid:15) z ) l i l j , (17)where the optical axesˆ m = ( m x , m y , m z ) = (cos ψ, sin ψ, , ˆ l = ˆ z × ˆ m = ( − sin ψ, cos ψ,
0) (18)lie in the plane of substrates (the x - y plane), the diagonal block-matrices, M and M ,vanish, and nondiagonal block-matrices are given by M = (cid:18) − q P /n z
00 1 (cid:19) , (19) M = n o (cid:18) − u a m x − u a m x m y − u a m x m y − u a m y − q P /n o (cid:19) , (20)where q P = q P ˆ x ; n z = √ (cid:15) z and n o = √ (cid:15) ⊥ are the principal refractive indices; u a =( (cid:15) (cid:107) − (cid:15) ⊥ ) /(cid:15) ⊥ is the parameter of in-plane anisotropy. In this case, the operator of evolutioncan be expressed in terms of the eigenvalue and eigenvector matrices, Λ ≡ diag( λ , λ , λ , λ )and V , as follows U ( τ, τ ) = exp { i M ( τ − τ ) } = V exp { i Λ ( τ − τ ) } V − , MV = VΛ , (21)where the eigenvector and eigenvalue matrices are of the following form: V = (cid:18) E EH − H (cid:19) , Λ = diag( Q , − Q ) , Q = diag( q e , q o ) (22)and can be computed from the relations given in Appendix B of Ref. [64]. C. Transfer matrix
In the ambient medium with (cid:15) ij = (cid:15) m δ ij , the general solution (15) can be expressed interms of plane waves propagating along the wave vectors with the tangential component (7).For such waves, the result is given by F m ( τ ) = V m ( q P ) (cid:18) exp { i Q m τ } exp {− i Q m τ } (cid:19) (cid:18) E + E − (cid:19) , (23) Q m = q m I , q m = (cid:113) n − q P , (24)where V m ( q P ) is the eigenvector matrix for the ambient medium given by V m ( q P ) = T rot ( φ P ) V m = (cid:18) Rt ( φ P )
00 Rt ( φ P ) (cid:19) (cid:18) E m − σ E m H m σ H m (cid:19) , (25) E m = (cid:18) q m /n m
00 1 (cid:19) , H m = (cid:18) n m q m (cid:19) , (26) Rt ( φ ) = (cid:18) cos φ − sin φ sin φ cos φ (cid:19) , (27) { σ , σ , σ } are the Pauli matrices σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . (28)From Eq. (23), the vector amplitudes E + and E − correspond to the forward and backwardeigenwaves with k + = k vac ( q m ˆ z + q P ) and k − = k vac ( − q m ˆ z + q P ), respectively. Figure 1shows that, in the half space z ≥ z after the exit face of the film with embedded emitters z = z , these eigenwaves describe the incoming and outgoing waves E + | z ≥ z = E ( L )out , E − | z ≥ z = E ( L )in = , (29)whereas, in the half space z ≤ z = − D before the entrance face of the glass substrate, thesewaves are given by E + | z ≤ z = E ( R )in = , E − | z ≤ z = E ( R )out . (30)The boundary conditions require the tangential components of the electric and magneticfields to be continuous at the boundary surfaces of the multi-layer structure: F L ≡ F ( z ) = F m ( z + 0) = V m ( q P ) (cid:32) E ( L )out E ( L )in (cid:33) , F R ≡ F ( z ) = F m ( z −
0) = V m ( q P ) (cid:32) E ( R )in E ( R )out (cid:33) . (31)In the standard light scattering (transmission/reflection) problem, we can use the bound-ary conditions (31) to rewrite the relation F L = U ( τ , τ ) · F R , τ i = k vac z i , (32)in the form (cid:32) E ( L )out E ( L )in (cid:33) = T (cid:32) E ( R )in E ( R )out (cid:33) (33)and introduce the transfer matrix T linking the amplitudes of the eigenwaves in the halfspaces z > z and z < z bounded by the faces of the multilayer structure. This matrixis the evolution operator U ( τ , τ ) in the basis of eigenmodes of the surrounding mediumwhich is given by T ( τ , τ ) ≡ T = V − ( q P ) U ( τ , τ ) V m ( q P ) = V − U rot ( τ , τ ) V m = (cid:18) T T T T (cid:19) , (34)where U rot ( τ, τ ) = T rot ( − φ P ) U ( τ, τ ) T rot ( φ P ) is the rotated operator of evolution. Thisoperator is the solution of the initial value problem (16) with M ( τ ) replaced with M rot ( τ ) = T rot ( − φ P ) M ( τ ) T rot ( φ P ). The scattering matrix relating the amplitudes of the incoming andoutgoing waves can be expressed in terms of the block matrices T ij giving the expressionsfor the transmission and reflection matrices [64]. We shall need the results for the case wherethe incident wave E inc = E ( L )in is coming from the half-space z > z : T L = T − , R L = T T − , (35)where T L and R L are the transmission and reflection matrices, respectively.Our concluding remark in this section is that, for the multi-layer structure that consistsof three layers depicted in Fig. 1, the evolution operator and the transfer matrix are givenby U ( τ , τ ) = U f ( h f ) U a ( h a ) U g ( h g ) , T ( τ , τ ) = T f ( h f ) T a ( h a ) T g ( h g ) , (36)where U f ( h f ) = U f ( τ , τ ) is the evolution operator for the film containing the nanorods, D f is the film thickness ( h f = k vac D f ); U a ( h a ) = U a ( τ , τ ) is the evolution operator for theazo-dye layer, D a is the layer thickness ( h a = k vac D a ); U g ( h g ) = U a ( τ , τ ) is the evolutionoperator for the glass substrate, D g is the thickness of the substrate ( h g = k vac D g ). D. Dyadic Green’s function and emission problem At F J ( k P ) (cid:54) = , the solution of non-homogeneous system F em ( τ ) = G ( τ ) F J ( k P ) (37)describing light radiation of the dipole emitter is expressed in terms of the dyadic (matrix-valued) Green’s function G ( τ ) that can be found by solving the following equation − i∂ τ G = M · G + I δ ( τ − τ em ) . (38)For uniformly anisotropic layer with the propagator given by Eqs. (21) and (22), we canuse the Fourier transform technique combined with the residue calculus (the poles at q o,e are shifted to the upper half of the complex plane: q o,e → q o,e + iδ + ) to obtain the followingexpression for the Green’s function: G ( τ ) = i V (cid:18) H ( τ − τ em )e i Q ( τ − τ em ) − H ( τ em − τ )e − i Q ( τ − τ em ) (cid:19) V − , (39)where H ( τ ) = , τ > / , τ = 00 , τ < τ ≤ τ ≤ τ is given by F f ( τ ) = U f ( τ, τ ) F + F em ( τ ) , (40)where the first term on the right hand side (general solution of the homogeneous systemwritten in the form given by Eq. (15)) represents the waves reflected from (and transmittedthrough) the boundaries of the film. The continuity conditions at the film boundaries τ = τ and τ = τ are F f ( τ ) = F L , F f ( τ ) = U a ( h a ) U g ( h g ) F R . (41)After eliminating F from Eq. (41), we have F L − U ( τ , τ ) F R = F em ( τ ) − U f ( τ , τ ) F em ( τ )= [ G ( τ ) − U f ( τ , τ ) G ( τ )] F J . (42)For the Green’s function given in Eq. (39), the relation (42) can be further simplified withthe help of the identity G ( τ ) − U f ( τ , τ ) G ( τ ) = i sign( τ − τ em ) U f ( τ , τ em ) . (43)We can now use Eqs. (29)– (32) to express the result in terms of the eigenwave amplitudes E ( L )out and E ( R )out as follows (cid:18) E ( L )out (cid:19) − T (cid:18) ( R )out (cid:19) = T em (cid:18) J H J E (cid:19) , (44) T em = i T f ( τ , τ em ) V − = i πk vac q m (cid:32) T (em)11 T (em)12 T (em)21 T (em)22 (cid:33) , (45)where q m /n m is the z -component of the unit wave vector ˆ k + = k + /n m k vac ; J H and J E aregiven by J H = − q P (cid:15) zz J z (cid:18) (cid:19) , J E = Rt ( − φ P ) J P . (46)In our final step, we deduce the expressions for the far-field amplitudes of the emitted(outgoing) waves, E ( f − f ) L and E ( f − f ) R from Eq. (44). The far-field asymptotic behavior of theamplitudes in a fixed direction ˆ r = r /r is known [65, 66] to be determined by the plane waveamplitude of the angular spectrum representation with ˆ k + = ˆ r . So, the far-field amplitudesof the radiated waves, E ( f − f ) L and E ( f − f ) R , are proportional to E ( L )out and E ( R )out multiplied bythe z -component of k + = k vac q m ˆ z + k P and k − = − k vac q m ˆ z + k P , respectively. The resultreads E em = (cid:32) E (em) p E (em) s (cid:33) ≡ E ( f − f ) L = − πik vac q m E ( L )out = W H J H + W E J E , (47) W H = T (em)11 − R L T (em)21 , W E = T (em)12 − R L T (em)22 , (48) E ( f − f ) R = 2 πik vac q m E ( R )out = T L ( T (em)21 J H + T (em)22 J E ) , (49)where the reflection and the transmission matrices, R L and T L , are defined in Eq. (35). Inwhat follows, the emitted wavefield (47) will be our primary concern.1 E. Orientational averaging
We shall assume that an ensemble of aligned NRs can be treated as a collection ofincoherently emitting and differently oriented dipoles and the total intensity of the emittersis a sum of the intensities. Orientation of a nanorod is specified by the tilt and azimuthalangles, θ c and φ c , giving the direction of the c -axis: ˆ c = (cos θ c cos φ c , cos θ c sin φ c , sin θ c ) andthe total intensities of the p- and s-waves can be obtained by averaging the intensities givenin Eq. (3) over orientation of the c -axis.More generally, the emitted wavefield can be described by the coherency matrix [66, 68] M = (cid:104) J exc (ˆ c ) E em ⊗ E ∗ em (cid:105) = (cid:32) (cid:104) J exc (ˆ c ) (cid:104)| E (em) p | (cid:105) q (cid:105) ˆ c (cid:104) J exc (ˆ c ) (cid:104) E (em) p [ E (em) s ] ∗ (cid:105) q (cid:105) ˆ c (cid:104) J exc (ˆ c ) (cid:104) E (em) s [ E (em) p ] ∗ (cid:105) q (cid:105) ˆ c (cid:104) J exc (ˆ c ) (cid:104)| E (em) s | (cid:105) q (cid:105) ˆ c (cid:33) , (50)where an asterisk stands for complex conjugation, can now be calculated using the matrixrelations (47) and (48). This matrix is given by M = W H (cid:104) J exc (ˆ c ) J H ⊗ J ∗ H (cid:105) W † H + W E Rt ( − φ P ) (cid:104) J exc (ˆ c ) J P ⊗ J ∗ P (cid:105) Rt ( φ P ) W † E , (51) (cid:104) J exc (ˆ c ) J H ⊗ J ∗ H (cid:105) = q P (cid:15) z (cid:18) (cid:19) (cid:104) J exc (ˆ c ) J (em) zz (cid:105) ˆ c , (52)[ (cid:104) J exc (ˆ c ) J P ⊗ J ∗ P (cid:105) ] αβ = (cid:104) J exc (ˆ c ) J (em) αβ (cid:105) ˆ c , (53)where a dagger will indicate Hermitian conjugation and (cid:104) ... (cid:105) ˆ c stands for orientational aver-ages.The result of orientational averaging is expressed in terms of the two symmetric matrices: C ( e ) αβ = (cid:104) c α c β (cid:105) ˆ c , C ( ex ) αβ = (cid:104) c c α c β (cid:105) ˆ c , (54)where c exc = (cid:0) ˆ c · ˆ e exc (cid:1) and ˆ e exc is the polarization unit vector of the exciting light. Orienta-tional ordering is described by the eigenvalues and the eigenvectors (the principal axes) ofthese matrices. Perfectly in-plane ordering presents the important special case with vanish-ing tilt angle, θ c = 0. We shall, however, consider a more general case with nonvanishing θ c and assume that the angles are statistically independent and the principal (alignment) axesare directed along the coordinate axes. The latter implies that the matrices (54) are bothdiagonal. It immediately follows that the matrix (53) is also diagonal (cid:104) J exc (ˆ c ) J P ⊗ J ∗ P (cid:105) = γ I + γ σ , (cid:104) J exc (ˆ c ) J (em) zz (cid:105) ˆ c = γ z (55)and the coherency matrix (51) can be written in the form of a linear combination: M = J em J exc [ γ W + γ (cos 2 φ P W − sin 2 φ P W ) + γ z W z ] , (56a) W = W E W † E , W i = W E σ i W † E , W z = q P (cid:15) z W H (cid:18) (cid:19) W † H , (56b) γ = 1 + u exc (cid:104) c (cid:105) ˆ c + u em (cid:2) (cid:104) c x + c y (cid:105) ˆ c + u exc (cid:104) c ( c x + c y ) (cid:105) ˆ c (cid:3) , (56c) γ = u em (cid:2) (cid:104) c x − c y (cid:105) ˆ c + u exc (cid:104) c ( c x − c y ) (cid:105) ˆ c (cid:3) , (56d) γ z = 1 + u exc (cid:104) c (cid:105) ˆ c + u em (cid:2) (cid:104) c z (cid:105) ˆ c + u exc (cid:104) c c z (cid:105) ˆ c (cid:3) . (56e)2Orientational averages that enter the coefficients of the linear combination (56a) can beexpressed in terms of the orientational parameters characterizing ordering of aligned NRs.For in-plane ordering, these parameters are as follows (cid:104) cos φ c (cid:105) ˆ c = 1 + p , (cid:104) sin φ c (cid:105) ˆ c = 1 − p , (cid:104) cos φ c sin φ c (cid:105) ˆ c = q, (57)where 0 ≤ q ≤ (cid:104) cos φ c (cid:105) ˆ c (cid:104) sin φ c (cid:105) ˆ c = (1 − p ) / − ≤ p ≤ orienta-tional (alignment) order parameter . Similar results for the averages over the tilt angle θ c characterizing out-of-plane deviations of the c -axis are given by (cid:104) sin θ c (cid:105) ˆ c = p z , (cid:104) cos θ c (cid:105) ˆ c = 1 − p z , (cid:104) cos θ c sin θ c (cid:105) ˆ c = q z , (58)where 0 ≤ q z ≤ p z (1 − p z ) and 0 ≤ p z ≤ III. RESULTS
The lightfield emitted by NRs is generally partially polarized and the far-field angulardistributions of its polarization parameters are determined by the coherency matrix M ( q P , φ P ) ≡ M ( θ, φ ) = 12 (cid:2) I ( θ, φ ) I + S ( θ, φ ) σ + S ( θ, φ ) σ + S ( θ, φ ) σ (cid:3) , (59) q P ≡ | k P | /k vac = n m sin θ, φ P ≡ φ, (60)where I ( θ, φ ) = I p ( θ, φ ) + I s ( θ, φ ) = Tr M ( θ, φ ) is the total intensity and { S , S , S } arethe Stokes parameters, evaluated as a function of the emission (detection) angle, θ , and theazimuthal angle, φ = φ P , (see Eq. (7)) that specifies orientation of the emission plane ( φ is the angle between the alignment axis and the emission plane). Though the polarizationstate of partially polarized radiation from nanoemitters such as NRs with the degree ofpolarization P = (cid:113) s + s + s = (cid:115) − M (Tr M ) , s i ≡ S i ( θ, φ ) /I ( θ, φ ) , (61)can be completely described using the Stokes parameters, there is a number of the technologi-cally important parameters widely used to characterize the anisotropy of photoluminescence.One of these parameters is the degree of linear polarization DOP = (cid:113) s + s = I max − I min I max + I min (62)where I max and I min are the maximum and minimum intensities of the curve representingthe experimentally measured intensity of light passed through the rotating polarizer placedin the emission beam path. Alternatively, the Stokes parameter s = I p − I s I p + I s (63)3and the related polarization ratios R ps = I p /I s , R sp = I s /I p , (64)are also used as convenient measures characterizing the anisotropy of emission in termsof the intensities of the p-polarized and s-polarized waves: I p ≡ I p ( θ, φ ) = M ( θ, φ ) and I s ≡ I s ( θ, φ ) = M ( θ, φ ). Note that the case where the degree of linear polarization is equalto the magnitude of s , DOP = | s | , occurs only when the Stokes parameter S = 2 Re M vanishes and the polarization azimuth of the polarization ellipse ψ p = 2 − arg( s + is ) (65)differs from zero and π/
2. The ellipticity of the polarization ellipse characterizing the po-larized part of emission (cid:15) ell = tan[2 − arcsin( s /P )] (66)is expressed in terms of the Stokes parameter S = 2 Im M .Equation (56a) shows that each element of the coherency matrix is a linear combinationof the three angular profiles defined by the matrices given in Eq. (56b). In the case ofunpolarized excitation with c replaced by c x + c y = 1 − c z , the expressions for the coefficientsof the linear combination can be obtained from the relations (56c)– (56e) in the followingform: γ = Q + u em Q , γ = u em p Q , γ z = (1 + u em ) Q − u em Q , (67a) Q = 1 + u exc (1 − p z ) , Q = 1 − p z + u exc (1 − p z − q z ) . (67b)An important point is that all the above discussed polarization characteristics depend onthe two ratios of the coefficients (67a)˜ γ = γ γ , ˜ γ z = γ z γ (68)that can be found as the fitting parameters when dealing with angular profiles obtainedfrom experimental data. Given the values of the ratios (68) and the emission anisotropyparameter u em , we can use the relations Q ≡ Q Q = 1 − p z + u exc ( p z (1 − p z ) − q z )1 + u exc (1 − p z ) = 2(1 + u em − ˜ γ z ) u em (2 + ˜ γ z ) , (69) p = (3 + u em )˜ γ u em − ˜ γ z (70)to estimate the alignment order parameters, p and p z .In what follows we concentrate on the four polarization parameters: the degree of linearpolarization (DOP) given by Eq. (62), the Stokes parameter s (see Eq. (63)), the polariza-tion azimuth, ψ p , (see Eq. (65)) and the ellipticity (cid:15) ell (see Eq. (66)). Figures 2– 7 presentthe results for these parameters evaluated as functions of the emission angle θ at differ-ent values of the azimuthal angle φ (the vector ( − sin φ, cos φ,
0) is normal to the emissionplane). Calculations are performed for the emission wavelength λ em = 590 nm assuming4 (a) DOP vs emission angle θ at p = 0 .
87 (b) s vs emission angle θ at p = 0 . Figure 2: Angular profiles of (a) DOP and (b) s computed at different values of theazimuthal angle φ for p = 0 . p z = 0 and u em = 3 . D a = 15 nm (the data are taken from Ref. [21]) are n ( a ) (cid:107) = 1 . n ( a ) ⊥ = 1 . D f = 375 nm) containing NRs with an aspect ratio 5 : 1 (the NR length is about 20 nmand the NR diameter is about 4 nm) [21] are n ( f ) (cid:107) = 1 . n ( f ) ⊥ = 1 . p z = 0 and the alignment axis directed along the x axis) [21], thelatter is normal to the in-plane optic axes of both the SD1 layer and the LCP film so thattheir orientation with respect to the emission plane is defined by the azimuthal angle ψ (seeEq. (18)) which is equal to π/ − φ .Figures 2 and 3 show the angular profiles computed for highly ordered NRs embeddedinto the LCP film using the value of the NR alignment order parameter p = 0 .
87 reported inRef. [21]. In addition, the value of DOP measured in [21] at θ = φ = 0 is about 0 .
62. Fromthis result, we have obtained the estimate for the emission anisotropy parameter: u em ≈ . u em used in our calculations.Referring to Fig. 2a, when the azimuthal angle φ does not exceed π/
4, DOP is generallya nonmonotonic function of the emission angle that increases at sufficiently large values of θ . At φ > π/
4, DOP monotonically decreases with θ .In the special case where θ = 0 and the detected lightwave is propagating along thenormal to the substrates, the value of DOP is independent of φ . As it can be seen fromFig. 3a, at θ = 0, the polarized part of the emitted light is linearly polarized and the ellipticityvanishes. In this case, rotation of the emission plane about the z axis by the azimuthal angle φ is equivalent to the rotation of the sample by the angle − φ . As a result, the polarizationplane is rotated by the same angle and, as is indicated in Fig. 3, the polarization azimuth ψ p equals − φ .Another consequence of such rotation is that, at θ = 0, the Stokes parameter s changesfrom DOP (0) = 0 .
62 to − DOP (0) = − .
62 as φ varies from zero to π/
2. It immediately5 (a) Ellipticity vs emission angle θ at p = 0 .
87 (b) Polarization azimuth vs θ at p = 0 . Figure 3: Angular profiles of (a) ellipticity and (b) polarization azimuth computed atdifferent values of the azimuthal angle φ for p = 0 . p z = 0 and u em = 3 . (a) DOP vs emission angle θ at p = 0 . s vs emission angle θ at p = 0 . Figure 4: Angular profiles of (a) DOP and (b) s computed at different values of theazimuthal angle φ for p = 0 . p z = 0 and u em = 3 . s + is = DOP e iψ p (71)that defines the DOP and the polarization azimuth given by Eqs. (62) and (65), respectively.This effect can be seen from the curves presented in Fig. 2b. This figure also demonstratesthat, in agreement with the identity (71), s (0) becomes negative when φ > π/
4. In thisregion, s is a monotonically increasing function of θ , whereas both the magnitude of s , | s | , and the DOP fall as the emission angle increases.Equation (71) shows that the magnitude of s is generally smaller than the DOP and thedifference between the DOP and s is dictated by the polarization azimuth ψ p . The curvespresented in Fig. 3b illustrate a noticeable increase in the polarization azimuth as θ becomessufficiently large.6 (a) Ellipticity vs emission angle θ at p = 0 . θ at p = 0 . Figure 5: Angular profiles of (a) ellipticity and (b) polarization azimuth computed atdifferent values of the azimuthal angle φ for p = 0 . p z = 0 and u em = 3 . φ = 0 and φ = π/
2, respectively), thelight appears to be linearly polarized and the ellipticity equals zero. At 0 < φ < π/
2, thisis, however, no longer the case and the ellipticity shows generally nonmonotonic variationswith θ . The ellipticity magnitude | (cid:15) ell | reaches its highest value at φ = π/
4. For p = 0 . . p = 0 .
4. An important point is that the most part ofthe above discussion being independent of the alignment order parameter remains valid forthe case of poorly aligned NRs. So, we will focus our attention on the differences introducedby changes in orientational ordering of NRs. (a) DOP vs emission angle θ at p = 0 (b) s vs emission angle θ at p = 0 Figure 6: Angular profiles of (a) DOP and (b) s computed at different values of theazimuthal angle φ for p = p z = 0 and u em = 3 . DOP (0) is reduced to 0 .
33. As is seen from Fig. 5a, thiseffect also manifests itself in an increase of the largest value of the ellipticity magnitudewhich is now about 0 .
17. Qualitatively, the common feature shared by all the angularprofiles calculated at p = 0 . p = 0 . (a) Ellipticity vs emission angle θ at p = 0 (b) Polarization azimuth vs θ at p = 0 Figure 7: Angular profiles of (a) ellipticity and (b) polarization azimuth computed atdifferent values of the azimuthal angle φ for p = p z = 0 and u em = 3 . p = 0. As is shown in Fig. 6, at p = 0 the values of boththe DOP and | s | experience significant reduction. For instance, DOP (0) is about 0 . .
51. Angular profiles for the polarization azimuth plotted in Fig. 7b demonstrate strongvariations and rapidly approach the vicinity of zero as the emission angle increases. Clearly,these features can be regarded as the effects of optically anisotropic environment.Our concluding remark in this section concerns the parity symmetry of the polarizationparameters under the change of sign of the emission and azimuthal angles: θ → − θ and φ →− φ . All the polarization parameters are even functions of θ : DOP ( − θ, φ ) = DOP ( θ, φ ), s ( − θ, φ ) = s ( θ, φ ), (cid:15) ell ( − θ, φ ) = (cid:15) ell ( θ, φ ) and ψ p ( − θ, φ ) = ψ ( θ, φ ). Similarly, when φ changes its sign, DOP and s remain intact: DOP ( θ, − φ ) = DOP ( θ, φ ) and s ( θ, − φ ) = s ( θ, φ ). In contrast, the ellipticity and the polarization azimuth are even functions of φ : (cid:15) ell ( θ, − φ ) = − (cid:15) ell ( θ, φ ) and ψ p ( θ, − φ ) = − ψ p ( θ, φ ). IV. DISCUSSION AND CONCLUSIONS
In this paper, we have studied the far-field angular distributions of the polarization pa-rameters characterizing the anisotropy of photoluminescence from orientationally orderedNRs placed inside an optically anisotropic multilayer system. By using a suitably modifiedversion of the transfer matrix method combined with the Green’s function technique wehave found the solution of the emission problem expressed in terms of the evolution op-erators (see Eqs. (44)– (48)). The emission and excitation anisotropy tensors (see Eq. (2)8and Eq. (5), respectively) that depend on the transition dipole moments, the exciton levelpopulations and the local field screening factors are then used to deduce the expression forthe orientationally averaged coherency matrix (50) of the emitted optical field.The results of our theoretical analysis are applied to the geometry of the photoalignmentmethod where aligned NRs are embedded into the LCP film placed on top of the photoalign-ing azo-dye layer [21]. By assuming perfectly in-plane ordering of nanorods and unpolarizedexcitation, we have computed the degree of linear polarization (DOP) (see Eq. (62)), theStokes parameter s (see Eq. (63)), the polarization azimuth (see Eq. (65)) and the ellipticity(see Eq. (66)) of the emitted light as functions of the emission and azimuthal angles, θ and φ , (see Eq. (60)) at different values of the alignment order parameter p (see Eq. (57)). Wehave found that the values of DOP and the order parameter measured in Ref. [21] can beused to obtain the estimate for the emission anisotropy parameter u em : u em ≈ . θ dependencies of the polarization parameters in differentlyoriented emission planes at varying value of p are presented in Figs. 2– 7. These profiles areshown to be determined by the two coefficients given by Eq. (68) that depend on the NRorientational averages and the emission/excitation anisotropy parameters, while the compo-nents of the three matrices given by Eq. (56b) define the angular dependent contributionsgoverned by the optical anisotropy of the LCP film and the photoaligning layer.We have evaluated the profiles for the three cases: (a) the film with highly ordered NRs( p = 0 . p = 0 . p = 0 . s so that their values for the disordered NRs are anorder of magnitude smaller than for the highly ordered NRs. By contrast, the largest valueof ellipticity significantly grows as the alignment order parameter decreases. The curvescomputed at p = 0 (see Figs. 6 and 7) demonstrate a pronounced nonmonotonic behaviorthat can be regarded as the effect of the optically anisotropic environment.It should be stressed that the emission and absorption properties of NRs embedded insurrounding dielectric media are generally influenced by the effects of dielectric confine-ment [43]. In our phenomenological approach, NRs are treated as radiating dipoles andthese properties are described by the emission and excitation anisotropy tensors. The di-electric confinement effects for dot-in-rods placed in optically anisotropic media has not beenstudied in any detail yet and analysis of such effects is well beyond the scope of this paper.We conclude this paper with the remark on the local field effect in the anisotropic LCPfilm. As it was discussed in Sec. II this effect will typically enhance the anisotropy of emissionand excitation leading to nonzero anisotropy parameters u em and u exc even if the transitionanisotropy is vanishing. For NRs with an aspect ratio 5 : 1 and (cid:15) em ≡ (cid:15) CdS = 5 .
23 [70]embedded in the LCP film with (cid:15) ( f ) (cid:107) = 2 .
62 and n ( f ) ⊥ = 2 .
16, the screening factors can beestimated assuming that the c -axis (ˆ c = ˆ x ) is normal to the optic axis of the film directedalong the y -axis. To this end we can use the well-known analytical results for an opticallyisotropic ellipsoid placed in the anisotropic medium [45] and evaluate the components, L x , L y and L z of the screening factor tensor (dyadic). In our case, this tensor is biaxial. Itsdiagonal elements are: L z ≈ . < L y ≈ . < L x ≈ .
93 giving the anisotropy ratios:( L x /L z ) ≈ .
33, ( L x /L y ) ≈ .
94 and ( L z /L y ) ≈ . p z = 0, the generalized form of this tensor9is as follows J em = J em ( I + u em ˆ c ⊗ ˆ c + u z ˆ z ⊗ ˆ z ) , (72)where I is the identity matrix. The effect of the local field giving the small negative valueof u z = ( L z /L y ) − ACKNOWLEDGMENTS
This work was partially supported by the Russian Science Foundation under grant 19-42-06302.
Appendix A: Equations for lateral components
In this section we discuss how to exclude the z -components of the electromagnetic field, E z and H z , that enter the representation (8), from the Maxwell equations (9). Our taskis to derive the closed system of equations for the lateral (tangential) components, E P and H P .After substituting Eq. (8) into the Maxwell equations (9), we use decomposition for thedifferential operator that enter Eq. (9) k − ∇ = ˆ z ∂ τ + i ∇ P , ∇ ⊥ P = ˆ z × ∇ P , (A1)where τ = k vac z ; ∇ P = − i k − (ˆ x ∂ x + ˆ y ∂ y ) ≡ ( ∇ x , ∇ y ) and ∇ ⊥ P = ( ∇ ⊥ x , ∇ ⊥ y ) = ( −∇ y , ∇ x ) , to recast Maxwell’s equations (9) into the following form: − i∂ τ [ˆ z × E P ] = µ H − ∇ P × E , (A2a) − i∂ τ H P = D + ∇ P × H + J , J ≡ πick vac J D , (A2b)where the explicit expressions for the last terms on the right hand side of the system (A2)are as follows ∇ P × E = − ∇ ⊥ P E z + (cid:0) ∇ ⊥ P · E P (cid:1) ˆ z , (A3a) ∇ P × H = − ∇ ⊥ P H z + (cid:0) ∇ P · H P (cid:1) ˆ z . (A3b)We can now substitute the electric displacement field and the current density of the dipole J = − πi µ D δ ( r − r ) written as a sum of the normal and in-plane components D = D z ˆ z + D P , J = J z ˆ z + J P , (A4)into Eq. (A2b) and derive the following expression for its z -component D z = (cid:15) zz E z + (cid:0) (cid:15) z · E P (cid:1) = − (cid:0) ∇ P · H P (cid:1) − J z , (A5)0where (cid:15) z = ( (cid:15) zx , (cid:15) zy ).From Eqs. (A2) and (A5), it is not difficult to deduce the relations H z = µ − (cid:0) ∇ ⊥ P · E P (cid:1) (A6) E z = − (cid:15) − zz (cid:2)(cid:0) (cid:15) z · E P (cid:1) + (cid:0) ∇ P · H P (cid:1) + J z (cid:3) (A7)linking the normal (along the z axis) and the lateral (perpendicular to the z axis) compo-nents.By using the relation (A7), we obtain the tangential component of the field (A4) D P = (cid:15) (cid:48) z E z + ε z · E P = ε P · E P − (cid:15) (cid:48) z (cid:15) − zz (cid:2)(cid:0) ∇ P · H P (cid:1) + J z (cid:3) (A8)where ε z = (cid:18) (cid:15) xx (cid:15) xy (cid:15) yx (cid:15) yy (cid:19) ; (cid:15) (cid:48) z = ( (cid:15) xz , (cid:15) yz ) and the effective dielectric tensor, ε P , for the lateralcomponents is given by ε P = ε z − (cid:15) − zz (cid:15) (cid:48) z ⊗ (cid:15) z . 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