Angular-Dependent Spontaneous Emission in Cholesteric Liquid

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Angular-Dependent Spontaneous Emission in Cholesteric Liquid Crystal Films Anna Margareta Risse, and Jürgen Schmidtke J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11134 • Publication Date (Web): 04 Jan 2019 Downloaded from http://pubs.acs.org on January 12, 2019

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The Journal of Physical Chemistry

Angular-Dependent Spontaneous Emission in Cholesteric Liquid Crystal Films ∗

Anna M. Risse and Jürgen Schmidtke

Chemistry Department, Faculty of Science, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany E-mail: [email protected]

1

ACS Paragon Plus Environment

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Abstract Due to the presence of a selective reection band, cholesteric liquid crystals (CLCs) strongly modify the emission of uorescent guest molecules. While emission along the helical axis is well understood, the emission properties at oblique angles with respect to the cholesteric helix have so far have not been explored in detail. We present a systematic study of the angular-dependent spontaneous emission properties of dyedoped cholesteric lms, including a full Stokes analysis of the emitted light. Variation of the cholesteric pitch allows us to probe the eect of the photonic band structure on spontaneous emission in a wide angular range. We nd exceptional angular-dependent intensity and polarization variations of the emitted light, and give experimental proof for the splitting of the rst-order reection band as well as its overlap with the secondorder band at large detection angles.

Introduction Cholesteric liquid crystals (CLCs, also known as chiral nematic liquid crystals) are selfassembled, one dimensional photonic crystals. They are formed by rod-like molecules, which spontaneously arrange themselves in a periodic helical order: Locally, the molecules have a preferred orientation, which is continuously twisted along a certain direction (as sketched in Figure 1a). For light propagation parallel to the helical axis, there exists the well-known selective reection banda photonic band gap for circularly polarized light with the same handedness as the chiral molecular order. As CLCs are soft, liquid crystalline media, the photonic properties can be easily tuned by various external stimuli like temperature variation, electric elds, or by photo chemistry. 1 CLCs have in recent years attracted considerable interest as materials for various photonic applications, like bistable displays, 2 tunable dielectric mirrors, 3 optical pulse compression, 4 slow light, 5 or planar optical elements. 6 While most applications so far use thermotropic, low molar mass materials, recently cholesteric systems based on cellulose nanocrystals have received great attention as well. 7 The pho2


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Figure 1: (a) Sketch of the cholesteric helix, denition of the cholesteric pitch p and denition of the emission angle θ. (b) Probing the angular-dependent photonic band structure by uorescent guest molecules: angular-dependent photonic band gaps (gray areas I-III) of samples with dierent cholesteric pitch overlap with the emission range of the uorescent guest molecules (red area) in dierent angular intervals. tonic band structure of CLCs strongly aects radiation spectra of emitters embedded in the CLC: At wavelengths inside the reection band, emission is suppressed, near the band edges it is enhanced due to the presence of resonant optical modes. CLCs doped with uorescent dyes can serve as incoherent sources of circularly polarized light, 8 andusing pulsed excitationcan be used as photonic band edge lasers; 9,10 furthermore, CLC-based single photon sources for polarized light, 11 as well as polarization modulation of inorganic emitters (quantum dots) dispersed in a CLC have been demonstrated. 12 Spontaneous emission along the helical axis has been thoroughly investigated, and is quantitatively understood. 13 In this paper, we present a comprehensive study of the angular-dependent emission properties of dye-doped CLCs. For propagation along the helical axis, there exists a single photonic band gap for circularly polarized light with the same handedness like the cholesteric order; the wavelengths of the band edges, λ1,2 = no,e p, depend on the ordinary and extraordinary refractive indices

no,e of the birefringent, quasi-nematic planes perpendicular to the helical axis, and on the cholesteric pitch p (i.e. the distance for a 360◦ turn of the nematic director, cf. Figure 1a). The angular-dependent optical properties of CLCs have been studied extensively by Berreman and Scheer, 14 Takezoe et al. 1517 and Oldano et al., 18 both experimentally and in simulations. These studies focussed on the polarization-dependent reection properties 3



of CLCs. The excellent agreement between measurements and simulations proved beyond doubt the validity of Oseen's optical model 19 of a spiraling dielectric ellipsoid as well as the local optical uniaxiality of the cholesteric medium. The main ndings of these reectance studies were a substantial broadening of the reection band with increasing detection angle with respect to the CLC's helical axis, strong angular variations of the polarization properties of the reected light, the emergence of a polarization-independent reection band at large angles, as well as the emergence of higher-order reection bands (which are absent for normal incidence) with a characteristic triplet structure. Angular-dependent measurements of emission in conventional one-dimensional crystals (i.e., periodic stacks of alternating dielectric layers) revealed a blueshift of the emission minimum and the band edge resonance peaks, in line with the angular-dependent blueshift of the rst-order photonic band gap. 20 Considering CLCs, the results from reectance studies promise much more diverse angular-dependent emission properties. However, so far experimental studies are sparse: Hara et al. 21 were the rst to perform emission measurements of uorescent guest molecules in a CLC, in a limited angular range (θ ≤ 30◦ ); they observed a blueshift of the emission minimum due to the blueshift of the selective reection band. More recently, Lee et al. 22 conrmed this nding (with improved spectral resolution). Penninck et al. 23 slightly extended the angular range (θ ≤ 40◦ ), however at the largest observation angles the band gap was blueshifted beyond the excitation wavelength. Umanskii et al. 24 extended the angular range further (θ ≤ 66◦ ), and indeed observed an extra emission dip indicating the emergence of a polarization-independent band gap. In the present study, we prepared four CLC mixtures with dierent cholesteric pitch (labeled S1, . . . , S4), resulting in an overlap of the photonic band gap(s) with the natural emission spectrum of the uorscent guest molecules in dierent wavelength intervals. This allows us to probe the eect of the CLC's photonic band structure on the emission in a wide angular range, as sketched in Figure 1b. Full Stokes analysis of the emitted light reveals strong angular-dependent variations of both emission intensity and emission polarization in the region of the rst- and second-order 4


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reection bands. We expect our results to be useful for a deeper understanding of the emission properties of CLC based lasers. So far, most studies have been concerned with laser emission along the cholesteric helix in systems with good match of the reection band with the dye's emission range. However, out-of-plane laser emission of cholesteric lms has been demonstrated as well: Lee et al. 22 found angular-dependent variations of the lasing wavelengths, andunder certain conditionscone-shaped emission patterns. Palto et al. 25 observed outof-plane lasing of dye-doped CLCs with substantial mismatch of the reection band and the dye's emission range, which they explained by the blueshifted band edge resonances for oblique wave propagation. More recently, Folcia et al. 26 studied in great detail the angular dependent emission characteristics of CLC-based lasers.

Methods Experimental A sketch of the experimental setup is shown in Figure 2a. The dye-doped sample lms are excited by a continuous-wave, frequency-doubled Nd:YAG laser (λ = 532 nm); the laser beam has lefthanded circular polarization (to minimize selective reection by the CLC) and is focussed on the sample lm, resulting in an excitation spot diameter of about 60 µm. To minimize refraction of the emitted light at the cell surface, the liquid crystal cell is centered in a circular glass vat (Ø = 60 mm), lled with glycerol as index-matching uid. The detection unit consists of an optical slit, a band edge lter blocking the excitation beam, and a combination of a rotatable, `superachromatic' quarter-wavelength retarder (Bernhard Halle Nachfolger GmbH, Berlin) and a polarizer for selective detection of circularly polarized emission contributions (for detecting linearly polarized emission contributions, the components are rearranged appropiately); nally, the light is fed via an optical ber to a spectrometer (Ocean Optics USB2000+UV-VIS-ES, spectral resolution 3 nm). In order 5



(a) sample

laser P λ�� P λ�� intensity adjustment

glycerol

θ

index matching vat

(b)

dI / dθ (arb. units)

spectrometer

laser

index matching vat

sample

3

(c)

2

∆θ ��

1 0 61.9

62.0

62.1

62.2

θ (deg) emission cone for angle θ

θ

∆θ (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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optical slit

1

(d) slit length slit width

0.1

0.01

0 10 20 30 40 50 60 70 80 90

θ (deg)

Figure 2: (a) Sketch of the experimental setup (P polarizer, λ/2, λ/4 retarders, F band edge lter). (b) Sketch of the emission cone for a certain detection angle θ, and of the tangential alignment of the optical slit. (c) Experimental angular resolution for the vertically aligned slit. (d) Angular resolution limitation ∆θ due to the length and width of the slit, respectively, calculated as a function of the detection angle θ: for θ < 20◦ resolution is limited by the slit length, and for θ > 20◦ resolution is limited by the slit width.

6


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to enable angular-dependent emission measurements, the detection unit is mounted on a rotation stage equipped with a stepper motor. In order to achieve a compromise between reasonable signal strength and angular resolution, an optical slit (150 µm×5 mm) was placed vertically (i.e. perpendicular to the rotation plane) at the front end of the detection unit. For detection angle θ = 0◦ , angular resolution is relatively poor (limited by the length of the slit), but improves with increasing θ (as the slit is oriented tangential to the emission cone determined by the angle θ, cf. Figure 2b). The angular resolution can be determined by scanning over the edge of an obstacle placed between emitter and detection unit. For vertical and horizontal orientation of the slit, we nd angular resolutions of 0.1◦ (Figure 2c) and 2.4◦ , respectively. With vertically aligned slit, the angular resolution of the setup improves monotonically in the range 0◦ ≤ θ ≤ 20◦ ; for larger angles it remains essentially constant and is determined by the slit width (see calculated angular resolution curve in Figure 2d). As the angular-dependent blueshift of the band gap approximately obeys Bragg's law (λR ∝ cos θ), poor angular resolution at small detection angles hardly aects the measurements, while at large angles, high resolution is essential for resolving any band-gap induced features in the emission spectra. A detailed discussion of angular-dependent overall emission intensity is beyond the scope of the present study. Hence, the intensity data were not corrected for the θ-dependent change in the solid angle subtended by the slit; at large angles, detected total intensities might be substantially aected by the diameter of the excited area of the sample lm. For correct Stokes analysis of emission (eq. 4), the relative detection sensitivities for the polarization pairs {H, V}, {+, −} and {RCP, LCP} were taken into account. For this, a liquid crystal cell with homeotropic surface anchoring (uniform orientation of the director perpendicular to the lm normal) was lled with a nematic mixture of the dye, and polarization-resolved emission along the lm normal (θ = 0◦ ) of this sample was measured. To minimize artefacts due to bleaching, in the angular-dependent scans all emission contributions necessary for the Stokes analysis where measured consecutively at each angle 7



(integration time for each spectrum: two seconds). All experiments were performed at room temperature (22◦ C).

Samples The samples are cholesteric mixtures of the nematic host HTW109100-100 (Jiangsu Hecheng Display Technology Co., Ltd, China), a chiral dopant (compound 1 in the paper by Heppke et al., 27 synthesized in-house), and the laser dye DCM (4-(dicyanomethylene)-2-methyl-6(4-dimethylaminostyryl)-4 H-pyran; supplied by Radiant Dyes). We prepared four samples

S1, . . . , S4 with decreasing concentrations of the chiral dopant (4.04, 3.39, 1.92, 0.83 weight%); for all samples, the dye concentration was about 0.5 weight-%. The dye-doped mixtures were lled in commercially available test cells (E.H.C. Ltd., Japan; nominal lm thickness 15 µm) by capillary action. The cell substrates are coated with an Indium-tin-oxide (ITO) layers and alignment layers enforcing homogeneous planar surface alignment of the director. After lling, the cells were sealed with glue. In the setup, the surface alignment of the CLC lms was horizontal (i.e., in the rotation plane of the detector).

Simulations For transmission simulations and for the calculation of the dispersion relations we used the

4 × 4 matrix method of Berreman. 28 It is a numerical method to simulate propagation of light in inhomogeneous media where the optical properties change only along the z direction (in our case, the CLC's helical axis). The method deals with waves generated by plane waves incident in the xz plane from an isotropic surrounding medium through an interface perpendicular to the z axis. Inside the inhomogeneous medium, the solutions are of the form ψ(z) exp [i(kx x − ωt)], where the four-component vector ψ(z) contains the x and y components of the electric and magnetic elds. At the core of the method lies the computation of the so-called propagation matrix F(z, s), which describes the relation between the electromagnetic elds at dierent positions along the z direction, ψ(z + s) = F(z, s)ψ(z). 8


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In our simulations, the repetition unit (corresponding to a half-turn of the director) of the CLC was approximated by a twisted stack of 200 birefringent layers. In the simulations, the refractive indices of the CLC and of the index matching uid were assumed to be wavelengthindependent. Including dispersion would only slightly aect the locations and widths of the reection bands in the transmission simulations, and would slightly aect the shape of the dispersion relations, without changing the overall patterns.

Photonic band structure Before presenting the experimental results, it is useful to briey recall the angular-dependent photonic properties of CLCs. Figures 3a,b show angular- and wavelength-dependent transmittance simulations for right- and lefthanded (RCP, LCP) illumination. Simulations were performed using the 4 × 4 method of Berreman, 28 assuming a CLC lm with 12 director turns along the lm normal, refractive indices no = 1.5 and ne = 1.8, and the surrounding medium has refractive index 1.4688 (which is the refractive index of glycerol 29 the index matching uid in our experimentsfor λ = 650 nm). The RCP reection band clearly shows up (transmission close to zero), and both continuously blueshifts and broadens with increasing angle of incidence. At large angles, gradually a narrow complete reection band emerges (almost zero transmission for both RCP and LCP illumination). At large angles, there is obviously a deviation from circular polarization for the selective reection band, as broad wavelength intervals of reduced transmission for both RCP and LCP illumination emerge. Higher order reection bands are absent for normal incidence, and they aect the transmittance only at large angles of incidence. The second-order band consists of three branches and is visualized in Figures 3d,e, which show the simulated transmittance of a CLC lm at large angles and short wavelengths, obtained for vertical and horizontal polarization (H and V) of the incident wave. The branches are labeled S2 , M2 and L2 (short-wavelength, middle, and long-wavelength branch). The three branches show dierent polarization behavior: The

9



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outer branches are sensitive to H polarization (S2 ) and V polarization (L2 ), respectively, while the middle branch (M2 ) shows up for both V and H polarized illumination. For large angles, there is an overlap of the rst-order short-wavelength branch (H polarized, labeled

S1 in Figure 3d) with the V polarized second-order branch L2 . Discussion of the eigenmodes of the cholesteric medium gives a more detailed insight into the angular-dependent photonic properties. A plane wave incident on the CLC can excite Bloch-like optical modes inside the CLC,

E(x, z) = exp(ikx x) × u(z) exp(ikz z)

(1)

where the plane of propagation is the xz plane, the z direction is parallel to the helical axis, and u(z) has the same periodicity as the CLC, i.e. u(z + p/2) = u(z). The dispersion relations of the eigenmodes has already been studied long ago by Sugita et al., 30 and in the following we briey recall their ndings. The dispersion relations are obtained by diagonalization of the Berreman propagation matrix F(0, p/2) (see

Methods for the denition

of F). Diagonalization yields four solutions for the eigenmodes: two pairs propagating (or decaying) along the +z and −z direction, respectively. Figure 3c shows dispersion relations of the eigenmodes for dierent values of kx , in the region of the rst-order reection band. The angles labeling the data correspond to the angle of incidence of a plane wave exciting the eigenmodes (assuming the refractive index n of the surrounding medium matches the average refractive index

n ¯=

p (n2o + n2e )/2

(2)

of the CLC). Depending on frequency and angle of incidence, the kz values are either real (undamped propagation), imaginary (evanescent wave, i.e. photonic band gap), or complex (real and imaginary part are both non-zero, i.e. damped propagation due to reection). For normal incidence (Figure 3c0◦ ) one eigenmode shows a photonic band gap (the 10


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Figure 3: Angular-dependent photonic properties of a CLC: (a,b) Simulated angular dependent transmittance of a CLC lm with right-handed helix, for left- and righthanded circularly polarized (LCP, RCP) illumination. (c) Dispersion relations of the Bloch-like eigenmodes in the region of the rst-order reection band, labeled with angles of incidence θ = 0◦ , 40◦ , 48◦ , 57◦ (see text). (d,e) Simulated angular dependent transmittance for horizontally and vertically polarized (H, V) illumination in the region of the second-order reection band. (f) Dispersion relations in the region of the second-order band, labeled with angles of incidence θ = 60◦ , 61◦ , 62.5◦ . In (c-f), the three branches of the rst- and second-order bands are labeled S1,2 , M1,2 , L1,2 (short-wavelength, middle, and large-wavelength branches). The dierently shaded areas in (c) and (f) indicate frequency intervals with dierent propagation behavior of the eigenmodes undamped propagation (kz real), evanescent wave (i.e. band gap, kz imaginary), and damped propagation due to reection (kz complex). The relative quantities λrel = λ/(¯ np), ωrel = n ¯ p/λ and kz,rel = kz /p are used (λ is the vacuum wavelength). 11



well-known selective reection band), while the other mode is propagating. At non-zero angles of incidence (or kx values), additionally a narrow polarization-independent band gap emerges close to the center of the selective reection band (Figure 3c40◦ ). Increasing the angle further, the polarization-independent band gap broadens, and eventually splits into two band gaps, separated by a frequency interval of attenuated propagation of both eigenmodes (Figure 3c48◦ ). At still larger angles, the reection band splits into three branches (Figure 3c57◦ ): a polarization-independent band with attenuated propagation, framed by polarization-sensitive band gaps; the three branches are separated by frequency intervals, where all eigenmodes propagate without attenuation. Figure 3f shows the angular-dependent behavior of the second-order band. It has a triplet structure with a central band of polarization-independent attenuated propagation, and outer polarization-sensitive band gaps (Figure 360◦ ). The three branches are well separated by frequency intervals where all modes propagate without attenuation. At large angles an overlap of the second-order long-wavelength branch with the rst-order short-wavelength branch occurs, resulting in a polarization-independent band gap (Figure 3f61◦ , 62.5◦ ).

Spontaneous emission Eect of the rst-order reection band Figure 4 shows the angular-dependent spontaneous emission of samples S1, S2 and S3 (with small, medium, and large cholesteric pitch, respectively), decomposed into left- and righthanded circularly polarized emission contributions. For each sample, the dye's emission spectrum (which would be smooth in an ordinary, isotropic solvent) is modied by the cholesteric medium in certain angular intervalswith increasing cholesteric pitch, we observe an overlap of the CLC's reection band with the natural emission spectrum of the dye at larger detection angles θ. Sample S1, having the smallest pitch, shows good overlap of the selective reection band 12


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Figure 4: Angular-dependent spontaneous emission, modied by the rst-order reection band: Left- and righthanded circularly polarized (LCP, RCP) emission contributions, obtained for the samples S1 (a,d), S2 (b,e), and S3 (c,f). Emission spectra were recorded in steps of 1◦ . The RCP and LCP plots for each sample have the same relative intensity scale. (gi) Selected emission spectra of the three samples; the vertical dashed lines indicate the location of band edges, the two insets in (i) show magnied sections of the spectra in the vicinity of the long-wavelength band edge.

13



with the dye's natural emission spectrum for low detection angles. The photonic band gap for right-handed circularly polarized light results in almost complete suppression of RCP emission, and the blueshift of the band gap with increasing detection angle clearly shows up in the angular-dependent RCP emission measurement (Figure 4d); on the other hand, the LCP emission (Figure 4a) is overall only little aected by the cholesteric medium. Exemplary emission spectra of sample S1 for three detection angles are shown in Figure 4g. Emission along the lm normal (θ = 0◦ ) shows the behavior well known from previous studies: 13 The LCP emission contribution (handedness of polarization opposite to that of the cholesteric helix) is hardly aected by the cholesteric medium and essentially resembles the emission spectrum of the dye in an optically isotropic solvent. On the other hand, the RCP spectrum is strongly modied by the presence of the photonic band gap: Inside the gap, which is centered at λ = 619 nm, emission is almost zero. At the long-wavelength band edge (λ = 664 nm) emission is strongly enhanced with respect to the LCP emission, due to the presence of band edge resonances. Close to the short-wavelength band edge (λ = 574 nm), RCP emission is suppressed with respect to the LCP emission. This can be attributed to enhanced re-absorption inside the sample. With increasing detection angle, there is a continuous blueshift of the band gap. The long-wavelength band edge resonance also slightly aects the LCP emission spectra, which show a weak emission maximum coinciding with the RCP band edge peak (see the example spectra in Figure 4g, obtained at 28◦ and 34◦ ). This can be attributed to the refractive index mismatch at the CLC/substrate interface (mainly due to the presence of the ITO layer), which results in (multiple) reections of the emitted light, where each reection changes polarization handedness. For angles larger than 43◦ , the band gap is blueshifted beyond the excitation wavelength, and the smooth spectra (both for LCP and RCP detection) essentially resemble the dye's emission in an ordinary, isotropic solvent (cf. Figures 4a,d). In case of sample S2 (medium pitch), the photonic band structure overlaps with the dye's emission range in the angular interval between about 30◦ and 70◦ (cf. Figures 4b,e). 14


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While the RCP emission contribution still shows a broad emission minimum (similar to sample S1 at lower detection angles θ), with increasing detection angle LCP emission shows striking modications: we clearly detect the emergence of a narrow LCP emission gap, located roughly in the center of the broad emission band gap; additionally, traces of the RCP band edge peaks become more clearly visible in the LCP emission contribution. As can be seen from the sample spectra in Figure 4h, the LCP band gap becomes more pronounced with increasing detection angle, the emission peaks framing the LCP emission minimum clearly indicating the presence of band edge resonances. With increasing detection angle, the notion of LCP and RCP band gaps becomes less and less justied: While for detection angles close to 0◦ , the selective reection band produces a broad, at wavelength interval with almost zero RCP emission (cf. Figure 4g), and hardly aects LCP emission, the two nested band gaps existing for oblique propagation aect to some degree both circularly polarized emission contributions. Sample S3, with even larger pitch, allows us to probe the CLC's photonic band structure at even larger angles. Figures 4c,f reveal modied emissionfor both RCP and LCP detectionfor detection angles larger than about 55◦ . At rst glance, the LCP data show one band of reduced emission, while the RCP data show two of them. As can be seen in the example spectrum for θ = 64◦ in Figure 4i, the pronounced LCP emission gap coincides with a RCP emission gap, resulting in a wavelength interval of almost complete emission suppression (between about 644 nm and 697 nm). Obviously, this is the center branch of the rst-order reection band; In Figure 4i66◦ , the center branch is slightly blueshifted, and framed by very pronounced emission peaks. The sharp drop in RCP emission in Figures 4i64◦ , 76◦ at short wavelengths (λ ≈ 582 nm and λ ≈ 554 nm, respectively) marks the short-wavelength edge of the rst-order short-wavelength branch. Spectra obtained at large angles reveal an additional weak emission peak at very large wavelengths (cf. insets in Figure 4i66◦ , 72◦ ), in the fringe region of the dye's emission range. This is obviously the long-wavelength band edge resonance of the long-wavelength branch of the reection band. 15



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Therefore, the results for samples S2 and S3 are qualitatively similar, however for sample S3, which allows us to probe the photonic band structure at larger detection angles, the rst-order band is much wider, and the central branch of polarization-independent emission suppression is fully evolved. This results in angular intervals, where emission spectra are almost completely dominated by a single emission peak (e.g. the RCP peak at 601 nm in the spectrum obtained at θ = 72◦ , cf. Figure 4i): In a previous study 13 we presented a quantitative model for spontaneous emission along the cholesteric helix, assuming that the total emission is the incoherent sum of emission into the two normal modes propagating along the detection direction. The two emission contributions are proportional to the photonic densities of states (DOS) and the average projections of the dye's transition dipole moment for emission on the polarization of the respective eigenmode. For propagation along the helical axis, these quantities can be calculated analytically, using the classic results of de Vries for light propagation in CLCs 31 andfor the calculation of the DOSusing the approach of Bendickson et al. 32 This model yielded an excellent quantitative description of emission along the cholesteric helix. Therefore, it seems promising to consider the emission at oblique angles as an incoherent sum of contributions of two normal modes as well. In order to decompose the total emission into normal modes, we performed a Stokes analysis of the emitted light. The starting point for analysis is the coherency matrix 33

 J=

hEH EH∗ i

hEH EV∗ i



1    2 hEV EH∗ i hEV EV∗ i

(3)

where EH,V are the horizontally and vertically polarized electric eld components and the brackets h. . . i denote the temporal average. The sum of the diagonal elements is the total intensity; if the H and V polarized eld contributions are uncorrelated, the o-diagonal elements vanish and we have incoherent superposition of H and V polarized radiation. Ex-

16


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perimentally, the coherency matrix can be constructed using the Stokes parameters

        

s0 = 1 s1 = (IH − IV )/(IH + IV )

(4)

s2 = (I+ − I− )/(I+ + I− ) s3

       = (IRCP − ILCP )/(IRCP + ILCP ) 

where IV,H and I+,− are the linearly polarized emission contributions with vertical, horizontal, ±45◦ orientation of polarization, and IRCP,LCP are the right- and lefthanded circularly polarized contributions (we consider the relative quantities s0,...,3 , as the detection sensitivities for the three polarization pairs {H, V}, {+, −}, {RCP, LCP} are slightly dierent). The normalized coherency matrix in terms of the normalized Stokes parameters reads

 J=



1  s0 + s1 s2 + is3    2 s2 − is3 s0 − s1

(5)

The decomposition of the emitted light into an incoherent sum of two normal modes with orhogonal polarizations now simply corresponds to the diagonalization of the experimentally determined coherency matrix: 34 the eigenvalues are the (relative) intensities of the two normal modes, and the eigenvectors are the Jones vectors characterizing their respective polarization. From the Jones vectors full information on the polarization ellipses can be extractedshape, orientation, and handedness (see e.g. the work of Dennis 35 ). Due to their orthogonality, the two polarizations have opposite handedness, the two polarization ellipses have the same ellipticity and their principal axes are rotated by 90◦ with respect to each other. For samples S1, S2 and S3 we performed a complete Stokes analysis of the emitted light in the respective angular intervals where the rst-order reection band traverses the dye's emission spectrum, and performed the decomposition into an incoherent pair of emission contributions. The results are expressed in terms of the relative intensities aR,L of the contributions with left- and righthanded polarization (aR + aL = 1) and two parameters 17



describing the shapes and orientations of the polarization ellipses (cf. Figure 5): The orientation of the polarization ellipses is characterized by the angles γR,L between the major axis and the H direction (−90◦ ≤ γR,L ≤ 90◦ , and |γR − γL | = 90◦ ). The shape factor b describes the ellipticity of the polarization: b is the relative length of the major half axis (0.5 ≤ b ≤ 1),

1 − b is the relative length of the minor half axis; values b = 0.5 and b = 1 correspond to circular and linear polarizations, respectively.

Figure 5: Parameters b and γ characterizing the shape and orientation of the polarization ellipses. The angular- and wavelength-resolved Stokes analysis for sample S1 in the angular range

0◦ ≤ θ ≤ 35◦ is shown in Figure 6a. Apart from the parameters aR,L , b, γL , also the total intensities IR,L of the two normal modes is shown, the indices R and L referring to the rightand lefthanded polarized mode, respectively. The selective reective band clearly shows up in the relative intensity contributions aR and aL , the band edges marked by very sharp jumps in relative intensity. The intensity data of the normal mode IL with left-handed polarization are smooth and featureless, resembling emission in an isotropic solvent, and the selective reection band almost completely suppresses the right-handed polarized contribution IR obviously, the decomposition into normal modes via Stokes analysis of the emitted light indeed corresponds to decomposition into the eigenmodes of the CLC. For angles up to about 20◦ , the normal modes have roughly circular polarization (shape factor b close to 0.5), while for larger angles drastic polarization variations emerge. The polarization ellipses are overall oriented roughly vertically or horizontally, with substantial tilts only occurring close to the band edges. Results for two selected detection angles are shown in Figure 6b. For θ = 0◦ , the results 18


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Figure 6: (a) Decomposition of the emission of sample S1 into normal modes: relative intensity contributions aR,L (top row) and total emission contributions IR,L (middle row) of the two normal modes, shape parameter b of polarization ellipses (bottom left), and orientation angle γL of the polarization ellipse of the eigenmode with left-handed polarization (bottom right). (b) Decomposition into normal modes at two angles: θ = 0◦ (left column) and θ = 25◦ (right column).

19



for IR and IL closely resemble the results for IRCP and ILCP in Figure 4g, as the normal modes have almost circular polarizations (b close to 0.5). The tilt angles γR,L show strong variations in the vicinity of the band edges, while theoretically 31 the polarization ellipses should be aligned with the director at the lm interface. This discrepancy can be explained by the refractive index mismatch of the CLC/substrate interface which results in a small coupling of the eigenmodes upon reection at the interface. As the polarizations of the eigenmodes are almost circular, a small coherent contribution of the respective mode with opposite polarization handedness can strongly aect the orientation angles γR,L . The data obtained for θ = 25◦ , also shown in Figure 6b, reveal strong variations of the polarization shape (parameter b) for wavelengths beyond the long-wavelength band edge, while inside the selective reection band, the polarizations are still roughly circular. The sharp peak of b at about 618 nm has no physical meaning: here, the intensities of the two normal modes intersect, which means that the emission is completely unpolarized, and therefore the decomposition into normal modes fails. The orientation angles γR,L suggest that the polarization ellipses are roughly aligned along the director at the lm surface, except for the vicinity of the long-wavelength band edge. However, here b approaches 0.5 (circular polarization), where the orientation angle is ill-dened. The normal mode decomposition of sample S2's emission is shown in Figure 7a, again in the angular interval where the CLC's rst-order reection band structure overlaps with the emission of the dye (in the gray areas, Stokes analysis fails due to low intensities). The nested pair of polarization sensitive band gaps clearly shows up as a narrow dark band in the intensity IL , and a broad band of almost zero emission IR , respectively. We observe strong angular- and wavelength-dependent variations of both the polarization shape factor and orientation angle. The shape of the polarization ellipse shows strong wavelength-dependent oscillations for short wavelengths, close to the short-wavelength edge of the band gap of the R polarized mode (cf. Figure 7b, data for θ = 43◦ ). The oscillations correspond to the interference fringes of the emission contributions IR,L , and are therefore probably due 20


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Figure 7: (a) Decomposition of the emission of sample S2 into normal modes: relative intensity contributions aR,L (rst column) and total emission contributions IR,L (second column) of the two normal modes, shape parameter b of polarization ellipses (top right), and orientation angle γL of the polarization ellipse of the eigenmode with left-handed polarization (bottom right). (b) Decomposition into normal modes at three angles, θ = 43◦ , 52◦ , 59◦ (from left to right).

21



to strong coupling of the CLC's eigenmodes due to multiple reections in the sample lm. Correspondingly, these oscillations are absent for wavelengths inside the wide band gap of the R polarized mode (cf. data in Figure 7b, obtained at three dierent angles θ). There are also no fringes in the b data at large wavelengths, beyond the long-wavelength band edge of the R polarized mode. Here polarizations are roughly linear (b ≈ 1), which reduces the coupling due to multiple reections. At the band edges of the R polarized mode, there are drastic changes in the shape of the polarization ellipses (in Figure 7b, the shape parameter b shows a kink in the data for θ = 43◦ , and a sharp rise in the data for θ = 59◦ , respectively). Inside the broad, R polarized band gap, there is a smooth variation of the polarization shape, without kinks or discontinuities occurring at the band edges of the nested, narrow L polarized band gap. In the angular region probed by sample S2, the polarization ellipses are in general tilted with respect to the director at the lm surface. Their orientations overall show a smooth wavelength-dependent variation; for large wavelengths, beyond the long-wavelength band edge of the R polarized mode, the (roughly linearly) polarized normal modes are essentially aligned parallel and perpendicular to the director at the lm surface, respectively. For wavelengths, where the polarizations are almost linear (b ≈ 1), the assignment of the two normal modes as being left- or righthanded polarized becomes ambiguous, which results in ips of the data between the L and R labelled curves for relative intensity, total intensity, and orientation angle. In Figure 7b, this happens for the data obtained at 43◦ in the short-wavelength region of strongly oscillating b, and in the data sets for the other two angles at large wavelengths, beyond the long-wavelength band edge of the R polarized mode. The normal mode decomposition of sample S3's emission at large detection angles is shown in Figure 8. At a rst glance, the angular-dependent behavior of the intensities IR,L is dicult to interpret, and again we nd strong wavelength- and angular-dependent variations of both the shape factor and orientation of the polarization ellipses (cf. Figure 8a). Figure 8b shows selected data obtained at four dierent angles, where dierent sections of the rstorder band overlap with the emission of the dye. For θ = 60◦ , emission is mainly aected by 22


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Figure 8: (a) Decomposition of the emission of sample S3 into normal modes: relative intensity contributions aR,L (rst column) and total emission contributions IR,L (middle column) of the two normal modes, shape parameter b of polarization ellipses (top right), and orientation angle γL of the polarization ellipse of the eigenmode with left-handed polarization (bottom right). (b) Decomposition into normal modes at four angles, θ = 60◦ , 63◦ , 68◦ , 75◦ (from left to right); the three branches of the rst-order band are marked by S1 , M1 and L1 , .

23



the small-wavelength branch. Here, the behavior for short wavelengths (strong oscillations of the shape factor and of the orientation angle close to the short-wavelength band edge), is similar to the results for sample S2 at lower detection angles (cf. Figure 7b43◦ ). Inside this branch, with increasing wavelength the polarizations of the normal modes continuously change from circular to linear. In the data for θ = 63◦ , the normal modes show a reversal of the handedness of their respective polarizations at λ = 652 nm (this is indicated by the color ip of the IL,R and

γL,R curves); the reversal of handedness happens via a continuous transition through linearly polarized states (b = 1). In the data obtained for θ = 68◦ , the same reversal occurs at

λ = 575 nm. In the long-wavelength branch of the reection band, another continuous reversal of the polarization handedness of the two normal modes occurs (in the data for

θ = 68◦ and θ = 75◦ , at λ ≈ 700 nm and λ ≈ 647 nm, respectively). We also observe the splitting of the reection band into a triplet of well-separated branches, predicted by the band structure calculations: In the data for θ = 63◦ , there is a narrow wavelength interval between the branches S1 (band gap for the R polarized mode) and M1 (emission suppression for both modes), where both modes contribute to emission. For θ = 63◦ , this results only in a small L polarized emission peak at 661.5 nm; increasing the detection angle, the distance between the branches S1 and M1 is predicted to increase. Indeed, the data for θ = 68◦ show pronounced L polarized emission in the wavelength interval

581 nm < λ < 596 nm separating the branches S1 and M1 . The band structure calculations predict a very small wavelength interval between the branches M1 and L1 (cf. Figure 3c, data for 57◦ ). Therefore, the corresponding L polarized emission peaks in the data for θ = 68◦ and

θ = 75◦ (at λ = 649 nm and λ = 573 nm, respectively) are very weak. To our knowledge, our uorescence measurements are the rst experimental proof of the triplet structure of the rst-order band at large angles (predicted 1982 in the band structure calculations of Sugita et al. 30 ).

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Eect of the second-order reection band

Figure 9: (a,d) Angular-dependent spontaneous emission of sample S4: (a) Left- and righthanded circularly polarized emission contributions, recorded in steps of 1◦ . (b,e) Linearly polarized emission contribution for large detection angles θ, in the region of the second order reection band (recorded in angular steps of 0.2◦ ). (c) Sample spectra obtained at detection angles θ = 68◦ , 74◦ , 81◦ , 85◦ . In (b,c,e), the branches of the second-order reection band are indicated (S2 , M2 , L2 ), as well as the short-wavelength branch of the rst-order band (S1 ). In order to study the eect of the second-order reection band on the emission properties, we prepared sample S4 with very low concentration of chiral dopant. The angular scan of RCP and LCP emission contributions (Figures 9a,d) is featureless in a wide angular range (apart from interference ripples), and only shows modications for θ ≥ 64◦ . A high-resolution scan for H and V emission contributions for large detection angles is shown in Figures 9b,d. Here, obviously the eect of the second-order reection band on emission is capturedthe bands with reduced emission closely resemble the pattern of reection bands in the simulated transmittance data shown in Figures 3d,e. Figure 9c shows selected spectra obtained at dierent detection angles, illustrating the modied emission due to the dierent branches of the second-order band: For θ = 68◦ , we observe a weak and narrow emission dip in H 25



Figure 10: (a) Decomposition of the emission of sample S4 into normal modes: relative intensity contributions aR,L (rst column) and total emission contributions IR,L (middle column) of the two normal modes, shape parameter b of polarization ellipses (top right), and orientation angle γL of polarization ellipse of the eigenmode with left-handed polarization (bottom right). (b) Decomposition into normal modes at four angles: θ = 67.5◦ , 71.5◦ , 81◦ , 83◦ (from left to right).

26


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emission, due to the short-wavelength branch of the second-order band. For θ = 74◦ , the middle branch clearly shows up for both polarizations. For θ = 81◦ , the long-wavelength branch strongly aects the V polarized emission contribution. For very large detection angles (cf. spectra for θ = 85◦ in Figure 9c), we observe the overlap of the second-order long-wavelength branch (gap in V polarized emission) and the rst-order short-wavelength branch (gap in H polarized emission). The normal mode decomposition of sample S4's emission is shown in Figure 10. Although the wavelength and angular dependence of the intensities IR,L (Figure 10a) closely resembles the high-resolution scan of H and V emission contributions shown in Figure 9a, the normal modes are overall not exactly linearly polarized. The alignment of the polarization ellipses is mostly along the V (or H) direction, except for wavelengths inside the middle branch (M2 ). There, some substantial tilt occurs (cf. data for θ = 71.5◦ in Figure 10b). In the data for

θ = 67.5◦ and 71.5◦ , shown in Figure 10b, the shape factor shows strong oscillations in the wavelength intervals separating the branches. For larger angles, the data become very noisy, because of the low intensities and multiple crossings of the intensity contributions IR,L .

Conclusions In conclusion, uorescent guest molecules are a sensitive probe of the complex polarizationdependent photonic band structure of cholesteric liquid crystals. In the angular intervals which provide overlap of the CLC's reection band with the emission range of the dye, we found strong variations of both emission intensity and emission polarization. Stokes analysis of the emitted light proofs the triplet structure of the rst order band at large angles. Using a sample with very large cholesteric pitch, we were able to study the eect of the second-order reection band, and obtained experimental proof for its partial overlap with the st-order band at large angles. In a previous study, 13 emission along the helical axis could be quantitatively described

27



assuming the emitted intensity to be an incoherent superposition of the two eigenmodes propagating along the lm normal. Assuming index matching of the surrounding medium, the normal modes as probed by emitters inside the lm are the same as those probed by light from an external source transmitted through the lm. For oblique angles, one can similarly expect a strong correspondence of the normal modes obtained by an emission measurement and the normal modes obtained by a transmission measurement. However, to our knowledge the latter haven't been determined experimentally so far. Apart from an experimental test, an alternative check of this correspondence would be a generalization of the model of our previous study 13 for oblique emission. However, modelling emission at oblique angles is more dicult, for two reasons: For propagation along the axis, the normal modes have constant polarizations (in a coordinate frame co-rotating with the director); this is not the case for oblique propagation (expressed by the periodic function u(z) in eq. 1). This results in a spatial variation of the coupling of the emitters to the normal modes. Modelling would require spatial (plus orientational) averaging of the projection of the dye's transition dipole moment on the polarizations of the normal modes. A second complication concerns the index matching of the surrounding medium: In case of emission along the helical axis, index matching prevents coupling of the eigenmodes, which would otherwise occur due to reection at the lm interface. Apparently, for oblique propagation no simple index matching condition exists, which would prevent coupling of the eigenmodes. This of course is also an argument against the popular method to derive the density of states in cholesteric lms (or CLC multilayer systems) from the transmission matrix, obtained e.g. by the Berreman method (for oblique transmission, the eigenpolariations of the lm correspond to a superposition of both eigenmodes inside the lm).

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Graphical TOC Entry

cholesteric liquid crystal angular-dependent emission

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Angular-Dependent Spontaneous Emission in Cholesteric Liquid

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