J.Y. Wang, J. Q. Wei, H. R. Xia, Y. G. Liu, and. M. H. Jiang, Cryst. Res. Technol. ... G. Gadret, J. Mangin, A. Brenier, G. Boulon, G. Aka, and D. Pelenc, Opt. Mater.
ISSN 1054660X, Laser Physics, 2011, Vol. 21, No. 7, pp. 1305–1312.
PHYSICS OF LASERS
© Pleiades Publishing, Ltd., 2011. Original Text © Astro, Ltd., 2011.
Refined Modeling of Angular Distributions of Linear Absorption and Fluorescence in Biaxial Crystals1 Y. Petita, b, *, S. Jolyb, P. Segondsb, and B. Boulangerb a
ICMCB–CNRS–UPR 9048: 87, Avenue du Docteur Schweitzer, 33608 PESSAC Cedex, France b Institut Néel CNRS/UJF, 25 rue des Martyrs, BP 166, F38402 Grenoble Cedex 9, France *email: Yannick.Petit@icmcbbordeaux.cnrs.fr Received October 31, 2010; in final form, November 14, 2010; published online June 4, 2011
Abstract—We provide a deeply detailed description of linear absorption and fluorescence angular distribu tions in biaxial media; we properly take into account the distributions of the polarization eigenmode vectors around the principal plane of the dielectric frame that contains the optical singularities of the optical axes. By introducing a nonzero angular aperture of experimental detection setups and a related angular integration of absorption and fluorescence angular distributions, we provide for the first time a complete interpretation and refined modeling of former experimental measurements. Such developments should be considered to perform relevant metrology of lowsymmetry biaxial materials for optics. DOI: 10.1134/S1054660X11130214 1
1. INTRODUCTION
Ongoing progress in crystal growth science still leads to innovating materials for optics, either with mature or promising potentials for applications such as laser and selfOPO devices [1, 2] or scintillation detectors based on photoinduced conductivity [3]. Among these materials, many of them are lowsym metry crystals, from the biaxial optical class, either from the orthorhombic or the monoclinic crystal sys tems. Before being able to take advantage of an opti mal exploitation of these new anisotropic crystals, one must perform relevant and exhaustive characteriza tions of their optical properties as linear absorption and fluorescence emission. Indeed, specific approaches are to be considered for such lowsymme try media, from methodological, metrological and theoretical points of view so as to define the best direc tions of propagation to get an optimal exploitation of absorption, fluorescence emission, or a combination of these two properties. Most of these approaches have already been discussed in our previous works, but a refined description is still necessary to interpret the fine structure of some former experimental measure ments of angular distributions of fluorescence in the monoclinic biaxial YCOB:Nd crystal. Recently, we have measured for the first time the angular distribution of linear absorption and sponta neous fluorescence of rare earth ions inserted in a biaxial laser host matrix, namely the YCOB crystal doped with Nd3+ ions [4]. These angular distributions give direct experimental access to those of the imagi nary part of the optical index n'. They strongly differ from the wellknown distributions of the real refractive index n in the principal planes of the dielectric frame, 1 The article is published in the original.
but also out of these planes [4, 5]. We could model these distributions with new analytical expressions in the principal planes, and with threedimensional numerical resolution of Maxwell equation of propaga tion out of these planes, in the case of lowsymmetry media [4–6]. These works showed that the symmetry elements of the absorption and the fluorescence angu lar distributions differ from the principal axes of the dielectric frame, and also from one to the other: this led us to introduce two new frames, namely the absorption frame and the fluorescence frame, whose principal axes correspond to the symmetry elements of linear absorption and fluorescence, respectively, and in which one can reach and thus exploit the optimal behavior of each of these optical properties [4]. We also showed an unexpected behavior specific to biaxial media, revealing the existence of lineic loci where absorption or fluorescence properties are polariza tionindependent with respect to the selected eigen mode of polarization [5]. In fact, further numerical work showed that such loci correspond to the geomet rical intersection of the two layers that depict the dou blelayer surface of the imaginary index n', leading to layerinversion zones with topology and symmetry properties strongly depending on the considered biax ial medium [6]. Finally, we recently reported for the first time the experimental study of the strong depen dence and rotation of the orientation of the absorption frame for seven resonant wavelengths of Nd3+ inserted in the monoclinic YCOB crystal matrix, providing a new inset into the richness of behavior of lowsymme try laser media [7]. In this proceeding paper, we provide new develop ments that afford to fully interpret and understand the influence of the optical singularities, along the optical axes, on the angular distributions of linear absorption
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or fluorescence in biaxial media. Firstly, we depict in detail the absorption and fluorescence angular distri butions in the principal plane of the dielectric frame that contains the optical singularities, i.e. the plane with the four umbilici: such angular distributions are described in relation either with the angular distribu tions of the ordinary or extraordinary refractive index sections or with those of the internal or external sheets of the wellknown surface of refractive indices. Sec ondly, we take into account the influence of the optical singularities from both parts of this considered princi pal plane. We numerically show for the first time the qualitative and quantitative difference of fluorescence angular distributions while one considers one single direction of propagation in the studied medium, or an angular integration along the propagation directions of a reallife beam, i.e. a diverging beam with a non zero angular aperture. Finally, these numerical con siderations lead us to perform a refined modeling of former experimental measurements of fluorescence angular distributions with the monoclinic YCOB:Nd laser crystal. These developments provide new under standing so as to realize relevant metrology of biaxial promising materials for optics applications. 2. THEORETICAL DESCRIPTION OF LINEAR ABSORPTION AND FLUORESCENCE ANGULAR DISTRIBUTIONS IN BIAXIAL MEDIA FOR UNIDIRECTIONAL BEAM PROPAGATIONS Linear optical properties of anisotropic laser mate rials are depicted by the angular distribution of the complex optical index nˆ (θ, ϕ) for any direction of propagation reported by the spherical angular coordi nates (θ, ϕ). This distribution nˆ (θ, ϕ) is obtained by solving the wave propagation equation, i.e., the fol lowing Maxell equation for the considered lightmat ter interaction [8]: 2 –1 nˆ ( θ, ϕ ) ( u ( θ, ϕ ) × u ( θ, ϕ ) × E ) + εˆ ε 0 E = 0 , (1)
where E is the propagating electric field, u(θ, ϕ) is the unit wave vector along the (θ, ϕ) direction of wave propagation, × is the vectorial product, ε0 is the vac uum dielectric permittivity, εˆ is the complex linear dielectric permittivity tensor that characterizes both linear propagation and linear spectroscopic properties as linear absorption or single photon fluorescence. The threedimensional resolution of Eq. (1) leads ± to two solutions nˆ (θ, ϕ) = n±(θ, ϕ) + jn'±(θ, ϕ) in the dielectric frame (X, Y, Z), where the real part n±(θ, ϕ) and the imaginary part n'±(θ, ϕ) undergo the following relations for any direction of propagation: by defini tion, n+(θ, ϕ) > n–(θ, ϕ); on the contrary, n'+(θ, ϕ) = + Im( nˆ (θ, ϕ)) = n n' + ( θ, ϕ ) (θ, ϕ) and n'–(θ, ϕ) =
–
Im( nˆ (θ, ϕ)) = n n' – ( θ, ϕ ) (θ, ϕ) do not follow systematic relation of order as already demonstrated [6]. In lowsymmetry laser materials as biaxial media, Eq. (1) is solved in the dielectric frame (X, Y, Z), where the complex linear dielectric permittivity tensor εˆ = ε + jε' is defined by the following secondrank polar tensor [8, 9]:
εˆ =
ε xx 0 0
ε 'xx ε 'xy ε 'xz
0 ε yy 0 + j ε 'yx ε 'yy ε 'yz 0 0 ε zz ε 'zx ε 'zy ε 'zz
(2)
where εxx, εyy, and εzz are the three principal values of the realpart tensor ε, which are related to the three principal refractive indices of the biaxial media: they verify ni = ε ii /ε 0 with i = x, y, z, with the convention nx < ny < nz. Note that by definition, this realpart ten sor ε is systematically diagonal in the dielectric frame of any biaxial crystal system. ε 'xx , ε 'yy , and ε 'zz are the diagonal values of the imaginarypart tensor ε'. Extra diagonal elements of ε' verify ε 'ij = ε 'ji , with i, j = x, y, or z [8, 9]: the extradiagonal elements are either all null elements or all nonzero elements in the case of orthorhombic or triclinic crystal systems, respectively. For monoclinic crystal systems, only one single pair ε 'ij = ε 'ji of extradiagonal elements is nonzero: as ' = ε zx ' ≠0 recently reported, we demonstrated that ε xz for symmetry orientation classes similar to that of YCOB:Nd [4]. Moreover, in standard conditions for laser materials, one can assume the low absorption hypothesis or its low fluorescence emission counter part, meaning in any case that εii Ⰷ ε 'jk with i, j, k = x, y or z [4, 6, 8]. While projected along the three principal axes of the dielectric frame (X, Y, Z), Eq. (1) provides a con tracted description of the following complex linear system of three coupled equations described in [4–6], whose threedimensional resolution leads to the determination of its two eigen values, namely the two ± complex optical indices nˆ (θ, ϕ): –1
2
2
( εˆ ij ε 0 + nˆ ( θ, ϕ ) ( u i u j ( 1 – δ ij ) – ( 1 – u i )δ ij ) )E j = 0,
(3) where i, j = x, y or z; Ei and ui stand for Cartesian coor dinates of the electric field E and those of the unit wave vector u in the dielectric frame (X, Y, Z), with ux = sin(θ)cos(ϕ), uy = sin(θ)sin(ϕ), and uz = cos(θ); δij stands for the Kroneker symbol such as δij = 1 when i = j, and δij = 0 when i ≠ j. Consequently, in all biaxial crystal systems, the refractive index surface n±(θ, ϕ) is composed of a two layer surface whose layers show four punctual contacts LASER PHYSICS
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in the XZ principal plane along the optical axes. At these four umbilici, one satisfies n+(θ = Vz, ϕ = 0°) = n–(θ = Vz, ϕ = 0°), where Vz = –2
–2
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–2
arcsin ( ( n x – n y )/ ( n z – n y ) ) is counted from the Z dielectric axis in the XZ plane [10]. Moreover, for any direction of propagation in biax ± ial media, the two optical indices nˆ (θ, ϕ) are associ ated to the two unit vectors that depict the eigenmode vectors of the electric field polarization, resulting from the following coupled linear system [10]: ±
2
± ± n ( θ, ϕ ) e i ( θ, ϕ ) = u i ( u ⋅ e ( θ, ϕ ) ) . 2 ± 2 ( n ( θ, ϕ ) – n i )
(4)
The determination of the two eigenmode unit vectors e±(θ, ϕ) only implies the three principal refractive indices, since we had assumed the weak absorption ' for i, j, k = and fluorescence properties with εii Ⰷ ε jk x, y or z, meaning that the imaginarypart ε' plays no significant role for the determination of these polar ization unit vectors. Note that these vectors e±(θ, ϕ) are generally not orthogonal. Out of the principal planes of the dielectric frame, e±(θ, ϕ) is the only way to define the polarization unit vectors. However, for propagation in the principal planes, one can also define the ordinary and extraordinary polarization unit vectors, eo(θ, ϕ) and ee(θ, ϕ) respectively, namely asso ciated to the ordinary and extraordinary complex opti o, e cal indices, nˆ (θ, ϕ) = no, e(θ, ϕ) + jn'o, e(θ, ϕ). The ordinary or extraordinary polarization unit vectors are respectively related to a polarization mode being orthogonal or tangent to the considered principal plane of the dielectric frame. In the XY plane, eo(θ = π/2, ϕ) and ee(θ = π/2, ϕ), respectively, correspond to e+(θ = π/2, ϕ) and e–(θ = π/2, ϕ), related to the opti o + cal indices nˆ (θ = π/2, ϕ) = nˆ (θ = π/2, ϕ) and e – nˆ (θ = π/2, ϕ) = nˆ (θ = π/2, ϕ). In the YZ plane, eo(θ, ϕ = π/2) and ee(θ, ϕ = π/2) respectively corre spond to e–(θ, ϕ = π/2) and e+(θ, ϕ = π/2), related to o – the optical indices nˆ (θ, ϕ = π/2) = nˆ (θ, ϕ = π/2) e + and nˆ (θ, ϕ = π/2) = nˆ (θ, ϕ = π/2). The situation in the XZ plane is more complex due to the existence of the four optical singularities related to the refractive indices, since one verifies eo(θ, ϕ = 0°) = e+(θ, ϕ = 0°) o and ee(θ, ϕ = 0°) = e–(θ, ϕ = 0°) with nˆ (θ, ϕ = 0°) = + e – nˆ (θ, ϕ = 0°) and nˆ (θ, ϕ = 0°) = nˆ (θ, ϕ = 0°) for –Vz < θ < Vz and π – Vz < θ < π + Vz while one gets eo(θ, ϕ = 0°) = e–(θ, ϕ = 0°) and ee(θ, ϕ = 0°) = e+(θ, ϕ = o – e 0°) with nˆ (θ, ϕ = 0°) = nˆ (θ, ϕ = 0°) and nˆ (θ, ϕ = + 0°) = nˆ (θ, ϕ = 0°) for the other propagation direc LASER PHYSICS
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tions in this principal plane. Optical axes are therefore associated to a polarization singularity with a π/2 rota tion of the e±(θ, ϕ) vectors [11]. All these properties of e±(θ, ϕ) and n±(θ, ϕ) are wellknown but, up to now, it is not yet the case for the related properties of their imaginary counterpart n'±(θ, ϕ), especially around the specific XZ plane, which is the scope of the article as shown by Fig. 1. Figure 1 provides illustrative sections of the imagi nary index surface close to the XZ plane for two dis tinct biaxial media. Figures 1a–1c describes a realistic theoretical medium from the orthorhombic crystal system with εxx/ε0 = 2.682, εyy/ε0 = 2.858, εzz/ε0 = ' /ε 0 = 6.878 × 10–5, ε yy ' /ε 0 = 7.776 × 10–5, 2.918, ε xx ε 'zz /ε 0 = 11.273 × 10–5, and ε 'i ≠ j /ε 0 = 0, while Figs. 1d–1f describes a medium from the monoclinic crystal system with the same parameters except an additional pair of extradiagonal nonzero elements ε 'xz /ε 0 = –4.015 × 10–5. Symmetry properties of sections drawn in Fig. 1 agree with that already discussed in [6]. Indeed, the three principal planes XY, YZ and XZ are symmetry mirror planes for the imaginary index surface n'±(θ, ϕ) of orthorhombic crystal systems, and Figs. 1a–1b illustrates that the XY and YZ planes are symmetry mirror planes. On the contrary, Figs. 1d–1e shows that YZ and XZ are no more mirror symmetry planes, so that the only remaining mirror symmetry plane of the imaginary index surface n'±(θ, ϕ) is the XZ plane in the considered case. Figures 1a–1d show the sections of the imaginary index surface in the XZ plane, from the ordinary and extraordinary description. These angular distributions are followed either with the ordinary unit vector eo(θ, ϕ) being orthogonal to the XZ plane, or with the extraordinary unit vector ee(θ, ϕ) being tangent to the XZ plane. Therefore, Figs. 1a–1d provide continuous angular distributions of the sections of imaginary index surface in the XZ plane, since such description includes no optical singularity. Figures 1b–1e lead to another description of the same sections of the imaginary index surface in the XZ plane, from the n'±(θ, ϕ) point of view. These angular distributions are now followed with the unit vector e±(θ, ϕ), where e+/–(θ, ϕ) is orthogonal/tangent to the XZ plane for angles such that –Vz < θ < Vz and π – Vz < θ < π + Vz, while e+/–(θ, ϕ) is tangent/orthogonal to the XZ plane for other directions of propagation. In this case, Figs. 1d–1e provide a nonusual angular dis tributions of the sections of imaginary index surface in the XZ plane, made of pieces of circular and bilobar distributions, leading to the first clear enlightening of optical singularities of imaginary index surfaces in the XZ plane. Besides, the wellknown optical singulari ties of e±(θ, ϕ) are related to the wellknown singular
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Fig. 1. Numerical simulations of angular distributions of linear absorption or fluorescence properties in the dielectric frame (X, Y, Z) of biaxial crystals for orthorhombic {(a)–(b)–(c)} and monoclinic {(d)–(e)–(f)} crystal systems; (a–d) circular ordinary and bibolar extraordinary sections of n'o(θ, ϕ = 0°) (grey) and n'e(θ, ϕ = 0°) (black), respectively, in the XZ plane; (b–e) same sections as for (a–d), but described here in terms of n'–(θ, ϕ = 0°) (grey) and n'+(θ, ϕ = 0°) (black), in the XZ plane; (c–f) same imaginary layers, but sections of n'–(θ, ϕ = 1°) (grey) and n'+(θ, ϕ = 1°) (black) layers, close to the XZ plane at ϕ = 1°. O.A. stands for optical axes. LASER PHYSICS
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ities of the refractive index surface n±(θ, ϕ), which cor respond to punctual discontinuities of the local gradi ent at the four umbilici directions. Therefore, Figs. 1b–1e show for the first time that the optical sin gularities of the refractive indices n±(θ, ϕ) along the optical axes lead to another type of singularities for the imaginary indices n'±(θ, ϕ): we report the first descrip tion that, for n'±(θ, ϕ) along the specific umbilici directions, there are four discontinuities at Vz with not only punctual discontinuities of the local gradient but also of the continuity of the n'±(θ, ϕ) angular distribu tions themselves, leading to a jump from a circular piece to a bibolar piece of the sections at Vz. Thus, we introduce here that discontinuities for n'±(θ, ϕ) are zeroorder discontinuities while they are firstorder discontinuities for n±(θ, ϕ). Figures 1c–1f depict the n'±(θ, ϕ) behavior out of the XZ principal plane, but close to it at ϕ = 1°. In this case, since there is no optical singularity of n±(θ, ϕ) out of the XZ plane, there is also no optical singularity of n'±(θ, ϕ). However, Figs. 1c–1f show the strong sen sitivity with respect to the polarization unit vector close to the XZ plane, at the vicinity of the optical sin gularities at Vz. Such sections at ϕ = 1° in polarized light corroborate the high angular variability of three dimensional angular distributions at the vicinity of umbilici [6]: note that this angular variability extends quiet far from the punctual umbilici, i.e., from several degrees out of the XZ plane. These strong gradients have thus a physical existence, and there are not numerical or experimental artifacts for example. These considerations traduce ideal experimental behavior related to strictly unidirectional beams: such considerations should thus be extended for reallife experiments including diverging beams with nonzero angular apertures, as discussed below. 3. ANGULARAVERAGED DESCRIPTION OF FLUORESCENCE ANGULAR DISTRIBUTIONS IN BIAXIAL MEDIA FOR DIVERGING BEAM PROPAGATIONS In reallife experiments, laser beams present a non zero angular aperture related to its divergence. For flu orescence measurements, spontaneous emission hap pens in all the directions of propagation, over the 4π steradians, with various efficiencies depending on the angular distribution of the spontaneous emission Ein stein coefficient in polarized light. In the case of spon taneous emission, such angular distribution is directly proportional to that of n'±(θ, ϕ). Note that fluores cence measurements in polarized light are always associated to a nonzero angular aperture resulting from the considered experimental detection setup. Therefore, for any direction of propagation (θ, ϕ) in which the emission of a fluorescent material is studied, there is an angular integration from the detection sys tem over a given fraction of the 4π steradians that cor LASER PHYSICS
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responds to an aperture of Δθ and Δϕ in spherical coordinates, Moreover, since fluorescence emission is to be studied in polarized light in biaxial laser media, one must consider a polarizer in front of the detector so as to selectively choose each of the two eigenmode polarization unit vectors e±(θ, ϕ). Let us note p(θ, ϕ) the orientation of the selected polarization through the polarized detection system for fluorescence collec tion along a (θ, ϕ) direction, with a (Δθ, Δϕ) angular aperture. In these conditions, we introduce that the collected spontaneous fluorescence is proportional to the polarizationprojected and angularintegrated dis tribution of n'±(θ, ϕ). This results into a fluorescence measurement being proportional to the value n 'exp (θ, ϕ; Δθ, Δϕ; p) that writes as following: n 'exp ( θ, ϕ; Δθ, Δϕ; p ) =
∫ ∫ [ n'
+
+
( θ', ϕ' ) ( e ( θ', ϕ' ) ⋅ p )
2
(5)
Δθ, Δϕ –
–
2
+ n' ( θ', ϕ' ) ( e ( θ', ϕ' ) ⋅ p ) ]δθ'δϕ'. Since it is clearly defined, it is then possible to qualita tively derive the behavior of n 'exp (θ, ϕ; Δθ, Δϕ; p), and therefore that of measurements of spontaneous fluo rescence emission. For reasonably limited angular aperture (Δθ, Δϕ) of detection, angular variability of n'±(θ, ϕ) layers is in general very limited, so that ' (θ, ϕ; Δθ, Δϕ; p) provides a direct measurement of n exp n'+(θ, ϕ) or n'–(θ, ϕ), depending on the selected eigen mode polarization vector e±(θ, ϕ). In the case of prop agation directions quiet close to the vicinity of the optical singularities, the variability of n'±(θ, ϕ) layers over the (Δθ, Δϕ) angular aperture is no more negligi ble [6], so that the experimental measurements n 'exp (θ, ϕ; Δθ, Δϕ; p) start to average and thus to differ from either the n'+(θ, ϕ) or n'–(θ, ϕ) medium proper ties. Finally, for directions of propagation (θ, ϕ) with (Δθ, Δϕ) apertures that incorporate one of the optical umbilici, fluorescence measurements integrate the zeroorder optical discontinuity of n'±(θ, ϕ), so that n 'exp (θ, ϕ; Δθ, Δϕ; p) clearly differs from n'+(θ, ϕ) or n'–(θ, ϕ) here, as corroborated by the first refined modeling shown below on Fig. 2. 4. REFINED MODELING OF FLUORESCENCE MEASUREMENTS WITH THE BIAXIAL MONOCLINIC YCOB:Nd CRYSTAL In [4], we had reported the first measurements of angular distributions of fluorescence in polarized light for a biaxial crystal, namely the monoclinic crystal YCa4O(BO3)3, doped with Nd3+ luminescent rare earth ions. This crystal YCOB:Nd belongs to crystal class CS (space group Cm, unit cell—parameters a =
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Fig. 2. Fluorescence angular distribution in polarized light at 1061 nm for the monoclinic biaxial crystal YCOB:Nb, for any propagation directions in the XZ plane. Ordinary and extraordinary distributions for experimental measure ments (grey stars and black stars, respectively) and their simulated counterparts (grey lines and black lines, respec tively) with our new theoretical description including a nonzero angular aperture for fluorescence collection.
8.076 Å, b = 16.02 Å, c = 3.53 Å, β = 101.23° between the a and c axes) [12]. The crystallographic plane orthogonal to b is a mirror symmetry plane, leading the dielectric axis Y to be parallel to b and the principal plane XZ to correspond to the crystallographic mirror symmetry plane. We had performed the first angular resolved spectroscopy of YCOB:Nd in polarized light, 4 by studying in the mirror plane XZ the 4I3/2 I11/2 3+ fluorescence emission of Neodymium ions Nd at 1061 nm that had been excited by constant absorption with the 4I9/2 (4F5/2 + 2H9/2) resonant transition at 812 nm. These experimental measurements of fluores cence angular distributions were performed with a YCOB:Nd sample shaped as a 7.1 mm diameter sphere that offered direct experimental access to any direc tion of propagation. These distributions had been modeled with analytical expressions that we had derived from the unidirectional beam propagation description. This former model could very properly fit the measured fluorescent measurements in the XZ plane for directions far from the umbilici, but it was not fully adapted to match with the measured fluores cence behavior close to these optical singularities as seen in [4, Fig. 3b]. Thanks to our theoretical developments from Eq. (5) in relation to nonzero angular aperture detec tion of fluorescence, we propose the first time refined analysis of the YCOB:Nd angular distributions in the XZ plane for the 1061 nm fluorescence. Figure 2 shows former experimental fluorescence angular distribu tions for an ordinarypolarized detection orthogonal to the XZ plane (grey stars), and for an extraordinary
polarized detection tangent to the XZ plane (black stars): these measurements were relative measure ments, without absolute scaling, so that only the ratio of these ordinary and extraordinary was measured. Figure 2 also shows the simulations for nonzero aper ture detection, for both ordinarypolarized and extraordinarypolarized detection in the XZ plane (grey line and black line, respectively). The agreement between these former experimental data and our new refined modeling is excellent as shown by Fig. 2. To obtain such fit, values of principal indices at 1061 nm 2
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were fixed at n x = εxx/ε0 = 2.682, n y = εyy/ε0 = 2.858, 2
n z = εzz/ε0 = 2.918 [13], while the nonnull ratios ε 'xx /ε 'yy , ε 'yy /ε 'yy = 1, ε 'zz /ε 'yy , ε 'xz /ε 'yy = ε 'zx /ε 'yy were free to adjust themselves while testing different (Δθ, Δϕ) collection apertures. The best results were obtained in the weak fluorescence hypothesis, for ε 'xx /ε 'yy = 0.7, ε 'yy /ε 'yy = 1.0, ε 'zz /ε 'yy = 2.0, ε 'xz /ε 'yy = ' /ε yy ' = –0.15, in good agreement with the values ε zx that had been previously published in [4]. Moreover, the optimum fit was related to an aperture (Δθ, Δϕ) = (3°, 7°), which is also in good agreement with the experimental angular aperture of (7°, 7°) associated to the 1 inch clear aperture of the 10 cm focal length col lection lens that was located at 10 cm after the studied spherical sample of YCOB:Nd. Such dissymmetry (Δθ, Δϕ) = (3°, 7°) with the two spherical coordinates is in direct relation with the strong astigmatism of the 812 nm pump laser source that was absorbed by YCOB:Nd. Finally, the description in terms of detection of flu orescent diverging beam has been illustrated here with a monoclinic biaxial crystal. However, the influence of optical singularities on experimental fluorescence angular distributions would be the same for any biaxial crystals, so that this work remains relevant for orthor hombic crystals and triclinic crystals a fortiori. The biaxial crystal system does not affect the existence of these singularities and thus the consecutive distortions of measured fluorescence angular distributions. Indeed, the biaxial crystal system only affects the sym metry of these distortions: for an orthorhombic crystal system, the four distortions close to the umbilici are symmetrical and equivalent; for a monoclinic crystal system, the four distortions are equivalent two by two, as visible on Fig. 2; and for an hypothetic triclinic crystal system, none of the four distortions would be equivalent anymore. LASER PHYSICS
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5. METHODOLOGICAL AND METROLOGICAL IMPLICATIONS OF REFINED MODELING OF LINEAR ABSORPTION AND FLUORESCENCE ANGULAR DISTRIBUTIONS FOR BIAXIAL CRYSTALS With appropriate modifications, the refined analy sis that is proposed here for spontaneous fluorescence can be adapted and extended to linear absorption or stimulated laser emission in biaxial crystals, since at the very close vicinity of the umbilici, even a slightly diverging beam—as a laser beam—can still integrate and spread over the optical singularities. For the presented angular distribution of fluores cence at 1061 nm, we had performed many measure ments so that we could directly plot the complete dis tribution in the XZ plane. We could therefore observe the experimental distortions at the vicinity of the opti cal singularities, and understand that the meaning of experimental measurements close to the umbilici was limited due to angularintegrating and polarization averaging during the fluorescence collection setup. For an isolated measurement close to the umbilici, there is no way to control if a measurement n 'exp (θ, ϕ; Δθ, Δϕ; p) is a real measurement of n'+(θ, ϕ) or n'–(θ, ϕ) values, or if n 'exp (θ, ϕ; Δθ, Δϕ; p) carries a non neg ligible deviation from these n'±(θ, ϕ) medium values. We highlight here for the first time that linear absorp tion or fluorescence measurements in biaxial crystals should always be performed far enough from the umbilici so as to provide meaningful experimental val ues. We advice that the larger the beam divergence is, the further one should measure from the umbilici. Appropriate metrology of absorption in biaxial should thus be performed with tunable laser sources instead of classical lamps in order to use sources with as weak divergence as possible, and thus to address a narrow range of propagation directions simultaneously [14]. Concerning spontaneous fluorescence studies, one should also care to fix the smallest angular aperture possible in the detection setup, to address a narrow range of propagation directions simultaneously. These methodological advices should prevent large mistakes in the metrology of orthorhombic, monoclinic and tri clinic biaxial crystals. 6. CONCLUSIONS We developed an improved treatment of angular distributions in polarized light for laser biaxial crystals, which details the influence of the light divergence in the metrology of spontaneous fluorescence properties. We proposed a refined analysis that led now to an excellent agreement with previously published angular distribution of spontaneous fluorescence at 1061 nm from the monoclinic biaxial crystal YCOB:Nd. These developments are relevant for any biaxial crystal, i.e. for the orthorhombic, the monoclinic and the triclinic LASER PHYSICS
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crystal systems. These developments could also be adapted for linear absorption and stimulated fluores cence emission in any biaxial crystals. We proposed detailed advices so as to improve biax ial crystal metrology, and thus to prevent experimental artefacts for the metrology of absorption, spontaneous and stimulated emission in these low symmetry mate rials. Therefore, we hope that we provide tools to improve the characterization of optical properties of biaxial innovative crystals, and thus that we contribute to reach an optimal exploitation of applicative poten tials of these materials for optics. Finally, these developments and precautions should be applied to any biaxial crystals for which the imaginary part of the linear permittivity is involved, as bulk laser crystals with Nd:(Gd,Y)Ca4O(BO3)3 [1, 2], Yb:KLu(WO4)2 [15], or Yb:LiGd6O5(BO3)3 [15], or thin disk lasers with Tm:KLu(WO4)2 obtained by epit axy [16], but also selfdoubling inorganic crystals with both luminescent and nonlinear optical properties [1,2] or semiorganic crystals with both stimulated Raman scattering and laser processes [17], photolu minescence with Pr:Lu2SiO5 [18], slowlight with Pr:Y2SiO5 [19], photorefractivity with Sn2P2O6 [20], and laserinduced photoconductivity related to scin tillation mechanisms [3]. We also assess that these developments are general with respect to the spectral range from the nearUV, visible, nearinfrared [7] to the midinfrared range [21], and may also be relevant elsewhere in the electromagnetic spectrum. REFERENCES 1. F. Mougel, K. Dardenne, G. Aka, A. KahnHarari, and D. Vivien, J. Opt. Soc. Am. B 14, 164 (1999). 2. P. Segonds, S. Joly, B. Boulanger, Y. Petit, B. Ménaert, and G. Aka, J. Opt. Soc. Am. B 26, 750 (2009). 3. M.F. Joubert, S. A. Kazanskii, Y. Guyot, J.C. Gâcon, and C. Pédrini, Phys. Rev. B 69, 165217 (2004). 4. Y. Petit, B. Boulanger, P. Segonds, C. Félix, B. Ménaert, J. Zaccaro, and G. Aka, Opt. Express 16, 7997 (2008). 5. S. Joly, Y. Petit, B. Boulanger, P. Segonds, C. Félix, B. Ménaert, and G. Aka, Opt. Express 17, 19868 (2009) . 6. Y. Petit, P. Segonds, S. Joly, and B. Boulanger, Materi als 3, 2474 (2010). 7. S. Joly, P. Segonds, B. Boulanger, Y. Petit, C. Félix, and B. Ménaert, Opt. Express 18, 19169 (2010). 8. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965). 9. J. F. Nye, Physical Properties of Crystals: Their Repre sentation by Tensors and Matrices (Oxford Univ., Oxford, 1985). 10. B. Boulanger and J. Zyss, “Nonlinear Optical Proper ties,” in International Tables for Crystallography, Vol. D: Physical Properties of Crystals, Ed. by A. Authier (Klu wer Academic, Dordrecht, Netherlands, 2004), Ch. 1.7.
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16. S. Vatnik, I. Vedin, M. C. Pujol, X. Mateos, J. J. Carva jal, M. Aguilo, F. Díaz, U. Griebner, and V. Petrov, Laser Phys. Lett. 7, 435 (2010). 17. A. A. Kaminskii, L. Bohatý, P. Becker, H. J. Eichler, H. Rhee, and J. Hanuza, Laser Phys. Lett. 6, 872 (2009). 18. J. P. Chaminade, V. Jubera, A. Garcia, P. Gravereau, and C. Fouassier, J. Opt. Adv. Mater. 2, 451 (2000). 19. M. Nikl, H. Ogino, A. Beitlerova, A. Novoselov, and T. Fukuda, Chem. Phys. Lett. 410, 218 (2005). 20. J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Man son, Phys. Rev. Lett. 95, 063601 (2005). 21. Z. G. Wang, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, Laser Phys. 20, 458 (2010).
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