1. Introduction

The potentiality of new laser crystals relies on a complete characterization of their absorption and fluorescence properties. These two effects are governed by the imaginary part of the complex permittivity $\widehat{\epsilon }$ that is described by a polar second rank tensor, the real part of $\widehat{\epsilon }$ being related to the refractive index. When laser crystals belong to cubic crystal classes, like Nd:YAG for example [1], the three main values of $\widehat{\epsilon }$ are equal, leading to optical properties that are independent to the crystal orientation. But, nowadays, there are promising laser materials that are anisotropic, like for example the uniaxial crystals Nd:YLiF₄ and Nd:YVO₄ [1], or the biaxial ones Yb:GdCOB [2] and Nd:YCOB [3]. For these crystals the tensors of the real and imaginary parts of $\widehat{\epsilon }$ can have up to three different main values each, so that the absorption and fluorescence angular distributions are expected to be anisotropic, as it is the case for the refractive index [4]. Here we theoretically and experimentally considered the case of a monoclinic crystal class, and Nd:YCOB is investigated. We report, for the first time to the best of our knowledge, that the angular distributions of the two polarization components of the absorption coefficient and of the fluorescence power exhibit singularities of the same kind than those of the biaxial index surface. Furthermore, we show that the main values of absorption and fluorescence are not along the principal axes of the dielectric frame, but respectively along the principal axes of two new frames tilted from the dielectric frame.

2. State of the Art : absorption of anisotropic high symmetry crystals

A complete description of absorbing anisotropic crystals had been reported by introducing a complex permittivity $\widehat{\epsilon }$=ε+jε′ in the equations of the propagation of light, where ε is related to the refractive index and ε′ governs absorption [4]. The discussion had been confined to anisotropic crystals belonging to high symmetry crystal classes, i.e. hexagonal, tetragonal, trigonal or orthorhombic classes. In these cases, the real and imaginary parts of $\widehat{\epsilon }$ are described by polar second rank tensors that are both diagonal in the dielectric frame [4]. The analysis was considerably simplified by assuming ε′≪ε. Then it relies on solutions of the complex Fresnel’s equation where real and imaginary parts can be analytically expressed. Here we focus on these solutions as a function of the spherical angle θ in the X-Z principal plane of the dielectric frame (X, Y, Z) of orthorhombic crystals, which are biaxial crystals. We consider the main values of the real and imaginary parts of the complex index n̂ that are defined from the main values of the complex permittivity tensor by: ${\widehat{n}}^2=\widehat{\epsilon }{\epsilon }_0^{-1}$, where ε ₀ is the free-space permittivity, leading to n ²=εε ^-1 ₀ and 2nn′=ε′ε ^-1 ₀ when n′≪n is assumed, n and n′ being the real and imaginary parts respectively of n̂. They write, according to the perpendicular (⊥) and parallel (//) polarization components with respect to the X-Z plane [4]:

where n₁, n₂ and n₃ are the three main values of the real part of n̂, i.e. the three main refractive indices, and n′₁, n′₂ and n′₃ are the three main values of the imaginary part of n̂. The (⊥) polarization is defined as orthogonal to the X-Z plane: it is thus collinear with the Y-axis. The (//) component is located in the X-Z plane. Equations (1) describe the (⊥) and (//) polarization components of the two sections of the well-known index surface of a biaxial crystal in the X-Z plane of the dielectric frame. They are given in Fig. 1(a) and show the two Optical Axes (OA), which correspond by pair to the four ombilics of the index surface. Equations (2) describe the (⊥) and (//) polarization components of the angular distributions of the imaginary part of n̂ in the X-Z plane. They are given in Fig. 1(b) and exhibit two directions (OA′) where the (⊥) and (//) polarization components are equal, as in the case of the index surface. Figures 1(a) and 1(b) indicate that the (⊥) polarized angular distribution is a circle for both the real and imaginary parts of n̂. But there is a difference between the (//) components of polarization, the angular distribution being elliptical for the real part, and “bilobar” for the imaginary one.

According to Eq. (1) and (2), a complete picture of the optical properties of absorbing anisotropic high symmetry crystals can be then obtained from measurements along the principal axes of the dielectric frame, which is completely linked up to the crystallographic frame for the high symmetry crystal classes mentioned above. Classically such measurements are performed with a polarized light beam properly focused in a polished slab oriented along the principal axes of the dielectric frame. It provides the determination of the main values of the refractive index and of the absorption, as well as the recording of their spectra. They can be discriminated by adjusting successively the orientation of the linear polarization of light along the (⊥) and (//) components associated with the considered direction of propagation. Note that fluorescence is also governed by the imaginary part of the complex index, so that the angular distributions of its (⊥) and (//) components are described by Eq. 2, in the case of high symmetry crystals.

 

Fig. 1. Angular distributions of the real part (a) and of the imaginary part (b) of the complex index in the X-Z plane of an orthorhombic crystal. V_Z and V℉_Z are the optical axes angles.

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3. Direct measurement of absorption and fluorescence in monoclinic crystals: method and experimental setup

Our goal was to perform direct measurements of the angular distributions of absorption and fluorescence of low symmetry crystals belonging to a monoclinic crystal class. A full access to the space is possible if the studied crystal is cut as a polished and oriented sphere. We report here such study for the monoclinic Nd:YCOB crystal cut as a sphere with a diameter of 7.44 mm. Nd:YCOB belongs to the Cm space group. Then the only symmetry element of this crystal is a mirror plane, m, which is perpendicular to the b-axis of the crystallographic frame (a, b, c). The angles between the crystallographic axes are: β_ab=90°, β_bc=90° and β_ac=120.26° [5]. The Y-axis of the dielectric frame (X, Y, Z) is parallel to the b-axis, while the X-and Z-axes are not collinear with the a- and b-axes, exhibiting the following angles: γ_aX=24.7° and γ_cZ=13.4°, these angles having no observable dependence with wavelength [5]. We studied in the Nd:YCOB sphere the absorption at 0.812 µm on the one hand, and the fluorescence at 1.061 µm in on the other hand when pumping the sphere with a laser source at 0.812 µm. These two wavelengths are relative to the Nd³⁺ ions doping the YCOB matrix.

 

Fig. 2. Top view of the experimental setup for both absorption (λ_i=0.812 µm) and fluorescence (λ_i=1.061 µm) angular distributions measurements. λ_pump=0.812 µm.

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We implemented the same experimental setup for both measurements, which top view is depicted in Fig. 2. It was coupled with a 10Hz-repetition rate CONTINUUM Panther Optical Parametric Oscillator (OPO) emitting a beam with a FWHM pulse duration of 5 ns which power is recorded on photodiode 1. The Nd:YCOB sphere was stuck along the Y-axis and placed on a rotation stage inserted behind the lens L₁. It was then possible to propagate the OPO beam parallel inside the sphere, successively over θ=360° in the X-Z plane that is also the symmetry mirror plane m of the crystal. The linear polarization of the OPO beam can be adjusted by using the half-wave plate L_H from the (⊥) component of polarization, along the Y-axis, to the (//) one oriented in the X-Z plane.

For the absorption measurements, we considered successively these two polarization components, and their transmission through the sphere was recorded on photodiode 2 coupled with lens L₂. For the fluorescence studies, the OPO beam was kept (⊥) polarized for any considered direction of propagation θ in the X-Z plane, in order to avoid any spatial walk-off or anisotropic effects. By this way, the power at 0.812 µm absorbed in the sphere remains the same for any direction of propagation. The pump beam was cut after the sample by a chromatic filter F (HR at 0.812 µm and HT at 1.061 µm) so that fluorescence power only is recorded through lens L₂ on photodiode 2. The (//) and (⊥) polarization components of absorption and fluorescence were discriminated by using a Glan-Taylor polarizer P.

The (//) and (⊥) components of the absorption coefficient at 0.812 µm, α _// and α _⊥ respectively, were determined from measurements of the total transmission coefficients, T _//,⊥, and of Fresnel transmission relative to the entrance and exit surfaces of the sphere, T ^F _//,⊥, according to ${\alpha }_{//,\perp }=-\frac{1}{L}\ln \left(\frac{T_{//,\perp }}{T_{//,\perp }^F}\right)$ where L is the sphere diameter. The (//) and (⊥) components of the power of fluorescence at 1.061 µm, P _// and P _⊥ respectively, were directly recorded and corrected by Fresnel transmission relative to the exit surface of the sphere. Absorption and fluorescence measurements were performed as a function of the propagation angle θ, successively over 360°, in the X-Z plane.

4. Experimental results and their modelization

Figure 3(a) gives the angular distribution of the (⊥) and (//) components of polarization of the absorption coefficient, α _⊥(θ) and α _//(θ), at 0.812 µm of Nd:YCOB. It is the first time, to the best of our knowledge, that this kind of measurement is performed. It shows that α _⊥(_θ) remains constant as a function of the direction of propagation θ, and α _//(θ) exhibits a bi-lobar angular distribution, which is in a very good agreement with the behaviour of the calculated patterns of Fig. 1(b) corresponding to orthorhombic crystals. But the symmetry axes of the extraordinary angular distribution of Fig. 3(a), labelled as X’_abs and Z’_abs, do not correspond to the axes X and Z of the dielectric frame that are the symmetry axes of the extraordinary pattern of Fig. 1(b). So for monoclinic crystals, which is the topics of our study, we have to define a new frame (X′_abs, Y′_abs, Z′_abs), the Y′_abs-axis being collinear with the Y-axis of the dielectric frame. We called the “Absorption Axes” (AA), the directions along which the (//) and (⊥) polarized absorption coefficients are equal, the angle between these two axes being written 2V_abs. We found the same type of angular distribution for the fluorescence at 1.061 µm, as shown in Fig. 3(b) where the powers are normalized to that of the (⊥) component emitted along the Z-axis, i.e. P _⊥(θ=0°). The angular distribution of the (⊥) polarized component of fluorescence defined as p _⊥(θ)=P _⊥(θ)/P _⊥(θ=0°) is described by a circle, while that of the (//) one, p _//(θ)=P _//(θ)/P _⊥(θ=0°), is a bi-lobar pattern, which in accordance with the fact that fluorescence is also governed by the imaginary part of the complex index, as for absorption. Fig. 3(b) also shows that the symmetry axes of the (//) fluorescence pattern, i.e. X′_fluo and Z′_fluo, are different than X′_abs and Z′_abs on the one hand, and than X and Z on the other hand, while Y′_fluo is collinear with Y′_abs and Y. The two axes for which the two components of fluorescence are equal are called the “Fluorescence Axes” (FA), the angle 2V_fluo between these two axes being different than the angle 2V_abs between the two “Absorption Axes” (AA).

 

Fig. 3. Perpendicular (⊥) (dots) and parallel (//) (squares) components of the absorption coefficient at 0.812 µm (a) and of the normalized fluorescence power at 1.061 µm (b) as a function of the angle θ in the X-Z plane of Nd:YCOB.

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Then our experimental results suggest the introduction of two specific principal frames for absorption and fluorescence of a monoclinic crystal, i.e. (X′_abs, Y′_abs, Z′_abs) and (X′_fluo, Y′_fluo, Z′_fluo) respectively, these frames being different and also not in coincidence with the dielectric frame (X, Y, Z). As a consequence, the imaginary part of the $\widehat{\epsilon }$ tensor is diagonal only in (X′_abs, Y′_abs, Z′_abs) for absorption, and in (X′_fluo, Y′_fluo, Z′_fluo) for fluorescence. Then the model developed in reference [4] for high symmetry crystals, describing the imaginary part of $\widehat{\epsilon }$ as diagonal in the dielectric frame, does not apply to monoclinic crystals. We thus wrote a new model able to describe our measurements, considering the imaginary part of $\widehat{\epsilon }$ to be not diagonal in the dielectric frame:

ε_xx, ε_yy and ε_zz are the three main values of the real part of the permittivity, and ε′_xx, ε′_yy, ε′_zz, ε′_xz=ε′_zx are the coefficients of the imaginary part. According to Eq. (3), the projection of the light propagation equation along the three principal axes of the dielectric frame (X, Y, Z) leads to the following single equation that relates the complex index n̂ to the different coefficients of the complex permittivity tensor, and that writes for any direction of propagation in the X-Z plane, i.e. for 0°<θ<360° ϕ=0°:

The angle θ is between the direction of propagation and the Z-axis. The resolution of Eq. (4) provides two complex solutions. Assuming n′≪n, the (⊥) and (//) components of the real part of the two solutions are given by Eq. (1), the same ones than for high symmetry crystals. But the equations relative to the (⊥) and (//) components of the imaginary part are specific to monoclinic crystals, and write in the X-Z plane:

The two Eq. (5) confirm circle and bi-lobar patterns for the angular distributions of the (⊥) and (//) components of the imaginary part of n̂ respectively. They were used to interpolate our experimental data, by assuming α _//,⊥(θ)=4πn′_//,⊥(θ)λ ^-1 for the absorption at 0.812 µm, and p _//,⊥=p _//,⊥(θ)/P⊥(θ=0°)=n′_//,⊥(θ)/n′_⊥(θ=0°) for the fluorescence at 1.061 µm, the fitting parameters being: ε′_xx, ε′_yy, ε′_zz and ε′_xz=ε′_zx.. The interpolations are shown in Fig. 3(a) and Fig. 3(b), and the agreement between the model and the experiment is very satisfying. The interpolation of absorption data allowed us to determine the tilt angle between the absorption frame (X′_abs, Y′_abs, Z′_abs) and the dielectric frame (X, Y, Z), i.e. δθ_abs=31.1±0.7°. We got δθ_fluo=-6.4±0.9° for the tilt angle between the fluorescence frame (X′_fluo, Y′_fluo, Z′_fluo) and the dielectric frame (X, Y, Z). The interpolation of our data with Eq. (5) also lead to 2V_abs=72°±4° and 2V_fluo=64°±3° as the angles between the two “Absorption Axes” (AA), and between the two “Fluorescence Axes” (FA), respectively. Note that they are also different from the angle between the two “Optical Axes” (OA), i.e. 2V_z=120.6° [5]. From the knowledge of the orientation of the absorption and fluorescence frames, we can determine the main values of the absorption and fluorescence in their specific frames. We got for the main values of the absorption coefficient at 0.812 µm: α ₁=2.1±0.8 cm^-1, α ₂=3.5±0.4 cm^-1 and α ₃=6.2±0.8 cm^-1, where the indices 1, 2 and 3 refer to X′_abs, Y′_abs and Z′_abs respectively. For the fluorescence at 1.061 µm we got for the main values: p₁=0.65±0.04, p₂=1.00±0.02 and p₃=2.01±0.04, where the indices 1,2 and 3 correspond to X′_fluo, Y′_fluo and Z′_fluo respectively. We see that these values are different than the diagonal values that are determined from measurements relative to the dielectric frame, i.e.: α₁=3.2±0.4 cm^-1, α₂=3.5±0.4 cm^-1 and α₃=5.1±0.6 cm^-1 for absorption, and p₁=0.67±0.02 cm^-1, p₂=1.00±0.02 cm^-1 and p₃=1.99±0.02 cm^-1, where the indices 1, 2 and 3 correspond in that case to X, Y and Z respectively. It corroborates that the main values of absorption and fluorescence can be obtained only along the principal axes of the absorption and fluorescence frames respectively, but not along those of the dielectric frame, as classically assumed prior to the present work.

5. Conclusion

We report, for the first time to the best of our knowledge, measurements and calculations relative to the angular distributions of the (⊥) and (//) components of the absorption and fluorescence in a monoclinic crystal. We investigated Nd:YCOB and showed that it is necessary to introduce new specific frames leading to the main values of absorption and fluorescence. The practical consequence to fully characterize absorption and fluorescence properties is that it is necessary to measure their main values in these frames instead of doing it in the dielectric frame. This feature is also of prime importance for devices purpose, for which it is necessary to optimize the absorption of the pump as well as the fluorescence. Then the choice of the direction in which the monoclinic laser material has to be cut must take into account the difference of symmetry axes of the two involved properties. This choice is even more complicated for self-doubling, which also occurs in Nd:YCOB, where a third property has also to be optimized, i.e. the phase-matching of second harmonic generation of the fluorescence, which principal frame is the dielectric frame. Finally, the part of our work devoted to absorption is of general interest for any monoclinic crystal, since absorption has to be considered and minimized in a lot of situations, like for example: second and third harmonic generations in un-doped YCOB [6], parametric up- and down-conversions in BiB₃0₅ (BiBO) [7], or photorefractivity in Sn₂P₂S₆ [8, 9].