1. Spectra measurement

Figure 5 shows the absorption spectra along its [100] [010] [001] orientations at room temperature. Along [010] and [001] directions, the baseline moves away, which may be generated by the large refractive indices along those two directions based on the measurement above. From Fig. 5, the intraconfiguration transitions of the Nd³⁺ ion and the bands’/states’ Stark split can be analyzed. The initial state of these transitions is all the ground ⁴I_9/2 state of the Nd³⁺ ion. All the final states are assigned and given in Table II. The absorption peaks are almost in the region of 400–900 nm. The transitions ⁴I_9/2–⁴F_5/2 and ⁴I_9/2–²H_9/2 are noteworthy, and all the line widths (full width at half maximum, or FWHM) are more than 17 nm at 808 nm, which are much larger than Nd:YVO₄ (2.0 nm).Reference Jensen, Ostroumov, Meyn, Huber, Zagumennyi and Shcherbalov⁶ The larger width may be generated by the following reasons: in the Nd:LaVO₄ crystal, the La possesses oxygen coordination number of 9. However, Nd has oxygen coordination number of 8 in NdVO₄ phase. When the Nd ions are doped in LaVO₄, a part of La ions are replaced by Nd. Usually, Nd coordinated with 8 oxygen atoms forms octahedron in most orthovanadate compounds, due to La atom is coordinated to 9 oxygen atoms in LaVO₄. Therefore, for Nd ions, there is a tendency from oxygen coordination number of 9 to 8, which would induce the variation of crystal field and inhomogeneously broadening in the spectra. Because the wave length of laser diode (LD) would change with its working temperature, it is suggested that a larger FWHM is suitable for pumping by a LD.

FIG. 5.

Absorption spectra along a, b, and c directions of crystal Nd:LaVO₄. (color online)

TABLE II.

Spectral parameters of Nd:LaVO₄.

2. Judd–Oflet theory analysis

Judd–Oflet (J–O) theoryReference Judd^18, Reference Ofelt¹⁹ is a very effective method in the analysis of radiative transitions of RE ions. According to J–O theory, only the transitions of ΔS = 0 between ^2S+1L_J states are permitted. However, the absorption transitions such as ⁴I_9/2–²H_9/2 and ⁴I_9/2–²P_1/2 for ΔS ≠ 0 between ^2S+1L_J states are also observed, shown in Fig. 5 and Table II. These transitions are relatively weak and were omitted during fitting J–O experimental parameters. At the same time, the background of the absorption spectra was subtracted during the fitting procedure.

According to J–O theory, the experimental oscillator strength $P_{\exp }$ can be obtained from the absorption spectra, which is expressed by:

(1)$$<![CDATA[$$\Gamma \, = \,\int {K({\rm{\lambda }})d{\rm{\lambda }}} \, = \,\int {D({\rm{\lambda }})d{\rm{\lambda }}} /(L\log e)\, = \,N_0 {{{\rm{\pi }}e^2 \overline {{\rm{\lambda }}^2 } } \over {mc^2 }}P_{\exp } (J \to J^\prime )\quad,$$

(2)$$<![CDATA[$${\rm{\sigma }}({\rm{\lambda }})\, = \,{{D({\rm{\lambda }})} \over {N_0 l\log e}}\quad,$$

where N ₀ is the concentration of Nd³⁺ (here, $N_0 = {{0.94 \times 4/100} \over V}$, V = 3.32 × 10⁻²⁸ cm³, so N ₀ = 0.11325 × 10²¹/cm³), L is the thickness, m, c, and e are electron mass, light speed, and electron charge, respectively. $\overline {\rm{\lambda }} $ is average transition wave length, which can be given by the gravity wave length of the absorption bands. $\Gamma $ is the integrated absorbance for each absorption band. $D({\rm{\lambda }})$ is the absorbance. $K({\rm{\lambda }})$ is the absorbance coefficient. ${\rm{\sigma }}_{\rm{\lambda {(k)} }} $ is the absorption cross section. The integrated absorbance $\Gamma $ and absorption cross section ${\rm{\sigma }}_{\rm{\lambda }} $ with the eight transitions are calculated as shown in Table II.

Based on the J–O theory, the measured line strengths are used to obtain the J–O intensity parameters Ω₂, Ω₄, and Ω₆ by fitting the set of equations from the corresponding transitions between $J$ and $J^′$ manifold in the following equation:

(3)$$<![CDATA[$$\Gamma \, = \,\int {K({\rm{\lambda }})d{\rm{\lambda }}\, = \,N_0 {{8{\rm{\pi }}^3 e^2 \overline {\rm{\lambda }} } \over {3\hbar c}}{{(n^2 + 2)^2 } \over {9n}}{1 \over {(2J + 1)}}} \times S_{\exp } (J \to J^\prime )} ,$$

(4)$$<![CDATA[$$S_{{\rm{cal}}} (J \to J^\prime ) = \mathop\sum\limits_{t = 2,4,6} {\Omega _t \left| {\left\langle {4f^n {\rm{\psi }}J\left\|| {U^{(t)} } \right\||4f^n {\rm{\psi }}^\prime J^\prime } \right\rangle } \right|} ^2 ,$$

where n is the refractive index (here, we use the mean refractive index of n ₁, n ₂, n ₃ for different wave lengths), h is the Planck constant, and J is the angular momentum of the ground state. U ^(t) (t = 2, 4, 6) are the matrix elements of unit tensor that are cited from Refs. Reference Carnal, Fields and Wybourne20 and Reference Carnal, Fields and Rajnak21.

The root-mean-square (rms) deviation between experimental and calculated line strengths is determined by:

(5)$$<![CDATA[$${\rm{rms}}\, = \,\sqrt {\mathop\sum\limits_{t = 1}^N {(S_{\exp } - S_{{\rm{cal}}} )^2 /(N - 3)} } \quad.$$

So, the rms error is 1.034 × 10⁻²¹ cm². After a least-square fitting of S _exp to S _cal, the three J–O intensity parameters were obtained: Ω₂ = 2.142 × 10⁻²⁰ cm², Ω₄ = 3.704 × 10⁻²⁰ cm², and Ω₆ = 2.948 × 10⁻²⁰ cm². The three J–O intensity parameters compared with other Nd-doped crystals are shown in Table III. The spectroscopic quality factor, $X = {\raise0.5ex\hbox{$\scriptstyle {\Omega _4 }$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle {\Omega _6 }$}}$, for the crystal Nd:LaVO₄ laser material was 1.25. The spectroscopic quality factor, however, has been found to vary from 0.22 to 1.5 for Nd³⁺ in different host materials.Reference Kaminskii²⁵ The large quality factor indicates that the crystal has a disorder structure, which also identifies that the wide absorption spectra are inhomogeneously broadened.

TABLE III.

The J–O intensity parameters and radiative lifetimes of Nd-doped crystals.

From the 4f–4f intensity model, the calculated oscillator strength of a transition between two multiplets is defined asReference Carnall, Crosswhite and Crosswhit²⁶:

(6)$$<![CDATA[$$P_{{\rm{cal}}} \, = \,\,{{8{\rm{\pi }}^2 mc} \over {3h\overline {\rm{\lambda }} (2J + 1)}}{{(n^2 + 2)^2 } \over {9n}}\, \times \,\mathop\sum\limits_{t = 2,4,6} {\Omega _t \left| {\left\langle {4f^n {\rm{\psi }}J\left\|| {U^{(t)} } \right\||4f^n {\rm{\psi }}^\prime J^\prime } \right\rangle } \right|} ^2 \quad,$$

(7)$$<![CDATA[$${\rm{rms}}P\, = \,\sqrt {\mathop\sum\limits_{t = 1}^N {\left| {p_{{\rm{cal}}} - p_{\exp } } \right|^2 /N\quad.} } $$

Then, the experimental oscillator strength P _exp and calculated oscillator strength P _cal were obtained. The rmsP error is obtained to be 3.302 × 10⁻⁷ cm². Table II lists the experimental and calculated line strength S _exp/S _cal and the oscillator P _exp/P _cal.

3. Fluorescence spectrum

The fluorescence spectrum of the crystal excited by 808 nm recorded at room temperature is shown in Fig. 6. In this spectrum, three intense emission peaks at 880, 1065, and 1347 nm are corresponding to the transitions ⁴F_3/2 → ⁴F_9/2, ⁴F_3/2 → ⁴F_11/2, and ⁴F_3/2 → ⁴F_13/2 of Nd³⁺ ions, respectively. The most intense line at 1065 nm with FWHM of 10 nm is also much larger than Nd:YVO₄ and Nd:YAG,Reference Zhang, Meng, Zhu, Wang, Chow and Lu^27, Reference Kaczmarker, Lukasiewicz, Giersz, Jablonski, Jabczynski, Swrkowicz, Galazka, Drozdowski and Kwasny²⁸ which indicates that this crystal is potentially applied for ultrashort pulsed lasers, especially in the femtosecond lasers.

FIG. 6.

Fluorescence spectrum of Nd:LaVO₄ at room temperature.

Using the obtained emission line strengths, the radiative decay rates A(J → J′) can be determined by the following equationsReference Krupke²⁹:

(8)$$<![CDATA[$$A(J \to J^\prime ) = {{64{\rm{\pi }}^4 e^2 } \over {3h\overline {\rm{\lambda }} ^3 }}{{n(n^2 + 2)^2 } \over 9}{1 \over {(2J + 1)}}S_{{\rm{cal}}} = {{8{\rm{\pi }}^2 e^2 n^2 } \over {mc\overline {\rm{\lambda }} ^2 }}P_{{\rm{cal}}} \quad,$$

(9)$$<![CDATA[$${\rm{\tau }}_{{\rm{rad}}} = {1 \over {\mathop\sum\limits_{J^\prime } {A(J \to J^\prime )} \,}}\quad,$$

(10)$$<![CDATA[$${\rm{\beta }}_{J^\prime } = {{A(J \to J^\prime )} \over {\mathop\sum\limits_{J^\prime } {A(J \to J^\prime )} }}\quad,$$

where A(J → J′) is the transition probability for the spontaneous emission per unit time, β_J’ is the fluorescence branching ratios, τ_rad is the radiative lifetime. The calculated parameters for different transition levels are listed in Table IV. The radiative lifetimes of other crystals are also listed in Table III.

TABLE IV.

Calculated luminescence parameters of Nd:LaVO₄ for the radiation ⁴I_9/2 → ⁴I_J’ transitions.

The stimulated emission cross section σ_p related to the radiative transition probability can be defined as:

(11)$$<![CDATA[$${\rm{\sigma }}_{\rm{p}} = {{A(J \to J^\prime ){\rm{\lambda }}_{\rm{p}} ^2 } \over {4{\rm{\pi }}^2 n^2 c\Delta v}}\quad,$$

where $\Delta v$ is the full frequency width at half maximum, λ_p is the vacuum wave length of the emission peak. The emission cross section σ_p of emission spectrum at 1065 nm is 6.5 × 10⁻²⁰ cm² much smaller than that of Nd:YVO₄, Nd:GdVO₄, and Nd:LuVO₄Reference Jiang, Zhang, Wang, Xia, Hu, Teng and Zhang^23, Reference Liu, Xia, Gao, Ran, Sun, Ling, Zhao, Zhang, Zhao and Wang^24, Reference Kaczmarker, Lukasiewicz, Giersz, Jablonski, Jabczynski, Swrkowicz, Galazka, Drozdowski and Kwasny²⁸ due to the wider inhomgeneously broadened fluorescence spectrum. Combining the long fluorescence lifetime, it can be found that this crystal possesses large energy storage ability, which overcomes the limitation of usual vanadates. It is noteworthy that the emission cross section σ_p of emission spectrum is 2.9 × 10⁻²⁰ cm² at 880 nm and it has larger fluorescence luminescence branch ratio (45.29%), which indicates that the crystal has potential application at 880 nm.