2.2.1. First, all of the emission spectrum data points can be expressed as a function with the independent variables and other parameters

Using the spontaneous transition probability , the photoluminescence transition intensity from E_I to E_F can be expressed as

where h is the Plank constant, ν_IF is the emission frequency from the initial state I to the final state F, its value is ν_IF = |E_I − E_F|/h and the corresponding emission wavelength λ_IF = hc/|E_I − E_F|n, c is the light velocity in vacuum, and σ is a constant that depends on the measurement conditions, which we call an instrumental constant. The absolute intensity in J/m³·s is photon energy emitted from the phosphor sample per unit volume per second, and generally is converted into electric current signal by a photoelectric detector such as a photomultiplier tube, whose converted coefficient is σ. It is too troublesome to demarcate σ due to the circumstances, which depend not only on excitation light intensity, phosphor position, pose and excitation area, activator concentration, but also on the sensor sensitivity and so on, so many effects should be considered that it is infeasible. On the other hand, it is unnecessary for an overwhelming majority of spectra measurements because the relative spectral intensity is enough in many cases. So it is very difficult to determine the absolute intensity due to the unknown quantity σ. Meanwhile, when the intensity contribution from magnetic dipole transition, which is usually weaker by a few of orders than that of electric dipole transition, is omitted, if is multiplied by an arbitrary constant s, then

The values of σ and s²σ cannot be discerned experimentally, indicating that the uncertainty of the instrumental constant results in the uncertainty of . Subsequently, it is difficult to directly determine the transition intensity parameters with the luminescence spectrum.

Equation (12a) can be expressed as

if σ and N_Ih are combined as a quantity, that is

It is obvious that σ_I depends not only on the instrumental constant σ but also on the population N_I in the initial state I. It is a great problem for spectroscopy to know the excited state population N_I, which further indicates that it is difficult to obtain the absolute fluorescence intensity , so in the full profile fitting, σ_I is fitted as a variable. The magnitude of ν_IF is about 10¹⁴–10¹⁵ s⁻¹ for the fluorescence in the wavelength range 100 mm–1000 nm, A_IF is about 1–10³ s⁻¹ for rare-earth ions, is about 10⁻¹ for a normalized emission spectrum, so the magnitude of σ_I is about 10⁻¹⁹–10⁻¹⁴ s. In order to keep all of the fitted parameters nearly on the same order, σ_I is written as

where C is an adjustable parameter manually, usually on the order 10⁻¹⁹–10⁻¹⁴ s, and is a parameter fitted by a computational routine.

Assuming that the transition line profile function from E_I to E_F is ϕ_IF(λ, λ_IF,w_IF) against λ with the full width at half maximum (FWHM) w_IF and peak center λ_IF, ϕ_IF(λ, λ_IF,w_IF) is normalized by

where ϕ_IF(λ,λ_IF,w_IF) may be a Lorenz, Gauss or Voigt function, and so on. If the line profile function is a Voigt function, w should be w_L and w_G, which are the FWHM of the Lorentz and Gauss components. Using the line profile function ϕ_IF(λ,λ_IF,w), the transition intensity from E_I to E_F in the whole measurement spectral range can be expressed as

The total intensity of a spectrum from all transitions can be expressed as

If the integral intensity of a spectral band from E_I0 to E_F0 is called , the relative emission intensity can be obtained as

If a spectrum is only from one initial state, 𝓡_IF = 1, equation (17a) can be simplified as

Now every data point in a spectrum is expressed as a function with the independent variables , 𝓡_IF or σ_I, w_IF and λ_IF, by Eqs. (16)–(17c). λ_IF can be directly obtained through line peaks. Usually a spectrum is composed of hundreds or thousands of experimental data points corresponding to the calculated point , so it is possible to determine the transition intensity parameters , w_IF and 𝓡_IF, as well as the Huang–Rhys factor S (described later), which are written in a variable vector x for convenience later. The is expressed as the following general form , i.e.,

2.2.2. Second, in order to remove arbitrariness of the parameters, a quantity associated with the ratio of the experimental to calculated fluorescence decaytime is incorporated into the fitting

According to Eqs. (17a)–(17c), although is independent of the instrumental constant σ when all of the transitions are from the same initial state, it keeps uncertainty. If the magnetic dipole contribution to transition intensity is omitted, let all of be multiplied by an arbitrary constant s,

That is, multiplied by an arbitrary constant s would not change the relative intensity . The uncertainty has not been removed although the instrumental constant σ has been removed from the relative intensities.

On the other hand, if all are multiplied by s, every fluorescence decaytime τ_j changes to τ_j/s² when the magnetic dipole transitions are omitted, so the fluorescence decaytime can be used as an “emendation standard” for to remove arbitrariness from the determination, so a quantity y(x), which depends on the fluorescence decaytime, can be introduced to fit and other parameters, is expressed as

where and are the observed and calculated energy lifetimes of the j_I-th crystal field energy levels, respectively. It can be seen that if , then y(x) → M/2, if , , then y(x) → ∞, so the minimal value of y(x) is zero only when . The observed is constant, and is proportional to , so cannot be scaled arbitrarily, due to the minimum limit of Eq. (19) and the arbitrariness of can be removed by incorporating y(x) into the optical transition intensity fitting.

To a great extent, the reliability of determining the parameters depends on the degree of consistency between and , which implies the follows.

1) is mainly from the radiative transition, and the nonradiative contribution can be omitted, and can generally be written as

where W is nonradiative transition probability. Let ,

and the relative measurement error of is . Further assuming that δ + 1 is all ascribed to , then with error should be , and the relative error for is . As an example, let δ = 0.1, , , if δ = 0.2, , . The nonradiative transition ratio for many optical transitions of rare-earth ions at low temperature is less than 0.1, so the error of the determined that is caused by nonradiative transition is small.

2) To what degree does the calculated trend to . The can be adjusted to be very close to in practice, with a difference that is usually less than 5%. As the above analysis, δ/(δ + 1)< 0.05, then δ < 0.05, , which is very small and can be omitted.

2.2.3. Third, the least-square fitting with a general iteration method is performed to solve the transition intensity and other parameters

In order to solve for variable x, least-squares fitting to the emission spectrum should be performed, which is generally to minimize the following difference value φ including y(x),

The recorded spectral intensity is a series of discrete wavelength and intensity values (λ₁, I(λ₁)), (λ₂, I(λ₂)), …, (λ_N, I(λ_N)), so equation (20) can be written as

where

The minimum of the normal equation of Eq. (20) can be obtained by

where

and n is the variable component number of the variables , w_IF and 𝓡_IF, as well as that of the Huang–Rhys factor S if necessary.

The x can be obtained by solving Eq. (23), which is a nonlinear equation system and can be solved by the Levenberg–Marquardt method, whose iteration formula is given by

where k stands for the k-th iteration, x^k is the value at the k-th iteration, μ_k is damping factor at the k-th iteration, and I is unit matrix.

3.3.1. Initial parameters estimation of

The wavefunctions, |Ii_α⟩ and |Ff_β⟩ mentioned previously, calculated by the superposition model parameters in Ref. [12], were used to calculate spectral transition intensity. The initial parameters were evaluated by the point charge model in the international system of units with

where ε₀ is the dielectric constant in vacuum, e is the charge of an electron, Z_L is the charge number of the L-th ligand, R_L is the distance between the central ion and the L-th ligand, is the spherical harmonic function, * stands for the complex conjugate operation, and Θ_L and Φ_L are angular coordinates of the L-th ligand. The dimension of is energy/L^t.

Another quantity to be evaluated is r^{t + 1}(n,l,n′,l′) in Ξ(t,k), whose dimension is L^k+1/energy. According to Eq. (8),

According to Krupke’s approximation,^[13]

In Krupke’s table,^[11] ⟨4f|r|5d⟩, ⟨4f|r³|5d⟩, ⟨4f|r⁵|5d⟩, ⟨4f|r⁴|4f⟩, ⟨4f|r⁶|4f⟩, ⟨4f|r⁸|4f⟩, Δ(5d) and Δ(n′g) are listed for Pr³⁺, Nd³⁺, Eu³⁺, Tb³⁺, Er³⁺ and Tm³⁺. In oxide compounds, Δ (5d) can be estimated by the work of Dorenbos.^[14] Goldner et al.^[15] give estimations of Δ(5d) and Δ(5g) for Yb³⁺, which are 21700 cm⁻¹, 51340–72340 cm⁻¹, respectively. The radial integrals and energy denominators Δ(5d) and Δ(5g) for Tm³⁺ and Eu³⁺ are also listed in Krupke’s work.^[16]

Based on these works, the values of ⟨4f|r^t|5d⟩ and ⟨4f|r^{t + 1}|4f⟩ used to estimate the values of Ξ(t,k) are shown in Table 1.

In the site symmetry C₂, there are twenty-five non-vanishing parameters , which are (k, t, p) = (0,1,0), (2,1,0), (2,3,2), (4,3,2), (2,3,0), (4,3,0), (2,3,−2), (4,3,−2), (4,5,4), (6,5,4), (4,5,2), (6,5,2), (4,5,0), (6,5,0), (4,5,−2), (6,5,−2), (4,5,−4), (6,5,−4), (6,7,6), (6,7,4), (6,7,2) (6,7,0), (6,7,−2), (6,7,−4), and (6,7,−6). The values of with p = 0 are real, the others are complex. There are twenty-five independent real parameters. The imaginary and real parts of with p > 0 and with p = 0 were chosen as the independent variables to be fitted. The estimated initial values are shown in Table 2.

3.3.2. Fitting intensity and discussion of results

The energy levels of Yb³⁺:GdTaO₄ are shown in Fig. 3. Due to the strong re-absorption, it is very difficult to obtain the accurate line intensity at about 973 nm from the transition E₅ → E₁, and this line would not be included in our full-profile fitting. On the other hand, the line of 973 nm is a zero-phonon line, on which no obvious phonon vibronic peak was observed, so it is reasonable that no phonon assisted transition was incorporated in the transition E₅ → E₁, whose electric dipole and magnetic transition probability can be calculated with and wavefunctions of Yb³⁺ in GdTaO₄, and the calculated energy lifetime of E₅ can be obtained. Correspondingly, the experimental photoluminescence decaytime of 1014 nm and excitation wavelength 972 nm at temperature 8 K was taken as the energy lifetime of E₅, so it is viable to take the energy lifetime of E₅ as a fitted quantity in Eq. (20).

The lattice relaxation shifted the emission wavelength by 5.3 cm⁻¹, so the energy level position of 10285 cm⁻¹ was changed and fixed as 10280 cm⁻¹ during the fitting procedure in order to keep the calculated transition wavelength consistent with the corresponding experimental line peaks. The vibronic peaks 1009.2 nm, 1020.0 nm, and 1038.6 nm were accounted for, their frequency center shifts are 88 cm⁻¹, 59 cm⁻¹, and 45 cm⁻¹ relative to the electronic state transition peaks 999.8 nm, 1013.4 nm, and 1033.3 nm, and were approximated by single-frequency phonons here, whose Huang–Rhys factors labeled by S₁, S₂, and S₃ were also fitted. On the other hand, the FWHM of the lines of 1000.3 nm, 1014 nm, and 1033 nm peaks with Lorentz profiles, which were labeled as w₁, w₂, and w₃, respectively, were also included in the fitting. In sum, the fitting variable x is composed of twenty-five components of , three Huang–Rhys factors S₁, S₂, S₃, and FWHM w₁, w₂, w₃, and σ_I, respectively.

A detailed Levenberg–Marquardt iteration algorithm with Eq. (25) is described in Ref. [17]. Here, the maximal iteration loop number, and searching damping factor loop number were set as 100 and 40, respectively.

The refractive index of Yb³⁺:GdTaO₄, which is necessary for calculating transition probabilities, is calculated by the equation^[18]

where A = 3.89546, B = 0.53175 μm², C = −0.30491 μm², and D = −0.05039 μm⁻².

After 88 iterations, the relative residual R is 11.75%, and the experimental and fitting curves are shown in Fig. 4. The calculated energy lifetime is 363.99 μs, which is close to the experimental value 364 μs. The fitted curves are very consistent with the experimental curves. The fitted values are shown in Table 2. Comparing the fitted with their initial values, the difference between the initial and fitted values is large, but the initial and final values of (k, t, p) = (2,1,0), (4,3,2), (2,3,0), (4,3,0), (4,5,0), (6,7,4), (6,7,2), and (6,7,0) keep the same sign.

The transition probabilities and line strengths obtained from the intensity fitting are shown in Table 3. It can be seen that the electric dipole transitions are dominant, the main component of magnetic dipole transition is the transition E₅ → E₁, whose ratio to electric dipole is 31%, E₅ → E₄ and E₅ → E₄ + phonon is 0.3%, while the others are far less than 1%. The fluorescence branch ratios are also listed in Table 4. The highest branch ratio, 39.5% is from the transition E₅ → E₂, and the second is E₅ → E₃, whose branch ratio is 24.7%. The emission cross section of E₅ → E₁ is calculated with the obtained and the Lorenz profile with FWHM 1.33 nm of 973 nm, and the other transitions E₅ → E₂, E₃, E₄ have also been calculated, and their curves are shown in Fig. 5. The emission cross sections of 973 nm, 1000 nm, 1009 nm, 1014 nm, 1019 nm, 1034 nm, 1039 nm are 2.1, 10.0, 6.0, 8.2, 1.8, 3.4, and 1.3 in 10⁻²⁰ cm², respectively. From Fig. 1(b), we can see that the experimental intensity of E₅ → E₁ is very weak at 8 K temperature, which is the result of reabsorption. As a comparison, the emission cross section of Yb³⁺:GdTaO₄ was calculated by the F–L equation^[19]

The emission cross sections obtained by this equation are also shown in Fig. 5. The emission cross sections of 973 nm, 1000 nm, 1009 nm, 1014 nm, 1019 nm, 1034 nm, 1039 nm are 2.6, 11.1, 6.8, 8.8, 2.0, 3.4, and 1.3 in 10⁻²⁰ cm². It can be seen that the emission cross sections obtained by F–L in the low wavelength range are less than the calculated values, which maybe results from the case that g(λ) is smaller than the value when there is reabsorption. The relative error ratios of the emission cross sections given by the F–L formula to calculated values by full-profile method are 23%, 11%, 13%, 7%, and 11% at the wavelength 973 nm, 1000 nm, 1009 nm, 1014 nm, and 1019 nm, respectively. This indicates that the emission cross sections obtained by the F–L formula are higher than the actual values in the low wavelength region and, otherwise, are the same as that in the long wavelength range.

Although the transition intensity parameters have been obtained, Judd–Ofelt transition intensity theory with the triplet parameters Ω_t (t = 2, 4, 6) cannot be applied at low temperatures because the populations in the stark levels of ²F_7/2 or ²F_5/2 are unequal. It can be seen that the photoluminescence of Yb³⁺in GdTaO₄ at 8 K are from the lowest stark level E₅ of ²F_5/2, and the transitions from the other stark levels of ²F_5/2 were not observed. So the approximation that the photoluminescence is caused by the multiple transitions of ²F_5/2 → ²F_7/2 is not valid, which can be shown as follows. The Ω_t (t = 2, 4, 6) calculated with the parameters are 0.27, 1.46, 6.67 in 10⁻²⁰ cm². The electric dipole transition line strength can be calculated by

in which the values of (⟨²F_7/2||U^t||²F_5/2⟩) (t = 2, 4, 6) are , , and . The calculated S_ed is 7.2 × 10⁻²⁰ cm², which is much greater than the 0.86 × 10⁻²⁰ cm² obtained from fitting to the emission spectrum of Yb³⁺:GdTaO₄. Apart from the unequal Boltzmann population of the stark levels of F_5/2, the relaxation from the higher stark levels to the lowest stark level E₅ of ²F_5/2 in Yb³⁺:GdTaO₄ results in a more unequal population. So only the transitions from the lowest stark level of ²F_5/2 to the stark levels of ²F_7/2 were observed experimentally.

When Yb³⁺:GdTaO₄ is used as a laser medium, it is expected that laser operation at the zero-phonon line 972 nm is not realizable due to strong reabsorption. If the laser operates as follows: Yb³⁺ is pumped to the excited level E₆, rapidly relaxed to E₅, and then transitions to the background state E₁, a three-level system forms, and a very high laser threshold is expected, so the optical transition of E₅ → E₁ is nonsense for laser operation. It can be seen that the emission cross sections in the range of 1000–1019 nm from E₅ → E₂, E₃ are high, and no great reabsorption occurs at low temperatures, the prospect of realizing laser operation in this channel is promising. When the working temperature of laser material increases, the thermal population of E₂ increases, and the reabsorption of 1000 nm increases, which result in the case that the strongest line is 1014 nm, rather than 1000 nm at room temperature, which is shown as Fig. 6. Table 4 lists the thermal populations of E₁–E₄ at 8 K, 100 K, 200 K, and 300 K. It can be seen that the population of the lowest level E₁ decreases, while the populations of the other levels increase gradually with increasing temperature. High thermal population of E₂–E₄ will result in difficulty of laser population inversion, so it is desirable for Yb³⁺:GdTaO₄ to operate at a low temperature.

At the same time, an increase in host temperature also results in broadening the FWHM of the emission spectrum, and a continuous emission band from 1001 nm to 1035 nm can be observed at room temperature (shown in Fig. 6) due to a remarkable increase in the strong phonon assist. The bandwidth of 34 nm is nearly equal to gain bandwidth 10.2 THz. The pulse width of a mode-locked laser can be estimated by^[20]

where is gain bandwidth, which gives Δτ_p ≅ 43 fs and favors the femtosecond pulse width. The E₆, as the pump level of a 970 nm laser diode (LD), E₅, as a laser up level, and E₂, E₃, E₄ as the terminal levels, of Yb³⁺:GdTaO₄ form a good quasi-four-level system of femtosecond laser or tuning laser from 1000–1034 nm. High power laser diodes of 970 nm are commercially available at good prices, so Yb³⁺:GdTaO₄ is a very promising laser material with the wide tuning laser wavelength of 1000–1034 nm or a ultra-short pulse laser pumped by LD, which can be used in many fields such as machining, gas detection, and so on.