1. INTRODUCTION

Coherent, narrowband optical emission from solar-pumped lasers is actively pursued in the context of renewable energies for magnesium energy cycling [1], hydrogen generation [2], or efficient photovoltaic conversion [3]. Owing to its low power density and broad optical spectrum, sunlight used for pumping laser media generally requires optical concentrators, which call for sun-tracking and cooling systems. Using a side-pumped ${\mathrm{Nd}}^{+3}$-doped fiber placed inside a greenhouse chamber, Masuda et al. [4] recently demonstrated a solar-pumped laser with unprecedently low threshold of oscillation. The optical fiber provides low losses at the lasing wavelength, while the side pumping geometry provides in principle unlimited power scalability with the fiber length without loss of beam quality. The small thickness of the core entails low absorption efficiency in the side-pumped geometry; this can be alleviated by placing the fiber inside an enclosure, which is a greenhouse chamber containing a wavelength-converting sensitizer and a dichroic reflective front mirror for trapping luminescent photons [5]. Such a design opens the possibility to eliminate concentrators and does away with sun-tracking and cooling systems. The proposed side-pumped design offers an attractive alternative to lasers powered by photovoltaics because it makes it possible to bypass the use of solar panels ($\text{efficiency}\approx 20\%$) and laser diodes ($\text{efficiency}\approx 50\%$) and thus achieve in principle higher efficiency at a lower cost. Moreover, a side-pumped geometry offers a greater power scalability than an end-pumped scheme, which is the only option for a design with a solar panel and laser diode pumping scheme.

Requirements for a low concentrated solar-pumped fiber laser are different from those of most fiber lasers. The latter use high-brightness laser diodes as a pump source, coupled into a double-clad fiber, with the core doped with, for example, quasi-four-level ${\mathrm{Yb}}^{3+}$ ions; they also operate well above the threshold of laser oscillation [6]. For low pump power density applications such as solar-pumped lasers, a four-level laser system is needed to prevent losses from ground-state absorption. Moreover, for broadband solar emission, high absorption in the visible region is desirable. These two factors explain the preference for ${\mathrm{Nd}}^{3+}$ over ${\mathrm{Yb}}^{3+}$ for such applications.

The ${\mathrm{Nd}}^{3+}$ glass laser operated at around $\lambda =1.1\mathrm{\mu m}$between the ${}^4{\mathrm{F}}_{3/2}$ and ${}^4{\mathrm{I}}_{11/2}$ Stark levels is considered a true four-level system owing to the large energy separation ($\approx 2000{\mathrm{cm}}^{-1}$ [7]) between the lower laser level with the ground level ${}^4{\mathrm{I}}_{9/2}$ compared to the thermal energy ($k_{B}T\approx 200{\mathrm{cm}}^{-1}$); this apparently guarantees that the population of the lower laser level is negligibly small, and therefore saturable losses should also be negligible. However, optical background losses of a good optical fiber are also exquisitely small, namely, on the order of $1{\mathrm{km}}^{-1}$ or less, and the question arises whether the thermal population of the lower laser level significantly affects the total fiber losses at the laser wavelength. Saturable losses arising from the excited ${\mathrm{Nd}}^{3+}$ population of the ${}^4{\mathrm{I}}_{11/2}$ level have a well-known Maxwell–Boltzmann dependence with temperature. Hence, it is possible to separate them from constant background losses by making temperature-dependent absorption measurements. Only background losses should subsist at low temperature. The optical fiber is a convenient geometry to probe low absorption materials owing to its long length and low background losses that would otherwise mask these saturable losses.

In this work, the temperature dependence of the absorption coefficient of the ${}^4{\mathrm{I}}_{11/2}\text{-}{}^4{\mathrm{F}}_{3/2}$ was measured in a ${\mathrm{Nd}}^{3+}$-doped aluminosilicate fiber at several temperatures ranging from $-34°\mathrm{C}$ to 50°C. The measured dependence of the peak absorption coefficient with temperature enabled the determination of the fraction of thermally excited ions into the lower laser level, which, combined with the knowledge of the ${\mathrm{Nd}}^{3+}$ concentration, enabled the determination of the effective absorption cross-section, ${\sigma }_a(\lambda )$, of the laser transition. We used our knowledge of ${\sigma }_a(\lambda )$ to reliably determine the emission cross section, ${\sigma }_e(\lambda )$, which is a key laser parameter since its product with the population inversion determines the amplification coefficient of an amplifier or oscillator. Published reports on the emission cross section of the ${}^4{\mathrm{F}}_{3/2}\text{-}{}^4{\mathrm{I}}_{11/2}$ transition of ${\mathrm{Nd}}^{3+}$-doped glasses have ignored ${\sigma }_a(\lambda )$ measurements for lack of availability, since slab geometries are too thin to allow the detection of tiny absorption losses at 1060 nm. Published reports have instead relied on the use of the $\beta -\tau$ method [8–10], which requires the lineshape function, branching ratios, ${\beta }_i$, of the possible terminal Stark level, $\mathrm{J}=9/2$, 11/2, 13/2, and 15/2; and radiative lifetime ${\tau }_{\mathrm{rad}}$. However, ${\tau }_{\mathrm{rad}}$ differs from the directly accessible fluorescence lifetime, ${\tau }_f$, due to nonradiative processes such as cooperative relaxation and multiphonon relaxation [11] or due to reabsorption trapping [12]; consequently, ${\tau }_f$ must be combined with quantum fluorescence efficiency measurements [13]. Since these measurements invariably have systematic errors, they are often supplemented with a Judd–Ofelt analysis [14–16], which requires analyzing several absorption bands from the ground level to enable useful estimation of the ${\beta }_i$ values and ${\tau }_{\mathrm{rad}}$.

The knowledge of the absorption cross section of the laser transition gives two advantages for an accurate determination of ${\sigma }_e$ compared to the former approach. First, it enables a direct estimate of the linestrength or radiative lifetime into the ${}^4{\mathrm{I}}_{11/2}$ manifold [17], which can be used in combination with the measured relative lineshape function to get the absolute ${\sigma }_e$ spectrum. Second, McCumber’s reciprocity principle [18,19] can be used, and this allows one to calculate ${\sigma }_e$ without the need to know the radiative lifetime, branching ratio, and quantum luminescence efficiency of the initial level. Hence, it provides another means for the determination of the ${\sigma }_e$, and thus enables a validation of the results obtained with the former method. Its accuracy in determining ${\sigma }_e$ primarily depends on the accuracy of ${\sigma }_a$, which is easier to measure, and on some knowledge of the position of the sublevels of each manifold. Since the application of McCumber’s principle require absorption spectra, it has been used only for transitions from the ground level, not for the characterization of higher Stark lower levels [10,19–22]. Here, we use McCumber’s principle to estimate ${\sigma }_e$ for the ${{}^4\mathrm{F}}_{3/2}\to {}^4{\mathrm{I}}_{11/2}$ laser level of ${\mathrm{Nd}}^{3+}$ doped in aluminosilicate glass. Small signal gain spectra obtained in an amplifier configuration are also presented at low ($-37°\mathrm{C}$) and room temperatures and compared to ${\sigma }_e$ since both parameters are proportional to each other.

The paper is organized as follows. In Section 2, we describe the materials, the experimental methods, and the theory used for the measurement of the temperature-dependent absorption and emission spectra. We also summarize the theory underlying the connection between the absorption spectra and ${\tau }_{\mathrm{rad}}$, as well as McCumber’s reciprocity principle. We describe the experimental procedure for the spectral gain distribution measurements in an amplifier configuration. We outline the Judd–Ofelt analysis, a technique that we use to validate our results. We conclude this section by a description of the experimental method used to determine a fluorescence lifetime that is free of reabsorption trapping artifacts. All experimental results are gathered in Section 3: emission cross sections calculated from the linestrength measurements are shown and compared with predictions from the McCumber’s reciprocity principle, with the spectral gain distribution obtained at room and low temperatures and with results from a Judd–Ofelt analysis. In Section 4, we summarize our results. We also provide estimates on how much the oscillation threshold and wavelength are impacted by operating the solar pumped fiber laser at a lower temperature.

2. METHODS, MATERIAL, AND EXPERIMENTAL PROCEDURE

A. Absorption Cross Section Measurements of the ${}^4{\mathrm{I}}_{11/2}\text{-}{}^4{\mathrm{F}}_{3/2}$ Transition

The active fiber is a 16-μm-core, $\mathrm{NA}=0.18$, ${\mathrm{Nd}}^{3+}$-doped aluminosilicate fiber supplied by Furukawa Denko Corp. The nominal average concentration of ${\mathrm{Nd}}^{3+}$ ions provided by the manufacturer is $N_{\mathrm{Nd}}=1.3\times {10}^{20}\mathrm{at}./{\mathrm{cm}}^3$, and the Al:Nd atomic concentration ratio estimated by electron probe x-ray microanalysis is 21:1. The transmittance spectrum, $T_{\lambda }$, of the active fiber was measured in the spectral range of the ${}^4{\mathrm{I}}_{11/2}\text{-}{}^4{\mathrm{F}}_{3/2}$ laser transition by injecting tens of μW of radiation from a fiber-coupled superluminescent diode (SLED) (Thorlabs, model S5FC) emitting in the range from $\lambda =950$ to 1170 nm into the fiber, and by measuring the output spectrum with an infrared spectrometer operating in the same spectral range (Ocean Optics, model NIRQuest). The absorption coefficient at room temperature was obtained by taking the ratio of transmittance spectra obtained at two different fiber lengths, $L_1=1.3\mathrm{m}$ and $L_2=200\mathrm{m}$, with the following formula:

The ${\mathrm{Nd}}^{+3}$ absorption cross section of the laser transition was obtained from the absorption coefficient by using the following procedure. First, we estimated the background absorption coefficient, ${\alpha }_{\mathrm{BG}}(\lambda )$, at both ends of the measured spectral range, where the ${\mathrm{Nd}}^{3+}$ cross section is negligible. ${\alpha }_{\mathrm{BG}}(\lambda )$ was then subtracted from the measured absorption spectrum $\alpha (\lambda )$ to get ${\alpha }_{\mathrm{Nd}}(\lambda )$. Next, the fraction of ions in the ${}^4{\mathrm{I}}_{11/2}$ manifold was estimated at room temperature using the Maxwell–Boltzmann distribution:

Table 1. Energy Levels of ${\mathrm{Nd}}^{3+}$ in Silicate Glass (taken from [7])

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The effective absorption cross section is obtained from

In order to compare our results with those predicted by the Judd–Ofelt theory, the transmission spectrum of the ground state ${}^4{\mathrm{I}}_{9/2}$ was also measured from 400 to 1000 nm by using a halogen lamp white light source coupled to a single-mode fiber (Ando Corp., Model AQ4305). The cut-back method was used to determine the absorption coefficient with 4- and 22-mm-long active fiber for all bands, except for the stronger ${}^4{\mathrm{G}}_{5/2}\text{-}{}^2{\mathrm{G}}_{7/2}$ band at around 580 nm, for which 4- and 12-mm-long active fiber lengths were used.

B. Determination of the Emission Cross Section

Using the principle of detailed balance applied to a two-state system in thermal equilibrium with a radiation field emitted from a blackbody, Einstein established that stimulated emission and absorption probabilities per unit energy density and per unit time were equal:

C. McCumber’s Reciprocity Principle

Both ${\sigma }_a(\lambda )$ and ${\sigma }_e(\lambda )$ are compared using McCumber’s reciprocity principle. These cross sections are effective cross sections, which differ from spectroscopic cross sections. The spectroscopic absorption and emission cross sections for a given pair of nondegenerate sublevels, ${\sigma }_{ij}$, are equal, and this is a direct consequence of the equality of Einstein coefficients in Eq. (4a). Effective cross sections are defined to relate the gain with the populations of the Stark manifolds, $N_1$ and $N_2$:

D. Gain Measurements under Pseudo-Solar Illumination

Gain was measured by illuminating a 85-m-long fiber with a xenon arc lamp (Model Seric XC-500ASS), whose emission spectrum is close to AM0 solar radiation and power density is about 75% of solar radiation. The fiber was placed in a greenhouse chamber similar to those described in [4,5]. It has totally reflecting walls and a dichroic front Bragg mirror that transmits light with $\lambda <570\mathrm{nm}$ and totally reflects wavelengths with $\lambda >570\mathrm{nm}$ at any incident angle from 0° up to 90°. Rhodamin 6G dye diluted in methanol in 0.3 mmol/l concentration was chosen as a sensitizer because of its high luminescent quantum yield, high absorption of sunlight, and good overlap of its luminescence spectrum with the strong ${}^2{\mathrm{G}}_{7/2}\text{-}{}^4{\mathrm{G}}_{5/2}$ absorption bands of ${\mathrm{Nd}}^{3+}$ between 570 and 600 nm. Laser gain experiments were similar to absorption measurements presented in Section 1. The differential gain, $\gamma (\lambda )$, is calculated with the formula

E. Judd–Ofelt Analysis

A Judd–Ofelt analysis [14–16] was performed by analyzing the linestrength of nine ${\mathrm{Nd}}^{3+}$ absorption bands obtained between $\lambda =400$ and 1100 nm. This enabled the estimation of the transition probabilities from the metastable ${}^4{\mathrm{F}}_{3/2}$ level to the four lower terminal Stark levels ${}^4{\mathrm{I}}_{9/2,11/2,13/2,15/2}$. According to the Judd–Ofelt model, the linestrength of an electric-dipole transition between initial $J$ and terminal manifolds can be written in the following form:

In contrast to other published works, the absorption band from the lower laser level ${}^4{\mathrm{I}}_{11/2}$ is also included in the analysis. Doing so, experimental linestrengths from the metastable ${}^4{\mathrm{F}}_{3/2}$ of the ground (${}^4{\mathrm{I}}_{9/2}$) and the lower laser (${}^4{\mathrm{I}}_{11/2}$) levels can be compared with estimates obtained with Judd–Ofelt analysis performed on all absorption bands. This analysis also allows us to estimate the transition probabilities of the other two terminal manifolds ${}^4{\mathrm{I}}_{13/2}$ and ${}^4{\mathrm{I}}_{15/2}$, which were not experimentally available. The summation of the transition probabilities over the four terminal manifolds gives the total transition probability $A$, from which the radiative lifetime, ${\tau }_{\mathrm{rad}}=A^{-1}$, is obtained.

F. Fluorescence Lifetime Measurements

Decay time measurements of the ${}^4{\mathrm{F}}_{3/2}$ level were carried out to check the estimates obtained thus far with direct measurements of the absorption of the ${}^4{\mathrm{I}}_{9/2}$ and ${}^4{\mathrm{I}}_{11/2}$ levels and with the Judd–Ofelt analysis. A piece of active fiber was spliced to an undoped fiber and placed at the center of a 2-inch-diameter integrating sphere (IS). A fast photodetector was placed at a 1” port of the IS. Light from a temperature-controlled fiber-coupled laser diode (LD) emitting at $\lambda =808\mathrm{nm}$ (model L808P1000MM from Thorlabs) in the continuous wave regime, controlled by a LD driver with 325-kHz bandwidth (model Arroyo Dr-4205), was injected into the fiber core. Two photodetectors were used for comparison: a silicon-based detector, model Advantest (Q82017A) and a germanium-based detector, model Advantest TQ82015. The overall response time of the LD driver and detector was measured at around 3 μs. The detector could detect luminescence emitted from most of the $4\pi$ sphere solid angle around the side and the tip of the active fiber.

It is well known that light trapped inside the active material by total internal reflection is likely to be reabsorbed by the ground level ${}^4{\mathrm{I}}_{9/2}$ of ${\mathrm{Nd}}^{3+}$ before being emitted again. This causes artificial lengthening of the lifetime [12] that creates systematic errors in the intrinsic fluorescence lifetime and makes the estimation of ${\tau }_{\mathrm{rad}}$ more difficult. In order to eliminate this effect, decay times were measured for various lengths of the active fiber ranging from 2 to 15 mm. As the fiber length becomes shorter, the probability of reabsorption decreases, and the effect vanishes at the limit where the material is optically thin. This approach is similar to the so-called pinhole method, proposed by Kuhn et al. [27] and revisited by Toci [28], wherein the decay time is measured for several values of pinhole diameter and results are extrapolated to zero-size pinhole diameter. The decay curve when LD light was cut was recorded and fitted to an exponential after removing the background offset.

3. RESULTS

A. Absorption and Emission Cross Sections

The transmission spectra at the laser transition band of the SLED obtained with short and long fibers and the corresponding absorption coefficient deduced from Eq. (1a) are shown in Fig. 1.

 

Fig. 1. Absorption spectrum of the laser transition measured at room temperature, obtained with Eq. (1a). Raw and filtered measurements in green and red, respectively. Corresponding transmission spectra of the SLED, obtained with short and long fibers, are shown in the inset.

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Cooling the fiber to $-34°\mathrm{C}$ enabled the elimination of most saturable absorption losses, leaving only the background losses, in the order of 1 to $2{\mathrm{km}}^{-1}$, as the dominant loss mechanism (Fig. 2). The evolution of the peak absorption of ${\mathrm{Nd}}^{3+}$ at $\lambda =1057\mathrm{nm}$ as a function of inverse temperature, after subtraction of the background absorption coefficient of ${\alpha }_{\mathrm{BG}}=1.25{\mathrm{km}}^{-1}$, is shown in Fig. 3. The slope of the function $\ln (\alpha -{\alpha }_{\mathrm{BG}})$ with $1/T$, which depends on the energy distance to ground level, confirms the Boltzmann character of the thermal population of the ${}^4{\mathrm{I}}_{11/2}$ level, namely, $N_1\propto \exp (-E_0/k_{B}T)$, and the $E_0\approx 2000{\mathrm{cm}}^{-1}$ value validates the estimate performed with Eq. (2) of $N_1/N_{\mathrm{Nd}}=7.8\times {10}^{-5}$ at room temperature. Absorption rapidly drops when the temperature is lowered, decreasing by one order of magnitude by lowering the temperature to $-34°\mathrm{C}$ from room temperature. The overall absorption cross section spectrum at room temperature, obtained from Eq. (3) after subtracting the background absorption, is shown in Fig. 4 (green). The measured $A_{{}^4\mathrm{F}_{3/2}\to {}^4\mathrm{I}_{11/2}}=753{\mathrm{s}}^{-1}$ value found from Eq. (7), combined with the measured normalized luminescence spectrum, enabled a determination of the effective emission cross section with Eq. (10) (Fig. 4, red curve).

 

Fig. 2. Absorption spectrum measured at various temperatures ranging from $-34°\mathrm{C}$ (bottom) to 50°C (top) by 12°C steps (solid lines). Absorption increases with temperature. The black dashed curve is the absorption spectrum measured at 23°C, used as a baseline to calculate other absorption spectra. The dotted line is the estimated background losses.

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Fig. 3. Evolution of the peak absorption value as a function of temperature.

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Fig. 4. Measured effective absorption (green) and emission (red) cross sections of the ${}^4{\mathrm{F}}_{3/2}\text{-}{}^4{\mathrm{I}}_{11/2}$ laser transition band of ${\mathrm{Nd}}^{3+}$-doped aluminosilicate optical fiber. The blue curve is deduced from the emission cross section using McCumber’s principle.

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The photon energy at which ${\sigma }_a={\sigma }_e$ is given from Eq. (17) and Table 1 by

 

Fig. 5. Effective cross section ratio (thick curve) versus wavenumber showing the simple exponential dependence between both quantities. The wavelength interval shown is from 1010 (right) to 1136 nm (left). The model of Eq. (17) is shown as a straight line (dashed).

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B. Gain Measurements When the Fiber Is Exposed to a Sunlight Simulator

An experimental measurement of the differential gain, that is, the difference between fiber gain with and without pseudosolar illumination, is shown in Fig. 6 at both low and room temperatures. The experimental maximum gain value is found to be slightly above 1060 nm, as expected from the emission cross-section spectrum. When illuminating the fiber, the fraction of ${\mathrm{Nd}}^{3+}$ ions excited in the metastable level is extremely small, so the population of the ground manifold remains constant, and so does $N_1$ since $N_1/N_0$ is a fixed fraction determined by temperature; hence, only $N_2$ appreciably changes in Eq. (11), and the differential gain is given by

 

Fig. 6. Experimental differential gain per unit length of Nd-doped fiber observed with and without illumination from an Xe lamp at room temperature (red solid line) and $-37°\mathrm{C}$ (green dashed line), and comparison with ${\sigma }_e$ obtained from linestrength measurements and the luminescence spectrum (blue dotted line).

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C. Judd–Ofelt Analysis and Lifetime Measurements

The obtained absorption spectrum in the visible and near IR and the energy level assignment are shown in Fig. 7. A comparison between experimental linestrengths of the various absorption bands of ${\mathrm{Nd}}^{3+}$ in the visible and near IR measured with Eq. (2), $S_{\exp }$, with those estimated by Judd–Ofelt analysis with Eq. (19), $S_{\mathrm{calc}}$, is shown in Table 2 together with the ${\mathrm{\Omega }}_i$ parameters found with a least-square regression. Transition probabilities and branching ratios from the ${}^4{\mathrm{F}}_{3/2}$ metastable level, and the corresponding radiative lifetime estimate, together with the experimentally measured fluorescence lifetime, are shown in Table 3. For both ${}^4{\mathrm{I}}_{9/2}$ and ${}^4{\mathrm{I}}_{11/2}$ levels, direct experimental values and those estimated from the fitting of all nine absorption bands are very similar. Measured fluorescence lifetimes are shown in Fig. 8 as a function of active fiber length; the near constancy of ${\tau }_f$ with fiber length indicates negligible reabsorption effects for the range of active fiber lengths used. Given the high ${\mathrm{Nd}}^{3+}$ concentration, nonradiative quenching effects taking place in ${\mathrm{Nd}}^{3+}$ ions are probably responsible for the observed difference between ${\tau }_{\mathrm{rad}}\approx 610\mathrm{\mu s}$ and ${\tau }_f\approx 450\mathrm{\mu s}$. This indicates how challenging it is to achieve a good estimate of ${\sigma }_e$ with the $\beta -\tau$ method, which requires knowledge of both the branching ratios and radiative lifetime.

Table 2. Integrated Cross Section and Linestrength Measurement^a

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Table 3. Calculated Transition Probabilities and Branching Ratios from ${}^4{\mathrm{F}}_{3/2}$ Metastable Level^a

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Fig. 7. Absorption spectrum of the optical fiber. Solid: obtained from 4- to 22-mm-long fiber; dashed: obtained from 4- to 12-mm-long fiber. The latter was used only to estimate absorption from the ${}^4{\mathrm{G}}_{5/2}\text{-}{}^2{\mathrm{G}}_{7/2}$ bands. The initial level is ${}^4{\mathrm{I}}_{9/2}$ for all bands except for band 1, shown in the insert, for which the initial level is ${}^4{\mathrm{I}}_{11/2}$.

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Fig. 8. Experimental decay times obtained for different lengths of active fiber with Si (circles) and Ge (stars) detectors.

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4. SUMMARY, DISCUSSION, AND CONCLUDING REMARKS

In summary, taking advantage of the long geometry of a ${\mathrm{Nd}}^{3+}$-doped aluminosilicate fiber, we were able to measure the extremely low absorption losses around the lasing wavelength at various temperatures and show that these losses are dominated by absorption from ${\mathrm{Nd}}^{3+}$ ions that are thermally excited to the ${}^4{\mathrm{I}}_{11/2}$ level at room temperature. The lowering of the temperature to $-34°\mathrm{C}$ made it possible to almost eliminate them and reduce the total fiber loss by one order of magnitude. The knowledge of the ${\mathrm{Nd}}^{3+}$ concentration and the energy distance of the lower laser level to the ground level enabled in turn accurate determination of the effective absorption cross section of the ${}^4{\mathrm{I}}_{11/2}\text{-}{}^4{\mathrm{F}}_{3/2}$ level. This information proved crucial for the accurate determination of the effective emission cross section of this transition for two reasons: it enabled the direct determination of the transition probability, $A_{{}^4\mathrm{F}_{3/2}\to {}^4\mathrm{I}_{11/2}}$; it also enabled the use of McCumber’s reciprocity principle to validate our calculation. To our knowledge, this is the first report of McCumber’s principle being used to assess the cross section of a transition not involving the ground level. Both methods yield absolute estimates and agree well with each other.

Gain measurements as a function of wavelength carried out with a source mimicking solar radiation were found to be very similar to the measured ${\sigma }_e(\lambda )$, except that the gain peak at around 1064 nm was not as prominent as predicted from, and blueshifted with respect to, the lineshape function. Figure 9 shows that McCumber’s calculation of ${\sigma }_e$ from ${\sigma }_a$ also suggest a smaller hump in this function than what the lineshape function predicts. The gain curve was found to remain almost unchanged when lowering the temperature, which indicates that the effective cross sections barely change with temperature. This is expected by the fact that the ${}^4{\mathrm{F}}_{3/2}$ level only has two sublevels that are not far apart, namely, a $142{\mathrm{cm}}^{-1}$ energy difference was measured for a similar glass, and the breadth of the ${}^4{\mathrm{I}}_{11/2}$ level is also small, in the order of $350{\mathrm{cm}}^{-1}$ [7].

 

Fig. 9. Comparison of the stimulated emission cross section distributions measured with Füchtbauer–Ladenburg relation (blue dotted line), McCumber reciprocity principle (green dashed line), and direct gain measurements (red solid line). The maximum measured gain value is less prominent and blueshifted relative to that predicted from Füchtbauer–Ladenburg but conforms to McCumber’s estimate.

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The Judd–Ofelt analysis produced an estimate of the transition probabilities to the ${}^4{\mathrm{I}}_{11/2}$ level consistent with that found from the absorption spectrum of the same transition. The large discrepancy of the radiative lifetime, ${\tau }_{\mathrm{rad}}$, and the experimentally accessible fluorescence lifetime, ${\tau }_f$, clearly showed that knowledge of the quantum fluorescence efficiency is needed together with the branching ratio when the $\beta -\tau$ method is employed. Our approach based on the direct determination of the transition probability and McCumber’s reciprocity principle does not have these drawbacks.

Knowledge of ${\sigma }_a(\lambda )$, ${\sigma }_e(\lambda )$, and background losses makes it possible to calculate the gain spectrum for any value of the excited population in the metastable ${}^4{\mathrm{F}}_{3/2}$ level, $N_2$, using Eq. (11) and to predict the oscillation threshold and the oscillation wavelength. Gain curves at room temperature are shown in Fig. 10 for various values of the ratio of $N_2/N_{\mathrm{Nd}}$. The positive gain threshold is first reached for $N_2/N_{\mathrm{Nd}}=20$parts per million (ppm) at $\lambda \approx 1100\mathrm{nm}$, which is the laser oscillation wavelength actually seen in experiments [4]. A similar calculation was done at 240 K, with the effective cross sections assumed to remain the same: gain curves are shown in Fig. 11 for the same values of $N_2/N_{\mathrm{Nd}}$. The estimated inversion threshold for positive gain is reduced by about one half compared to room temperature. The lasing wavelength is also expected to shift to shorter wavelength, $\lambda =1064\mathrm{nm}$, although the gain at around $\lambda =1100\mathrm{nm}$ is almost as large. Hence, operating the solar-pumped fiber laser in cold environments, such as at high altitude on an aircraft or near the poles, is an attractive option for coherent, narrowband light generation without electrical power supply.

 

Fig. 10. Calculated gain as a function of wavelength for various values of excited population at $T=296\mathrm{K}$. Results are shown, from bottom to top, for $N_2/N_{\mathrm{Nd}}=0$, 10, 20, …, 60 ppm. Positive gain threshold is reached for $N_2/N_{\mathrm{Nd}}\approx 20\mathrm{ppm}$ at ${\lambda }_{\text{laser}}\approx 1100\mathrm{nm}$.

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Fig. 11. Calculated gain as a function of wavelength for various values of excited population at 240 K ($-33°\mathrm{C}$). Results are shown, from bottom to top, for $N_2/N_{\mathrm{Nd}}=0$, 10, 20,…, 60 ppm (solid line) and 11 and 12 ppm (dashed line). Positive gain threshold is $N_2/N_{\mathrm{Nd}}\approx 11\mathrm{ppm}$. Laser emission occurs at a wavelength close to the maximum of the emission cross section of ${\lambda }_{\text{laser}}=1064\mathrm{nm}$.

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