Abstract

The neodymium-doped optical fiber operated at 1.1 μm is a very promising material for the solar-pumped laser without concentrator because of its strong absorption bands in the visible region and its extremely low optical losses. It is generally considered a true four-level system owing to the large energy gap of the lower level of the laser transition to the ground level. In this study, the exquisitely small thermally excited population in the I11/24 Stark level is shown to be primarily responsible for the absorption losses at the laser wavelength at room temperature. Thanks to its long geometry, the absorption cross section and linestrength of the laser transition could be directly measured, allowing easier estimation of the emission cross section than with usual methods relying on fluorescence decay time and quantum efficiency measurements, or a Judd–Ofelt analysis. Our measurements are corroborated by McCumber’s reciprocity principle. The small-signal gain spectrum measured in an amplifier experiment matches well with the emission cross section. Order-of-magnitude loss reduction is demonstrated by lowering the temperature to 34°C, implying substantial reduction of the laser oscillation threshold in cold solar-pumping environments.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 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 (efficiency20%) and laser diodes (efficiency50%) 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 Yb3+ 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 Nd3+ over Yb3+ for such applications.

The Nd3+ glass laser operated at around λ=1.1μmbetween the F3/24 and I11/24 Stark levels is considered a true four-level system owing to the large energy separation (2000cm1 [7]) between the lower laser level with the ground level I9/24 compared to the thermal energy (kBT200cm1); 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 1km1 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 Nd3+ population of the I11/24 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 I11/24-F3/24 was measured in a Nd3+-doped aluminosilicate fiber at several temperatures ranging from 34°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 Nd3+ concentration, enabled the determination of the effective absorption cross-section, σa(λ), of the laser transition. We used our knowledge of σa(λ) to reliably determine the emission cross section, σe(λ), 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 F3/24-I11/24 transition of Nd3+-doped glasses have ignored σa(λ) 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 βτ method [810], which requires the lineshape function, branching ratios, βi, of the possible terminal Stark level, J=9/2, 11/2, 13/2, and 15/2; and radiative lifetime τrad. However, τrad differs from the directly accessible fluorescence lifetime, τf, due to nonradiative processes such as cooperative relaxation and multiphonon relaxation [11] or due to reabsorption trapping [12]; consequently, τf 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 [1416], which requires analyzing several absorption bands from the ground level to enable useful estimation of the βi values and τrad.

The knowledge of the absorption cross section of the laser transition gives two advantages for an accurate determination of σe compared to the former approach. First, it enables a direct estimate of the linestrength or radiative lifetime into the I11/24 manifold [17], which can be used in combination with the measured relative lineshape function to get the absolute σe spectrum. Second, McCumber’s reciprocity principle [18,19] can be used, and this allows one to calculate σe 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 σe, and thus enables a validation of the results obtained with the former method. Its accuracy in determining σe primarily depends on the accuracy of σa, 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 [10,1922]. Here, we use McCumber’s principle to estimate σe for the F43/2I11/24 laser level of Nd3+ doped in aluminosilicate glass. Small signal gain spectra obtained in an amplifier configuration are also presented at low (37°C) and room temperatures and compared to σe 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 τrad, 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 I11/24-F3/24 Transition

The active fiber is a 16-μm-core, NA=0.18, Nd3+-doped aluminosilicate fiber supplied by Furukawa Denko Corp. The nominal average concentration of Nd3+ ions provided by the manufacturer is NNd=1.3×1020at./cm3, and the Al:Nd atomic concentration ratio estimated by electron probe x-ray microanalysis is 21:1. The transmittance spectrum, Tλ, of the active fiber was measured in the spectral range of the I11/24-F3/24 laser transition by injecting tens of μW of radiation from a fiber-coupled superluminescent diode (SLED) (Thorlabs, model S5FC) emitting in the range from λ=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, L1=1.3m and L2=200m, with the following formula:

α(λ)=ln(Tλ(L1)Tλ(L2))/(L2L1).
The transmittance spectrum was measured at various temperatures ranging from 37°C up to 50°C. To achieve low temperatures, the active fiber was placed into an enclosure containing methanol and dry ice, that is, CO2 with solidification temperature T=78.5°C, surrounded by blocks of polystyrene for thermal insulation; warm water was used to achieve higher temperatures. The temperature was monitored with a thermocouple placed nearby the fiber. In order to avoid cutting and splicing a new fiber at each temperature, changes in absorption, δα(λ), with respect to the room temperature value, Tref, were measured at various temperatures, T, by taking the ratio of transmission spectra:
δα(λ)=ln(Tλ(Tref)Tλ(T))/L,
with a fiber length L=85m. The absolute absorption coefficient at a given temperature was estimated by adding the measured absorption variation, δα(λ), to the reference spectrum obtained at room temperature.

The 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, αBG(λ), at both ends of the measured spectral range, where the Nd3+ cross section is negligible. αBG(λ) was then subtracted from the measured absorption spectrum α(λ) to get αNd(λ). Next, the fraction of ions in the I11/24 manifold was estimated at room temperature using the Maxwell–Boltzmann distribution:

N1NNd=i=16exp(E1i/kBT)i=16exp(E1i/kBT)+i=15exp(E0i/kBT),
where N1 is the concentration in the lower level of laser transition (I11/24), E0i (respectively E1i) are the energy positions of the five (six) sublevels of the ground (lower level of laser transition) manifold I9/24 (I11/24) with respect to the ground sublevel. Energy position of the sublevels are taken from [7] and are shown in Table 1. The partition functions are also shown because they will be useful when comparing absorption and emission cross sections with McCumber’s reciprocity principle.

Tables Icon

Table 1. Energy Levels of Nd3+ in Silicate Glass (taken from [7])

The effective absorption cross section is obtained from

σa=αNd/N1.
The fraction calculated with Eq. (2) was validated by temperature-dependent measurements of absorptions, which were fitted with a Boltzmann factor to estimate the energy gap between the lower laser level and the ground state.

In order to compare our results with those predicted by the Judd–Ofelt theory, the transmission spectrum of the ground state I9/24 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 G5/24-G7/22 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:

B12=B21B,
and connected to the rate of spontaneous emission, A21=1/τ21, by the following formula:
A21=8πhv123c3B,
where hv12 is the energy separation between the two levels [23]. Einstein relations, which remain valid in quantum electrodynamics, were later reformulated in terms of induced dipole matrix elements, absorption and emission cross sections, oscillator strength, and radiative lifetime and also extended by Fowler and Dexter to luminescent centers [17]. Trivalent rare-earth elements in free space have 2J+1-fold degeneracy that is split by the crystal field when placed inside a host. The integrated absorbance is related to the linestrength, S, by
bandσa(λ)dλ=2π2e2λ03cε0h(2J+1)n[(n2+2)29]SF3/24I11/24,
where λ0 is the mean wavelength of the absorption band, J is the total angular momentum of the initial level (here, J=11/2), n is the bulk refractive index, and the factor (n2+2)2/9 represents the local field correction for the ion in a dielectric medium under the tight binding approximation [17]. The transition probability for spontaneous emission for a transition from initial level F3/24 to final level I11/24 is given by [8]
AF3/24I11/24=16π3e23ε0h(2J+1)λ03n[(n2+2)29]SF3/24I11/24,
where J=3/2. Combining Eqs. (5) and (6) gives an estimate of AF3/24I11/24:
AF3/24I11/24=(2J+12J+1)8πcn2λ04bandσa(λ)dλ.
The effective emission cross section is obtained by using Füchtbauer–Ladenburg equation, which, for isotropic centers, is [10,24]
σe(λ)=1τradλ48πn2cg(λ),
where τrad is the radiative lifetime, n is the refractive index of the host material, c is the speed of light in vacuum, and g(λ) is the normalized lineshape function, defined as the probability of spontaneously emitted photon per unit wavelength bandwidth and per unit time in all four terminal Stark levels: J=15/2, 13/2, 11/2, and 9/2. Since
AF3/24I11/24=βF3/24I11/24τrad,
where βF3/24I11/24 is the branching ratio into the I11/24manifold, we obtain
σe(λ)=AF3/24I11/24λ48πn2cgI11/24(λ),
where gI11/24 is the normalized lineshape function into the J=11/2 level only. The refractive index values used were obtained from the knowledge that the optical fiber cladding is pure silica taken from [25] and the knowledge of numerical aperture NA=0.18 with the formula NA=ncore2nclad2. The lineshape function was measured using a photon-flux-calibrated fiber coupled spectrometer (Ocean Optics, model NIRQuest). Therefore, the measurement of the emission cross section involves the measurement of the absorption cross section and relative emission spectrum in photon flux units of only one Stark level.

C. McCumber’s Reciprocity Principle

Both σa(λ) and σe(λ) 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, σij, 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, N1 and N2:

γ(λ)=N2σe(λ)N1σa(λ)αBG(λ).
Assuming thermal equilibrium within each manifold, the effective absorption and emission cross sections are related to the spectroscopic cross sections σij by the following formulas:
σa(λ)=i=16j=12σij(λ)exp((E1iE11)/kBT)Z1,
and
σe(λ)=i=16j=12σij(λ)exp((E2jE21)/kBT)Z2,
where
Z1=i=16exp((E1iE11)/kBT),
and
Z2=j=12exp((E2jE21)/kBT),
are the partition functions, kB is the Boltzmann’s constant, and the i (respectively j) index runs through the 6 (respectively 2) sublevels of the lower I11/24 (respectively higher F3/24) manifold:
σa(λ)σe(λ)=Z2Z1i=16j=12σij(λ)exp((E1iE11)/kBT)i=16j=12σij(λ)exp((E2jE21)/kBT).
Only transitions between sublevels 1i and 2j whose resonant energy coincides with photon energy within the linewidth, ΔEij, of the transition 1i2j take significant part in the summations. When ΔEijkBT, we may replace with good approximation EijE2jE1i by the photon energy, that is, Eijhc/λ, and express the exponential in the numerator as
exp((E1iE11)kBT)exp((E2jE21)kBT)exp((EZLhv)/kBT),
giving a simple exponential relation between σa(λ) and σe(λ) [19]:
σa(λ)σe(λ)Z2Z1exp((hcλEZL)kBT),
where EZLE21E11 is the zero-phonon line, and Z1 and Z2 are the partition functions of each Stark multiplet, whose expression is given in Table 1.

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 λ<570nm and totally reflects wavelengths with λ>570nm 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 G7/22-G5/24 absorption bands of Nd3+ between 570 and 600 nm. Laser gain experiments were similar to absorption measurements presented in Section 1. The differential gain, γ(λ), is calculated with the formula

γ(λ)=ln(Tλ1Tλ2)/L,
where L=85m, and Tλ1 and Tλ2 are the measured transmitted spectra of the SLED with and without pseudosolar illumination. However, an important difference with the absorption measurements takes place because a significant amount of luminescence is produced by excited Nd3+ ions, which adds up to the SLED signal, especially at the wings of the spectral range, where the SLED signal is weaker. This luminescence signal was removed by separately measuring the signal with the SLED turned off and the xenon lamp turned on. Negligible luminescence was produced by the SLED itself because of low SLED power and low absorption by the fiber in the 1050 nm range.

E. Judd–Ofelt Analysis

A Judd–Ofelt analysis [1416] was performed by analyzing the linestrength of nine Nd3+ absorption bands obtained between λ=400 and 1100 nm. This enabled the estimation of the transition probabilities from the metastable F3/24 level to the four lower terminal Stark levels I9/2,11/2,13/2,15/24. 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:

Si=Ω2[U(2)]i2+Ω4[U(4)]i2+Ω6[U(6)]i2,
where [U(i)]2 are the doubly reduced unit tensor operators calculated in the intermediate coupling approximation. Their numerical values have been calculated and are listed in [26]. The Ωi coefficients arise from the odd-symmetry crystal-field terms that allow the transition between 4fN levels, which otherwise would be forbidden by Laporte’s rule. Note that all lines will share the same set of {Ω2,Ω4,Ω6} values. Since nine absorption bands are available, this is an overdetermined linear system with nine equations and three unknowns (Ωi). The coefficients Ωi can be determined by a linear least-square regression that minimizes the total least square error between experimental linestrengths, calculated with Eq. (5), and those estimated with Eq. (19). A 10% error on each Si measurement and no correlation between band measurements are assumed.

In contrast to other published works, the absorption band from the lower laser level I11/24 is also included in the analysis. Doing so, experimental linestrengths from the metastable F3/24 of the ground (I9/24) and the lower laser (I11/24) 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 I13/24 and I15/24, 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, τrad=A1, is obtained.

F. Fluorescence Lifetime Measurements

Decay time measurements of the F3/24 level were carried out to check the estimates obtained thus far with direct measurements of the absorption of the I9/24 and I11/24 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 λ=808nm (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 4π 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 I9/24 of Nd3+ 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 τrad 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.

 figure: 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.

Download Full Size | PPT Slide | PDF

Cooling the fiber to 34°C enabled the elimination of most saturable absorption losses, leaving only the background losses, in the order of 1 to 2km1, as the dominant loss mechanism (Fig. 2). The evolution of the peak absorption of Nd3+ at λ=1057nm as a function of inverse temperature, after subtraction of the background absorption coefficient of αBG=1.25km1, is shown in Fig. 3. The slope of the function ln(ααBG) with 1/T, which depends on the energy distance to ground level, confirms the Boltzmann character of the thermal population of the I11/24 level, namely, N1exp(E0/kBT), and the E02000cm1 value validates the estimate performed with Eq. (2) of N1/NNd=7.8×105 at room temperature. Absorption rapidly drops when the temperature is lowered, decreasing by one order of magnitude by lowering the temperature to 34°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 AF3/24I11/24=753s1 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).

 figure: Fig. 2.

Fig. 2. Absorption spectrum measured at various temperatures ranging from 34°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.

Download Full Size | PPT Slide | PDF

 figure: Fig. 3.

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

Download Full Size | PPT Slide | PDF

 figure: Fig. 4.

Fig. 4. Measured effective absorption (green) and emission (red) cross sections of the F3/24-I11/24 laser transition band of Nd3+-doped aluminosilicate optical fiber. The blue curve is deduced from the emission cross section using McCumber’s principle.

Download Full Size | PPT Slide | PDF

The photon energy at which σa=σe is given from Eq. (17) and Table 1 by

Eσa=σe=EZL+kBTln(Z1Z2)9559cm1,
or λσa=σe1046nm. This value corresponds well to the crossing points obtained in Fig. 4. McCumber’s principle, which connects σa(λ) to σe(λ), is confirmed by plotting the ratio of the logarithm of the experimental cross sections, ln(σa/σe), as a function of 1/λ, Fig. 5. The slope matches very well the theoretical value of hc/kBT, with T=296K.

 figure: Fig. 5.

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).

Download Full Size | PPT Slide | PDF

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 Nd3+ ions excited in the metastable level is extremely small, so the population of the ground manifold remains constant, and so does N1 since N1/N0 is a fixed fraction determined by temperature; hence, only N2 appreciably changes in Eq. (11), and the differential gain is given by

Δγ(λ)=ΔN2σe(λ).
The obtained spectral distribution accordingly matches σe(λ) quite well. The gain curve was also found not to significantly vary over the investigated temperature range.

 figure: Fig. 6.

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°C (green dashed line), and comparison with σe obtained from linestrength measurements and the luminescence spectrum (blue dotted line).

Download Full Size | PPT Slide | PDF

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 Nd3+ in the visible and near IR measured with Eq. (2), Sexp, with those estimated by Judd–Ofelt analysis with Eq. (19), Scalc, is shown in Table 2 together with the Ωi parameters found with a least-square regression. Transition probabilities and branching ratios from the F3/24 metastable level, and the corresponding radiative lifetime estimate, together with the experimentally measured fluorescence lifetime, are shown in Table 3. For both I9/24 and I11/24 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 τf with fiber length indicates negligible reabsorption effects for the range of active fiber lengths used. Given the high Nd3+ concentration, nonradiative quenching effects taking place in Nd3+ ions are probably responsible for the observed difference between τrad610μs and τf450μs. This indicates how challenging it is to achieve a good estimate of σe with the βτ method, which requires knowledge of both the branching ratios and radiative lifetime.

Tables Icon

Table 2. Integrated Cross Section and Linestrength Measurementa

Tables Icon

Table 3. Calculated Transition Probabilities and Branching Ratios from F3/24 Metastable Levela

 figure: Fig. 7.

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 G5/24-G7/22 bands. The initial level is I9/24 for all bands except for band 1, shown in the insert, for which the initial level is I11/24.

Download Full Size | PPT Slide | PDF

 figure: Fig. 8.

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

Download Full Size | PPT Slide | PDF

4. SUMMARY, DISCUSSION, AND CONCLUDING REMARKS

In summary, taking advantage of the long geometry of a Nd3+-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 Nd3+ ions that are thermally excited to the I11/24 level at room temperature. The lowering of the temperature to 34°C made it possible to almost eliminate them and reduce the total fiber loss by one order of magnitude. The knowledge of the Nd3+ 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 I11/24-F3/24 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, AF3/24I11/24; 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 σe(λ), 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 σe from σa 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 F3/24 level only has two sublevels that are not far apart, namely, a 142cm1 energy difference was measured for a similar glass, and the breadth of the I11/24 level is also small, in the order of 350cm1 [7].

 figure: Fig. 9.

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.

Download Full Size | PPT Slide | PDF

The Judd–Ofelt analysis produced an estimate of the transition probabilities to the I11/24 level consistent with that found from the absorption spectrum of the same transition. The large discrepancy of the radiative lifetime, τrad, and the experimentally accessible fluorescence lifetime, τf, clearly showed that knowledge of the quantum fluorescence efficiency is needed together with the branching ratio when the βτ 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 σa(λ), σe(λ), and background losses makes it possible to calculate the gain spectrum for any value of the excited population in the metastable F3/24 level, N2, 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 N2/NNd. The positive gain threshold is first reached for N2/NNd=20parts per million (ppm) at λ1100nm, 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 N2/NNd. 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, λ=1064nm, although the gain at around λ=1100nm 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.

 figure: Fig. 10.

Fig. 10. Calculated gain as a function of wavelength for various values of excited population at T=296K. Results are shown, from bottom to top, for N2/NNd=0, 10, 20, …, 60 ppm. Positive gain threshold is reached for N2/NNd20ppm at λlaser1100nm.

Download Full Size | PPT Slide | PDF

 figure: Fig. 11.

Fig. 11. Calculated gain as a function of wavelength for various values of excited population at 240 K (33°C). Results are shown, from bottom to top, for N2/NNd=0, 10, 20,…, 60 ppm (solid line) and 11 and 12 ppm (dashed line). Positive gain threshold is N2/NNd11ppm. Laser emission occurs at a wavelength close to the maximum of the emission cross section of λlaser=1064nm.

Download Full Size | PPT Slide | PDF

Funding

Natural Sciences and Engineering Research Council of Canada (NSERC) (Discovery Grant).

Acknowledgment

J.-F. B. is grateful for financial support from Toyota Motor Corporation.

REFERENCES

1. T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006). [CrossRef]  

2. T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014). [CrossRef]  

3. Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014). [CrossRef]  

4. T. Masuda, M. Iyoda, Y. Yasumatsu, and M. Endo, “Low concentrated solar-pumped laser via transverse excitation fiber-laser geometry,” Opt. Lett. 42, 3427–3430 (2017). [CrossRef]  

5. M. Endo and J.-F. Bisson, “Positive gain observation in a Nd-doped active fiber pumped by low-concentrated solar-like xenon lamp,” Jpn. J. Appl. Phys. 51, 022701 (2012). [CrossRef]  

6. J. Nillson and D. N. Payne, “High-power fiber lasers,” Science 332, 921–922 (2011). [CrossRef]  

7. M. M. Mann and L. G. De Shazer, “Energy levels and spectral broadening of neodymium ion in laser glass,” J. Appl. Phys. 41, 2951–2957 (1970). [CrossRef]  

8. W. F. Krupke, “Induced emission cross-sections in neodymium laser glasses,” IEEE J. Quantum Electron. 10, 450–457 (1974). [CrossRef]  

9. R. R. Jacobs and M. J. Weber, “Dependence of the F3/24->I11/24 induced-emission cross section for Nd3+ on glass composition,” IEEE J. Quantum Electron. 12, 102–111 (1976). [CrossRef]  

10. B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998). [CrossRef]  

11. L. A. Riseberg and M. J. Weber, “Relaxation phenomena in rare-earth luminescence,” in Progress in Optics XIV, E. Wolf, ed. (Elsevier, 1976), pp. 89–159.

12. M. A. Noginov, “Reabsorption trapping of luminescence in laser crystals: enhancement of energy storage and upconversion,” Appl. Opt. 36, 4153–4158 (1997). [CrossRef]  

13. J. A. Caird, A. J. Ramponi, and P. R. Staver, “Quantum efficiency and excited-state relaxation dynamics in neodymium-doped phosphate laser glasses,” J. Opt. Soc. Am. B 8, 1391–1403 (1991). [CrossRef]  

14. B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962). [CrossRef]  

15. G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962). [CrossRef]  

16. B. M. W. Walsh, “Judd–Ofelt theory: principles and practices,” in Advances in Spectroscopy for Lasers and Sensing (Springer, 2006), Chap. 21.

17. W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962). [CrossRef]  

18. D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964). [CrossRef]  

19. S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992). [CrossRef]  

20. R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002). [CrossRef]  

21. M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002). [CrossRef]  

22. R. M. Martin and R. S. Quimby, “Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses,” J. Opt. Soc. Am. B 23, 1770–1775 (2006). [CrossRef]  

23. A. Einstein, “Zur quanten theorie der strahlung,” Phys. Zeit. 18, 121–128 (1917) (in German).

24. P. F. Moulton, “Spectroscopic and laser characteristics of Ti:Al2O3,” J. Opt. Soc. Am. B 3, 125–133 (1986). [CrossRef]  

25. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]  

26. W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965). [CrossRef]  

27. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. 32, 1908–1910(2007). [CrossRef]  

28. G. Toci, “Lifetime measurements with the pinhole method in the presence of radiation trapping: I—theoretical model,” Appl. Phys. B 106, 66–71 (2012). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
    [Crossref]
  2. T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
    [Crossref]
  3. Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
    [Crossref]
  4. T. Masuda, M. Iyoda, Y. Yasumatsu, and M. Endo, “Low concentrated solar-pumped laser via transverse excitation fiber-laser geometry,” Opt. Lett. 42, 3427–3430 (2017).
    [Crossref]
  5. M. Endo and J.-F. Bisson, “Positive gain observation in a Nd-doped active fiber pumped by low-concentrated solar-like xenon lamp,” Jpn. J. Appl. Phys. 51, 022701 (2012).
    [Crossref]
  6. J. Nillson and D. N. Payne, “High-power fiber lasers,” Science 332, 921–922 (2011).
    [Crossref]
  7. M. M. Mann and L. G. De Shazer, “Energy levels and spectral broadening of neodymium ion in laser glass,” J. Appl. Phys. 41, 2951–2957 (1970).
    [Crossref]
  8. W. F. Krupke, “Induced emission cross-sections in neodymium laser glasses,” IEEE J. Quantum Electron. 10, 450–457 (1974).
    [Crossref]
  9. R. R. Jacobs and M. J. Weber, “Dependence of the F3/24->I11/24 induced-emission cross section for Nd3+ on glass composition,” IEEE J. Quantum Electron. 12, 102–111 (1976).
    [Crossref]
  10. B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
    [Crossref]
  11. L. A. Riseberg and M. J. Weber, “Relaxation phenomena in rare-earth luminescence,” in Progress in Optics XIV, E. Wolf, ed. (Elsevier, 1976), pp. 89–159.
  12. M. A. Noginov, “Reabsorption trapping of luminescence in laser crystals: enhancement of energy storage and upconversion,” Appl. Opt. 36, 4153–4158 (1997).
    [Crossref]
  13. J. A. Caird, A. J. Ramponi, and P. R. Staver, “Quantum efficiency and excited-state relaxation dynamics in neodymium-doped phosphate laser glasses,” J. Opt. Soc. Am. B 8, 1391–1403 (1991).
    [Crossref]
  14. B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
    [Crossref]
  15. G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
    [Crossref]
  16. B. M. W. Walsh, “Judd–Ofelt theory: principles and practices,” in Advances in Spectroscopy for Lasers and Sensing (Springer, 2006), Chap. 21.
  17. W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
    [Crossref]
  18. D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964).
    [Crossref]
  19. S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
    [Crossref]
  20. R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
    [Crossref]
  21. M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
    [Crossref]
  22. R. M. Martin and R. S. Quimby, “Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses,” J. Opt. Soc. Am. B 23, 1770–1775 (2006).
    [Crossref]
  23. A. Einstein, “Zur quanten theorie der strahlung,” Phys. Zeit. 18, 121–128 (1917) (in German).
  24. P. F. Moulton, “Spectroscopic and laser characteristics of Ti:Al2O3,” J. Opt. Soc. Am. B 3, 125–133 (1986).
    [Crossref]
  25. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
    [Crossref]
  26. W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965).
    [Crossref]
  27. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. 32, 1908–1910(2007).
    [Crossref]
  28. G. Toci, “Lifetime measurements with the pinhole method in the presence of radiation trapping: I—theoretical model,” Appl. Phys. B 106, 66–71 (2012).
    [Crossref]

2017 (1)

2014 (2)

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

2012 (2)

M. Endo and J.-F. Bisson, “Positive gain observation in a Nd-doped active fiber pumped by low-concentrated solar-like xenon lamp,” Jpn. J. Appl. Phys. 51, 022701 (2012).
[Crossref]

G. Toci, “Lifetime measurements with the pinhole method in the presence of radiation trapping: I—theoretical model,” Appl. Phys. B 106, 66–71 (2012).
[Crossref]

2011 (1)

J. Nillson and D. N. Payne, “High-power fiber lasers,” Science 332, 921–922 (2011).
[Crossref]

2007 (1)

2006 (2)

R. M. Martin and R. S. Quimby, “Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses,” J. Opt. Soc. Am. B 23, 1770–1775 (2006).
[Crossref]

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

2002 (2)

R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
[Crossref]

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[Crossref]

1998 (1)

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[Crossref]

1997 (1)

1992 (1)

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

1991 (1)

1986 (1)

1976 (1)

R. R. Jacobs and M. J. Weber, “Dependence of the F3/24->I11/24 induced-emission cross section for Nd3+ on glass composition,” IEEE J. Quantum Electron. 12, 102–111 (1976).
[Crossref]

1974 (1)

W. F. Krupke, “Induced emission cross-sections in neodymium laser glasses,” IEEE J. Quantum Electron. 10, 450–457 (1974).
[Crossref]

1970 (1)

M. M. Mann and L. G. De Shazer, “Energy levels and spectral broadening of neodymium ion in laser glass,” J. Appl. Phys. 41, 2951–2957 (1970).
[Crossref]

1965 (2)

I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
[Crossref]

W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965).
[Crossref]

1964 (1)

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

1962 (3)

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
[Crossref]

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
[Crossref]

W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

1917 (1)

A. Einstein, “Zur quanten theorie der strahlung,” Phys. Zeit. 18, 121–128 (1917) (in German).

Baasandash, C.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Barnes, N. P.

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[Crossref]

Bisson, J.-F.

M. Endo and J.-F. Bisson, “Positive gain observation in a Nd-doped active fiber pumped by low-concentrated solar-like xenon lamp,” Jpn. J. Appl. Phys. 51, 022701 (2012).
[Crossref]

Caird, J. A.

Carnall, W. T.

W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965).
[Crossref]

Chase, L. L.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

De Shazer, L. G.

M. M. Mann and L. G. De Shazer, “Energy levels and spectral broadening of neodymium ion in laser glass,” J. Appl. Phys. 41, 2951–2957 (1970).
[Crossref]

Dexter, D. L.

W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

Di Bartolo, B.

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[Crossref]

Digonnet, M. J. F.

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[Crossref]

Einstein, A.

A. Einstein, “Zur quanten theorie der strahlung,” Phys. Zeit. 18, 121–128 (1917) (in German).

Endo, M.

T. Masuda, M. Iyoda, Y. Yasumatsu, and M. Endo, “Low concentrated solar-pumped laser via transverse excitation fiber-laser geometry,” Opt. Lett. 42, 3427–3430 (2017).
[Crossref]

M. Endo and J.-F. Bisson, “Positive gain observation in a Nd-doped active fiber pumped by low-concentrated solar-like xenon lamp,” Jpn. J. Appl. Phys. 51, 022701 (2012).
[Crossref]

Falquier, D. G.

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[Crossref]

Fields, P. R.

W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965).
[Crossref]

Fowler, W. B.

W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

Fredrich-Thornton, S. T.

Hasegawa, K.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Ichikawa, T.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Ichiki, A.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Iida, Y.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Iizuka, H.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Ikuta, K.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Ito, H.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Iyoda, M.

Jacobs, R. R.

R. R. Jacobs and M. J. Weber, “Dependence of the F3/24->I11/24 induced-emission cross section for Nd3+ on glass composition,” IEEE J. Quantum Electron. 12, 102–111 (1976).
[Crossref]

Judd, B. R.

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
[Crossref]

Kajino, T.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Karita, T.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Kränkel, C.

Krupke, W. F.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

W. F. Krupke, “Induced emission cross-sections in neodymium laser glasses,” IEEE J. Quantum Electron. 10, 450–457 (1974).
[Crossref]

Kühn, H.

Kway, W. L.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Malitson, I. H.

Mann, M. M.

M. M. Mann and L. G. De Shazer, “Energy levels and spectral broadening of neodymium ion in laser glass,” J. Appl. Phys. 41, 2951–2957 (1970).
[Crossref]

Martin, R. M.

Masuda, T.

McCumber, D. E.

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Mizuno, S.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Mohamed, M. S.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Mori, Y.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Motohiro, T.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Moulton, P. F.

Murphy-Chutorian, E.

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[Crossref]

Nakamura, K.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Nillson, J.

J. Nillson and D. N. Payne, “High-power fiber lasers,” Science 332, 921–922 (2011).
[Crossref]

Nishikawa, Y.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Noginov, M. A.

Ofelt, G. S.

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
[Crossref]

Ogata, Y.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Payne, D. N.

J. Nillson and D. N. Payne, “High-power fiber lasers,” Science 332, 921–922 (2011).
[Crossref]

Payne, S. A.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Petermann, K.

Peters, R.

Quimby, R. S.

R. M. Martin and R. S. Quimby, “Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses,” J. Opt. Soc. Am. B 23, 1770–1775 (2006).
[Crossref]

R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
[Crossref]

Ramponi, A. J.

Riseberg, L. A.

L. A. Riseberg and M. J. Weber, “Relaxation phenomena in rare-earth luminescence,” in Progress in Optics XIV, E. Wolf, ed. (Elsevier, 1976), pp. 89–159.

Saiki, T.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Sakurai, Y.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Satoh, Y.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Smith, L. K.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Staver, P. R.

Takeda, Y.

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

Taniguchi, S.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

Toci, G.

G. Toci, “Lifetime measurements with the pinhole method in the presence of radiation trapping: I—theoretical model,” Appl. Phys. B 106, 66–71 (2012).
[Crossref]

Tuji, M.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Uchida, S.

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Walsh, B. M.

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[Crossref]

Walsh, B. M. W.

B. M. W. Walsh, “Judd–Ofelt theory: principles and practices,” in Advances in Spectroscopy for Lasers and Sensing (Springer, 2006), Chap. 21.

Weber, M. J.

R. R. Jacobs and M. J. Weber, “Dependence of the F3/24->I11/24 induced-emission cross section for Nd3+ on glass composition,” IEEE J. Quantum Electron. 12, 102–111 (1976).
[Crossref]

L. A. Riseberg and M. J. Weber, “Relaxation phenomena in rare-earth luminescence,” in Progress in Optics XIV, E. Wolf, ed. (Elsevier, 1976), pp. 89–159.

Wybourne, B. G.

W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965).
[Crossref]

Yabe, T.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Yasumatsu, Y.

Yoshida, K.

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

G. Toci, “Lifetime measurements with the pinhole method in the presence of radiation trapping: I—theoretical model,” Appl. Phys. B 106, 66–71 (2012).
[Crossref]

Appl. Phys. Lett. (1)

T. Yabe, S. Uchida, K. Ikuta, K. Yoshida, C. Baasandash, M. S. Mohamed, Y. Sakurai, Y. Ogata, M. Tuji, Y. Mori, and Y. Satoh, “Demonstrated fossil-fuel-free energy cycle using magnesium and laser,” Appl. Phys. Lett. 89, 261107 (2006).
[Crossref]

IEEE J. Quantum Electron. (4)

W. F. Krupke, “Induced emission cross-sections in neodymium laser glasses,” IEEE J. Quantum Electron. 10, 450–457 (1974).
[Crossref]

R. R. Jacobs and M. J. Weber, “Dependence of the F3/24->I11/24 induced-emission cross section for Nd3+ on glass composition,” IEEE J. Quantum Electron. 12, 102–111 (1976).
[Crossref]

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[Crossref]

Int. J. Sustainable Green Energy (1)

T. Saiki, S. Uchida, T. Karita, K. Nakamura, Y. Nishikawa, S. Taniguchi, and Y. Iida, “Recyclable metal air fuel cells using sintered magnesium paste with reduced mg nanoparticles by high-repetitive ns pulse laser ablation in liquid,” Int. J. Sustainable Green Energy 3, 143–149 (2014).
[Crossref]

J. Appl. Phys. (4)

Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, A. Ichiki, and T. Motohiro, “Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064 nm,” J. Appl. Phys. 116, 014501 (2014).
[Crossref]

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratio, cross-sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[Crossref]

R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
[Crossref]

M. M. Mann and L. G. De Shazer, “Energy levels and spectral broadening of neodymium ion in laser glass,” J. Appl. Phys. 41, 2951–2957 (1970).
[Crossref]

J. Chem. Phys. (2)

W. T. Carnall, P. R. Fields, and B. G. Wybourne, “Spectral intensities of the trivalent lanthanides and actinides in solution. I. Pr3+, Nd3+, Er3+, Tm3+, Yb3+,” J. Chem. Phys. 42, 3797–3806 (1965).
[Crossref]

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (3)

Jpn. J. Appl. Phys. (1)

M. Endo and J.-F. Bisson, “Positive gain observation in a Nd-doped active fiber pumped by low-concentrated solar-like xenon lamp,” Jpn. J. Appl. Phys. 51, 022701 (2012).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (3)

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
[Crossref]

W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Phys. Zeit. (1)

A. Einstein, “Zur quanten theorie der strahlung,” Phys. Zeit. 18, 121–128 (1917) (in German).

Science (1)

J. Nillson and D. N. Payne, “High-power fiber lasers,” Science 332, 921–922 (2011).
[Crossref]

Other (2)

B. M. W. Walsh, “Judd–Ofelt theory: principles and practices,” in Advances in Spectroscopy for Lasers and Sensing (Springer, 2006), Chap. 21.

L. A. Riseberg and M. J. Weber, “Relaxation phenomena in rare-earth luminescence,” in Progress in Optics XIV, E. Wolf, ed. (Elsevier, 1976), pp. 89–159.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

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.
Fig. 2.
Fig. 2. Absorption spectrum measured at various temperatures ranging from 34 ° 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.
Fig. 3.
Fig. 3. Evolution of the peak absorption value as a function of temperature.
Fig. 4.
Fig. 4. Measured effective absorption (green) and emission (red) cross sections of the F 3 / 2 4 - I 11 / 2 4 laser transition band of Nd 3 + -doped aluminosilicate optical fiber. The blue curve is deduced from the emission cross section using McCumber’s principle.
Fig. 5.
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).
Fig. 6.
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 ° C (green dashed line), and comparison with σ e obtained from linestrength measurements and the luminescence spectrum (blue dotted line).
Fig. 7.
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 G 5 / 2 4 - G 7 / 2 2 bands. The initial level is I 9 / 2 4 for all bands except for band 1, shown in the insert, for which the initial level is I 11 / 2 4 .
Fig. 8.
Fig. 8. Experimental decay times obtained for different lengths of active fiber with Si (circles) and Ge (stars) detectors.
Fig. 9.
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.
Fig. 10.
Fig. 10. Calculated gain as a function of wavelength for various values of excited population at T = 296 K . Results are shown, from bottom to top, for N 2 / N Nd = 0 , 10, 20, …, 60 ppm. Positive gain threshold is reached for N 2 / N Nd 20 ppm at λ laser 1100 nm .
Fig. 11.
Fig. 11. Calculated gain as a function of wavelength for various values of excited population at 240 K ( 33 ° C ). Results are shown, from bottom to top, for N 2 / N Nd = 0 , 10, 20,…, 60 ppm (solid line) and 11 and 12 ppm (dashed line). Positive gain threshold is N 2 / N Nd 11 ppm . Laser emission occurs at a wavelength close to the maximum of the emission cross section of λ laser = 1064 nm .

Tables (3)

Tables Icon

Table 1. Energy Levels of Nd 3 + in Silicate Glass (taken from [7])

Tables Icon

Table 2. Integrated Cross Section and Linestrength Measurement a

Tables Icon

Table 3. Calculated Transition Probabilities and Branching Ratios from F 3 / 2 4 Metastable Level a

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

α ( λ ) = ln ( T λ ( L 1 ) T λ ( L 2 ) ) / ( L 2 L 1 ) .
δ α ( λ ) = ln ( T λ ( T ref ) T λ ( T ) ) / L ,
N 1 N Nd = i = 1 6 exp ( E 1 i / k B T ) i = 1 6 exp ( E 1 i / k B T ) + i = 1 5 exp ( E 0 i / k B T ) ,
σ a = α Nd / N 1 .
B 12 = B 21 B ,
A 21 = 8 π h v 12 3 c 3 B ,
band σ a ( λ ) d λ = 2 π 2 e 2 λ 0 3 c ε 0 h ( 2 J + 1 ) n [ ( n 2 + 2 ) 2 9 ] S F 3 / 2 4 I 11 / 2 4 ,
A F 3 / 2 4 I 11 / 2 4 = 16 π 3 e 2 3 ε 0 h ( 2 J + 1 ) λ 0 3 n [ ( n 2 + 2 ) 2 9 ] S F 3 / 2 4 I 11 / 2 4 ,
A F 3 / 2 4 I 11 / 2 4 = ( 2 J + 1 2 J + 1 ) 8 π c n 2 λ 0 4 band σ a ( λ ) d λ .
σ e ( λ ) = 1 τ rad λ 4 8 π n 2 c g ( λ ) ,
A F 3 / 2 4 I 11 / 2 4 = β F 3 / 2 4 I 11 / 2 4 τ rad ,
σ e ( λ ) = A F 3 / 2 4 I 11 / 2 4 λ 4 8 π n 2 c g I 11 / 2 4 ( λ ) ,
γ ( λ ) = N 2 σ e ( λ ) N 1 σ a ( λ ) α BG ( λ ) .
σ a ( λ ) = i = 1 6 j = 1 2 σ i j ( λ ) exp ( ( E 1 i E 11 ) / k B T ) Z 1 ,
σ e ( λ ) = i = 1 6 j = 1 2 σ i j ( λ ) exp ( ( E 2 j E 21 ) / k B T ) Z 2 ,
Z 1 = i = 1 6 exp ( ( E 1 i E 11 ) / k B T ) ,
Z 2 = j = 1 2 exp ( ( E 2 j E 21 ) / k B T ) ,
σ a ( λ ) σ e ( λ ) = Z 2 Z 1 i = 1 6 j = 1 2 σ i j ( λ ) exp ( ( E 1 i E 11 ) / k B T ) i = 1 6 j = 1 2 σ i j ( λ ) exp ( ( E 2 j E 21 ) / k B T ) .
exp ( ( E 1 i E 11 ) k B T ) exp ( ( E 2 j E 21 ) k B T ) exp ( ( E Z L h v ) / k B T ) ,
σ a ( λ ) σ e ( λ ) Z 2 Z 1 exp ( ( h c λ E Z L ) k B T ) ,
γ ( λ ) = ln ( T λ 1 T λ 2 ) / L ,
S i = Ω 2 [ U ( 2 ) ] i 2 + Ω 4 [ U ( 4 ) ] i 2 + Ω 6 [ U ( 6 ) ] i 2 ,
E σ a = σ e = E Z L + k B T ln ( Z 1 Z 2 ) 9559 cm 1 ,
Δ γ ( λ ) = Δ N 2 σ e ( λ ) .

Metrics