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Probing the erbium ion distribution in silica optical fibers with fluorescence based measurements
Journal of Non-Crystalline Solids 357 (2011) 3847–3852
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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Probing the erbium ion distribution in silica optical fibers with fluorescence based measurements F. Sidiroglou a,⁎, A. Roberts b, G.w. Baxter a a b
Optical Technology Research Laboratory, Centre for Telecommunications and Microelectronics, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia School of Physics, University of Melbourne, Victoria 3010, Australia
a r t i c l e
i n f o
Article history: Received 10 June 2011 Received in revised form 19 July 2011 Available online 17 August 2011 Keywords: Optical fibers; Erbium ion distribution; Confocal fluorescence microscopy; Fiber characterization; Fiber lasers and amplifiers
a b s t r a c t Accurate determination of the rare earth dopant distribution in optical fibers enhances our understanding of the fiber manufacture process and enables further improvement in the design of fiber based products such as optical fiber lasers and amplifiers. Here a simple theoretical model consisting of an ensemble of rate equation systems, characteristic of the most likely electronic transitions that take place in the vicinity of erbium (Er 3+) doped silica glasses, is developed and solved. Through this theoretical study it is established that information about the relative Er3+ ion distribution in fibers can be inferred by simply monitoring the backscattered fluorescence signal originating from the de-excitation of specific energy levels in the investigated samples. Following these theoretical studies a fluorescence intensity confocal optical microscopy (FICOM) scheme was employed to investigate the Er3+ ion distribution profiles in a range of silica optical fibers. The validity of the proposed theoretical model was confirmed through a comparison of the Er3+ ion distribution profiles acquired using the FICOM technique and those obtained from the application of a powerful analytical ion probe. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction Photonic devices, such as optical fibers, fiber lasers, and fiber amplifiers play an important role in today's society. Their capabilities have found use in a vast range of scientific and industrial environments amongst which the communications field has been influenced the most. In particular, although optical communication technologies were initially introduced to improve traditional information exchange, they have recently become a key factor behind the tremendous growth in internet traffic, and optical technologies will be even more important in enabling and supporting the future expansion of internet traffic. Optimizing the design and fabrication of such devices has become an ongoing challenge to optical scientists and engineers. For example, it has been demonstrated that the properties and performance of Er 3+ doped fiber lasers and amplifiers (EDFA) are related to a number of parameters such as the fiber glass material, the waveguide characteristics, and the distribution profile of the Er3+ ions. Accurate knowledge of the latter has been found to be vital for the optimal design and operation of these devices [1–4]. Various techniques have therefore been developed and investigated in order to acquire information related to the distribution of the dopants in the fibers.
⁎ Corresponding author. E-mail address: [email protected] (F. Sidiroglou).
To date, two different approaches have been employed to measure the distribution of Er 3+ or other rare earth (RE) ions in optical fibers. In the first case, the concentration of dopants and their distribution, together with the refractive index profile (RIP) and other parameters are usually measured in the fiber preform, prior to the fiber drawing process. Some of the most important techniques employed to reveal the location of dopants in bulk samples such as fiber preforms, are based on the utilization of methods that are commonly used in the field of analytical chemistry and material science. These techniques include secondary ion mass spectroscopy [5], inductively coupled plasma atomic emission spectroscopy [6], and X-ray microprobe analysis [7]. The distribution of dopants in fiber preforms has also been examined using the electron probe microanalysis (EPMA) method and its counterpart systems [8,9]. The resultant dopant concentration and distribution is then scaled down to match the drawn fiber dimensions assuming that there are no defects affecting the resulting fiber profile during the drawing process. This assumes that the profile remains unchanged during the drawing process. Huntington et al. [10] has shown that this assumption is not always valid for the determination of the RIP. This is believed to be the consequence of various effects such as the diffusion of elements that takes place because of the high temperature environment created during the drawing process. As a result it is not always valid to simply relate the dopant concentration profile extracted from preform measurements to fiber dimensions. In the second approach, measurements are made directly on the drawn fiber. Once again a number of the techniques utilized for the
0022-3093/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.07.024
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analysis of the RE distribution in optical fibers have been borrowed from the analytical chemistry and materials science field. Such techniques include nano-secondary ion mass spectroscopy (NanoSIMS) [11], time of flight secondary ion mass spectroscopy (ToF-SIMS) [12], transmission electron microscopy [13] and Raman microscopy [14]. Although these techniques can offer adequate spatial imaging resolution for the direct investigation of optical fibers and they can offer information about the distribution of other dopants within the fiber core region, they normally require complex and time-consuming sample preparation and the use of high cost instrumentation. In an attempt to overcome such limitations, a number of optical imaging schemes have been applied over the years in order to provide information about the RE ion distribution in the core of optical fibers directly from the investigation of their cleaved endface [15–17]. With the exception of the work of Petreski et al. [15], where a fluorescence lifetime confocal optical microscopy technique was developed for the investigation of praseodymium-doped optical fibers, the other optical based systems were capable of providing information about the Er 3+ ion distribution in optical fibers with the application of a fluorescence intensity confocal optical microscopy scheme [16,17]. In the latter cases, the intensity of the backscattered fluorescence originating from the de-excitation of the 2H11/2 and 4S3/2 upper energy levels (Fig. 1) was monitored and taken as an indication of the local dopant concentration of the Er 3+ ions at a point. By scanning the excitation beam across the center of the fiber core, two-dimensional images or line-scans of the relative Er 3+ ion distribution were acquired. However, the acquired intensity profiles were assumed to be directly related to the relative Er 3+ ion distributions without establishing that the resulting backscattered fluorescence signal was indeed proportional to the erbium ion concentration. Care must be taken regarding the reliability of the intensity based measurements given that cooperative effects in RE doped fibers may yield incorrect values of dopant concentration [18]. In this work, the issue of the dynamics of different emission lines observed in Er 3+ doped optical fibers and their relationship to the total Er 3+ ion concentration is addressed by developing and solving a system of rate equations characteristic to the most likely electronic transitions observed in a Er 3+ doped silica glass. This exploration validates the use of direct pumping of the 2H11/2 (at 514 nm, as shown in some of our previous work [17]) or the 4F7/2 (at 488 nm in the work of [16]) levels with the subsequent detection of the backscattered fluorescence signal (around 550 nm) from the de-excitation of the 4S3/2 level as a measure of the relative Er3+ ion distribution in optical fibers. Furthermore, we extend
Γ Γ Γ α
Γ
α
Fig. 1. Partial energy level diagram for Er3+ in silica illustrating a six energy level system. Solid arrows represent radiative transitions, dashed arrows non-radiative decay and dotted arrows two ion interactions.
the relevance of this theoretical model as applied in our previous work with the use of a confocal optical microscope [17] to our more recent efforts in trying to improve spatial resolution using a Near-field Scanning Optical Microscope (NSOM). 2. Theoretical modeling The main focus of this article is to identify the specific electronic transitions that take place in Er 3+ doped silica glasses, which could be related directly to the concentration or distribution of the erbium ions within the core of optical fibers. The issue remains whether fluorescence intensity based measurements could possibly be related to the erbium dopant profile. As long as the process of de-excitation of an energy level is free from any possible inter-ionic cross-relaxation or upconversion effects then it can be assumed that the intensity of the emitted signal is proportional to the population of ions at that point. For radiative events during which the excited ions are returning to the ground state 4I15/2, there are three energy levels that have been identified as potential levels in which de-excitation can occur and whose intensity of emitting signal could be taken as an indication of the ion population. These levels are (Fig. 1): • the 4I13/2 state, where radiative decay back to the ground state from this level yields a fluorescence band centered at around 1530 nm, • the 4I11/2 state where de-excitation from this level occurs with the simultaneous emission of photons near 980 nm, • and the 4S3/2 state from which the ions can return back to the ground state with the concurrent emission of light at 550 nm. Depending on the associated energy level, excitation of the ions can be achieved with either resonant or near-resonant pumping and the collection of all energy transfers and radiative events can be explained by treating each system as a n-level system, where n the number of energy levels involved. In the case of the well-known metastable state 4I13/2, the ions could be excited using a near resonant pumping scheme with a 980 nm source, which excites the ions first to the 4I11/2 level before non-radiatively relaxing to the next lower level 4I13/2 from which they can then return back to the ground state by radiative emission centered on 1530 nm. It can be postulated that the intensity of this emission could be taken as an indication of the erbium dopant distribution while the excitation source is scanned across the endface of an optical fiber as long as there are no inter-ionic cross-relaxation or upconversion effects that can affect the relationship between the measured fluorescence signal and the investigated dopant concentration. However, the dynamics of the 4I13/2 state are strongly affected by various types of ion to ion interactions and in particular by excited state absorption (ESA) [19]. Therefore the emission intensity is not a direct measure of the relative erbium ion concentration. With the exception of the 1480 nm pumping scheme that does not suffer from ESA when used to pump Er 3+ doped optical fibers, ESA has been observed to originate from the 4I13/2 level for most of the other exhibited wavelengths used for pumping Er 3+ doped fibers [19]. If a scheme where the signal from the de-excitation of the 4I11/2 energy level is used as a tool to investigate the erbium ion distribution, crossrelaxation or upconversion effects can once again alter the dynamics of the system so that such an approach will not be accurate. Particularly in the most conventional case, where a 980 nm source is used to resonantly excite the ions at that level, ESA is known to take place in such a way that the already excited ions can be promoted to the 4F7/2 upper state. Furthermore, since most of the ions will instead relax to the metastable 4I13/2 level – by either non-radiative or radiative processes – before they return to the ground state, the ability to obtain concentration measurements based on the intensity of the detected backscattered fluorescence signal becomes even less realistic.
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On the other hand, if ion distribution measurements are carried out based on the detected signal from the de-excitation of the 4S3/2 state, non-radiative processes such as multiphonon decay are the predominant electronic transitions that have to be incorporated in theoretical models rather than any inter-ionic events as were encountered in the other cases. As it has been demonstrated by Desurvire et al. [20,21], the relaxation process between the 2H11/2 and 4 I13/2 (Fig. 1) energy levels is strongly dominated by fast non-radiative decay. In fact, the non-radiative transition processes for the upper energy levels are described by rates that on average are, almost by a factor of one thousand or larger, more probable than from those that describe radiative decay (Table 1). As a result, even in the presence of possible inter-ionic upconversion events in the lower energy levels, their contribution to the dynamics of the radiative transitions originating from the upper levels such as that of the 4S3/2 state is expected to be insignificant. The same argument can be drawn for the case where ESA promotes ions from the metastable 4I13/2 energy level to the 2H9/2 higher energy level. It is expected that the excited ions will return through a series of non-radiative processes to the lower energy states without affecting the dynamics of the radiative events originating from the de-excitation of the 4S3/2 state. Consequently it may be assumed that the intensity of the backscattered fluorescence signal originating from the de-excitation of the 4S3/2 energy level is proportional to the ion concentration. This statement is better demonstrated by investigating the rate equations that characterize all radiative and non-radiative events in the case of the six energy level diagram of Fig. 1. This diagram demonstrates the more realistic case where two ion interaction and ESA are also considered in the ensemble of events that describe the change in the ion populations for each level. A near-resonant pumping scheme at 514 nm is used as the source for the excitation of the ions from the ground state. Absorption of the pump (at a pump rate of RG) promotes the ground state ions to the 2H11/2 upper energy level. With the exception of a small fraction of excited ions that return back to the ground state through radiative decay (i.e. 2H11/2 → 4I15/2, 4S3/2 → 4I15/2 and 4S3/2 → 4I13/2), the vast majority of the ions will decay through a fast non-radiative process through lower energy levels until they reach the 4I13/2 metastable state. While at this energy level, most of the ions will return through radiative decay to the ground state. However, part of the ions can be promoted through ESA to the 2H9/2 upper energy level at a rate which according to the work of Laming et al. [22] is equal to: RE = 0:5RG :
ð1Þ
In addition, the ion population of the 4I13/2 level can be quenched due to an ion pair up-conversion process. Such a process involves two neighboring ions in the 4I13/2 state: one of the ions transfers its energy to the other, producing one up-converted 4I9/2 ion and one ground state ion. The up-converted ion quickly decays to the 4I13/2 state. The consequence is the loss of one excited erbium ion. This event has been modeled for erbium doped silicate fibers [23–26] and is described by a characteristic up-conversion coefficient (α), which is expressed in units of m 3/s. Table 1 Decay rates of electronic transitions encountered in erbium doped silica as depicted by the energy level diagram of Fig. 1. Transition
Type
Constant
Decay rate
Reference
2
Non-radiative decay Radiative decay Non-radiative decay Radiative decay Radiative decay Non-radiative decay Radiative decay
Γ43 P40 Γ32 P30 P31 Γ21 P10
~ 107 s− 1 ~ 3 × 103 s− 1 ~ 0.14 × 107 s− 1 ~ 103 s− 1 ~ 4 × 102 s− 1 ~ 2 × 108 s− 1 ~ 90 s− 1
[20] [20] [20] [20] [21] [20] [21]
H11/2 → 4S3/2 2 H11/2 → 4I15/2 4 S3/2 → 4I13/2 4 S3/2 → 4I15/2 4 S3/2 → 4I13/2 4 I9/2 → 4I13/2 4 I13/2 → 4I15/2
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The rates of change of the ion densities for each level of the energy level diagram of Fig. 1 take the form: dN5 = RE N1 −Γ54 N5 dt
ð2Þ
dN4 = RG N0 + Γ54 N5 −P40 N4 −Γ43 N4 dt
ð3Þ
dN3 = Γ43 N4 −P30 N3 −P31 N3 −Γ32 N3 dt
ð4Þ
dN2 1 2 = Γ32 N3 + αN1 −Γ21 N2 2 dt
ð5Þ
dN1 2 = Γ21 N2 + P31 N3 −P10 N1 −αN1 −RE N1 dt
ð6Þ
dN0 1 2 = P40 N4 + P30 N3 + P10 N1 + αN1 −RG N0 2 dt
ð7Þ
where Ni is the ion population for each level i (with i = 0, 1, 2, 3, 4 and 5), Γij represents the non-radiative decay between two energy levels i and j and Pij represents the radiative decay between two energy levels i and j. RG is the pump absorption rate for the ground state ions and can be determined as follows. RG = Pin
λ σabs hc Aspot
ð2:8Þ
where Pin and λ are the input power and wavelength of the excitation source, Aspot is the size of the beam spot, σabs is the absorption cross section for the ground state ions when an excitation at λ = 514 nm is used, while h and c are Plank's constant and the speed of light respectively. At each instant the overall ion density ρ is given by the sum of the ion densities at each level ρ = N0 + N 1 + N 2 + N 3 + N 4 + N 5
ð2:9Þ
which can be simplified to ð2:10Þ
ρ = N 0 + N1 + N2 + N3 + N4
,since at any instance the majority of the total ions exist between levels 4 I15/2 to 2H11/2. The units for the ion density at each level i is expressed in ions/m3, while the units for the pump rate as well as for the radiative and non-radiative decays are that of s− 1. By solving the above system of rate equations (Eqs (2)–(10)) for conditions of equilibrium (i.e. dNi/dt = 0), information regarding the relationship between the ion density at each level Ni and the total ion population ρ can be obtained (Appendix). Alternatively we can numerically solve the above set of equations (and those presented in the Appendix) using the values displayed in tables Table 1 and Table 2 and produce a number of graphical displays expressing the rate of change for the population of every energy state level (Ni) as a function of the overall ion density ρ and the characteristic up-conversion coefficient α. Table 1 summarizes the constant values that describe all ion transitions used in this 6 energy Table 2 Information about the experimental conditions used in solving the system of rate Eqs. (12)–(16). Parameter
Value
Λ Pin Aspot σabs Ρ Α
514 nm 2 mW 1 μm2 ~ 2 × 10− 25 m2 100–7500 ppm 4 × 10− 24–10− 22 m3/s
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F. Sidiroglou et al. / Journal of Non-Crystalline Solids 357 (2011) 3847–3852 Table 3 Fabrication characteristics of the erbium doped fiber samples investigated in this work.
Fig. 2. The ion population (N1) in the 4I13/2 state as a function of the total ion density (ρ) and the up-conversion coefficient. N1 and ρ are given in ions/m3.
level system, while Table 2 contains the parameters for which the above equations were solved. The ion density at each energy level can thus be evaluated and plotted as a function of the total ion population (for an erbium ion concentration between 100 and 7500 ppm) and the up-conversion coefficient (4 × 10 − 24 to 10 − 22 m 3/s). Fig. 2 and Fig. 3 represent the ion population in the 4I13/2 and 4S3/2 energy levels as a function of ρ and α. As the probability for two neighboring ions in the 4I13/2 level to interact increases, the ion density (N1) in 4I13/2 ceases to be linearly proportional to the total ion population of the system (Fig. 2). In contrast, the ion population N3 of the 4S3/2 energy state remains linearly dependent to the total ion population (Fig. 3) even in the presence of possible inter-ionic upconversion events in the lower energy levels. This demonstrates that the intensity of the backscattered fluorescence signal originating from the de-excitation of the 4S3/2 energy level will be proportional to the ion concentration, at a pump power and spot size typical of those used for measuring the erbium ion distribution in optical fibers as described in previous work [17] published by the authors.
Fiber
Sample code
Application type
Estimated Er3+ ion conc. (ppm)
Host composition
1 2 3 4 5
EDFA Er34 EDF-PCF Er10 Er35
Amplification fiber Experimental fiber Photonic crystal fiber laser Experimental fiber Experimental fiber
100 300 1000 2500 7600
Si, Si, Si, Si, Si,
Ge, Al, Ge, Al, Al, P Ge, Al, Ge, Al,
P P P P
and 7600 ppm were studied. In all cases the fiber preform was manufactured using the MCVD technique in conjunction with solution doping, with the silica core matrix also incorporating aluminum, phosphorous and germanium as network modifiers. Based on the above theoretical model, information about the way the Er 3+ ions are distributed in the core region of these fibers was obtained with the application of the FICOM technique [17]. The validity of this imaging scheme and its theoretical basis were then confirmed through a direct quantitative comparison of the data acquired using the FICOM [17] and NanoSIMS [10] techniques respectively. 4. Results Normalized transverse profiles displaying the relative Er 3+ ion distribution in the core region of the investigated fibers using the FICOM and NanoSIMS schemes are co-plotted in the series of graphs displayed in the following figures (Fig. 4 and Fig. 5). 5. Discussion A good agreement is evident between the extracted distribution profiles with the two different techniques. Any apparent discrepancies between the obtained data with the two different schemes are mainly a
3. Experimental procedures A total of five Er 3+ doped silica optical fibers (Table 3) having estimated erbium ion concentrations that ranged between 100 ppm
Fig. 3. The ion population (N3) in the 4S3/2 state as a function of the total ion density (ρ) and the up-conversion coefficient. N3 and ρ are given in ions/m3.
Fig. 4. Normalized transverse profiles of the relative Er3+ ion distribution in optical fibers, acquired from the application of the FICOM (continuous line) and NanoSIMS (dashed line) techniques.
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imaging set ups. A good quantitative agreement was evident, confirming the validity of the micro-fluorescence scheme and its associating theoretical model presented in this work. Appendix A At conditions of equilibrium (i.e. dNi/dt = 0), Eqs. (2)–(5) take the form, ð1Þ
Γ54 N5 = RE N1 ⇒ Γ54 N5 = 0:5RG N1
3+
Fig. 5. Normalized transverse profile of the relative Er ion distribution in Er35, acquired from the application of the FICOM (continuous line) and NanoSIMS (dashed line) techniques.
N4 = RG
ð11Þ
ðN0 + 0:5N1 Þ Γ43 ≫P40 ðN + 0:5N1 Þ ⇒ N4 = R G 0 Γ43 ðP40 + Γ43 Þ
N3 = RG Γ43
ðN0 + 0:5N1 Þ ðP40 + Γ43 ÞðP30 + P31 + Γ32 Þ
Γ43 ≫P40 &Γ32 ≫P30 + P31
⇒ N3 = RG
ð12Þ ðN0 + 0:5N1 Þ Γ32
ð13Þ direct consequence of the two following reasons. First, measurements with the two techniques were acquired from different sections along the length of each fiber, suggesting that possible elemental redistributions induced during the preform collapse process or the fiber draw process may result in slight variations in the distribution profiles along the length of the fiber. Second, from the obtained cross section analysis of each sample with the two techniques, it was found that most samples were to some extent non-symmetrical. Since there is no guarantee that the distribution profiles in each case have been extracted along the same coordinates, it is expected that such minor inconsistencies may exist between the two sets of data. Overall, information about the relative Er 3+ ion distribution in silica optical fibers was successfully obtained from the application of the FICOM technique according to the theoretical studies presented in this work. The good agreement that exists between the data acquired through the application of the optical and analytical systems constitutes a confirmation of the validity of the fluorescence based imaging technique and its associating theoretical model. From a spatial resolution point of view, both systems provided data at an estimated lateral resolution of the order of 0.4 μm and exhibited a repeatability for multiple measurements over the same region that deviated on average around 2.2% for the FICOM and 2.0% for the NanoSIMS. However, the major advantage of the FICOM method in comparison with the NanoSIMS configuration lies in its simplicity and ease of use. In contrast to the NanoSIMS arrangement, which requires a relatively expensive and not-widely available system, as well as time consuming and complex operation, the fluorescence based method represents a much cheaper and practical alternative. In addition, sample measurements can be obtained directly from the freshly cleaved endface of optical fibers without the need for special sample preparation techniques. 6. Conclusion Advances in the use of rare-earth doped silica optical fiber lasers and amplifiers in telecommunications and a range of industrial applications depend critically on an accurate knowledge of the rareearth dopant distribution within the core of the optical fiber or device. Here, a simple theoretical study of the spectroscopic characteristics of Er3+ doped silica optical fibers has established that information about the Er3+ ion distribution in optical fibers can be obtained by exciting the Er3+ ions using a 514 nm source and then monitoring the intensity of the backscattered fluorescence signal originating from the radiative transition of ions between the 4S3/2 upper energy level and the ground state. In order to test the practical validity of these theoretical studies, a FICOM was employed to analyze the relative Er3+ ion distribution in a range of optical fibers. The latter were also investigated with the use of a secondary ion probe and a direct quantitative comparison was performed between the information obtained with these two different
2
N2 =
1 N1 ðN + 0:5N1 Þ α + RG 0 2 Γ21 Γ21
ð14Þ
while from Eqs. (6) and (7) we get 2
ð1Þ& ð13Þ 1 N1 α + N1 ðRE + P10 Þ−N3 ðP31 + Γ32 Þ = 0 ⇒ 2 RG Γ32 ≫P31 1 N12 ðN + 0:5N1 Þ α + N1 ð0:5RG + P10 Þ−RG 0 ðP31 + Γ32 Þ = 0⇒ 2 RG Γ32 2
N0 =
1 N1 P N α + 10 1 : 2 RG RG ð15Þ
By substituting Eqs. (11)–(15) in the equation for the total ion density ρ and solve for N1 we get: 1 N12 P N 1 N2 ðN + 0:5N1 Þ α + 10 1 + N1 + α 1 + RG 0 2 RG 2 Γ21 Γ21 RG + RG
ðN0 + 0:5N1 Þ ðN + 0:5N1 Þ + RG 0 = ρ⇒ Γ32 Γ43
1 1 1 P 1 1 1 2 + + + αN1 + N1 10 + 1 + RG ðN0 + 0:5N1 Þ RG 2 RG Γ21 Γ21 Γ32 Γ43 1 1 1 + + ≅0 Γ21 Γ32 Γ43 ð16Þ −ρ = 0⇒ 1 1 1 P 2 + N1 10 + 1 −ρ = 0⇒ + αN1 2 RG Γ21 RG sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi P10 P10 1 1 − +1 + + 1 + 2αρ + RG Γ21 RG RG N1 = 1 1 α + RG Γ21
In a similar manner and by replacing Eq. (16) in the set of Eqs. (12)–(15) we can establish a relationship for the population of every energy state level (Ni) as a function of the overall ion density ρ and the characteristic up-conversion coefficient α. References [1] E. Desurvire, J.L. Zyskind, C.R. Giles, Design optimization for efficient erbiumdoped fiber amplifiers, Journal of Lightwave Technology 8 (11) (1990) 1730–1741. [2] M. Ohashi, Design considerations for an Er3+-doped fiber amplifier, Journal of Lightwave Technology 9 (9) (1991) 1099–1104. [3] P. Myslinski, D. Nguyen, J. Chrostowski, Effects of concentration on the performance of erbium-doped fiber amplifiers, Journal of Lightwave Technology 15 (1) (1997) 112–120.
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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Probing the erbium ion distribution in silica optical fibers with fluorescence based measurements F. Sidiroglou a,⁎, A. Roberts b, G.w. Baxter a a b
Optical Technology Research Laboratory, Centre for Telecommunications and Microelectronics, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia School of Physics, University of Melbourne, Victoria 3010, Australia
a r t i c l e
i n f o
Article history: Received 10 June 2011 Received in revised form 19 July 2011 Available online 17 August 2011 Keywords: Optical fibers; Erbium ion distribution; Confocal fluorescence microscopy; Fiber characterization; Fiber lasers and amplifiers
a b s t r a c t Accurate determination of the rare earth dopant distribution in optical fibers enhances our understanding of the fiber manufacture process and enables further improvement in the design of fiber based products such as optical fiber lasers and amplifiers. Here a simple theoretical model consisting of an ensemble of rate equation systems, characteristic of the most likely electronic transitions that take place in the vicinity of erbium (Er 3+) doped silica glasses, is developed and solved. Through this theoretical study it is established that information about the relative Er3+ ion distribution in fibers can be inferred by simply monitoring the backscattered fluorescence signal originating from the de-excitation of specific energy levels in the investigated samples. Following these theoretical studies a fluorescence intensity confocal optical microscopy (FICOM) scheme was employed to investigate the Er3+ ion distribution profiles in a range of silica optical fibers. The validity of the proposed theoretical model was confirmed through a comparison of the Er3+ ion distribution profiles acquired using the FICOM technique and those obtained from the application of a powerful analytical ion probe. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction Photonic devices, such as optical fibers, fiber lasers, and fiber amplifiers play an important role in today's society. Their capabilities have found use in a vast range of scientific and industrial environments amongst which the communications field has been influenced the most. In particular, although optical communication technologies were initially introduced to improve traditional information exchange, they have recently become a key factor behind the tremendous growth in internet traffic, and optical technologies will be even more important in enabling and supporting the future expansion of internet traffic. Optimizing the design and fabrication of such devices has become an ongoing challenge to optical scientists and engineers. For example, it has been demonstrated that the properties and performance of Er 3+ doped fiber lasers and amplifiers (EDFA) are related to a number of parameters such as the fiber glass material, the waveguide characteristics, and the distribution profile of the Er3+ ions. Accurate knowledge of the latter has been found to be vital for the optimal design and operation of these devices [1–4]. Various techniques have therefore been developed and investigated in order to acquire information related to the distribution of the dopants in the fibers.
⁎ Corresponding author. E-mail address: [email protected] (F. Sidiroglou).
To date, two different approaches have been employed to measure the distribution of Er 3+ or other rare earth (RE) ions in optical fibers. In the first case, the concentration of dopants and their distribution, together with the refractive index profile (RIP) and other parameters are usually measured in the fiber preform, prior to the fiber drawing process. Some of the most important techniques employed to reveal the location of dopants in bulk samples such as fiber preforms, are based on the utilization of methods that are commonly used in the field of analytical chemistry and material science. These techniques include secondary ion mass spectroscopy [5], inductively coupled plasma atomic emission spectroscopy [6], and X-ray microprobe analysis [7]. The distribution of dopants in fiber preforms has also been examined using the electron probe microanalysis (EPMA) method and its counterpart systems [8,9]. The resultant dopant concentration and distribution is then scaled down to match the drawn fiber dimensions assuming that there are no defects affecting the resulting fiber profile during the drawing process. This assumes that the profile remains unchanged during the drawing process. Huntington et al. [10] has shown that this assumption is not always valid for the determination of the RIP. This is believed to be the consequence of various effects such as the diffusion of elements that takes place because of the high temperature environment created during the drawing process. As a result it is not always valid to simply relate the dopant concentration profile extracted from preform measurements to fiber dimensions. In the second approach, measurements are made directly on the drawn fiber. Once again a number of the techniques utilized for the
0022-3093/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.07.024
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analysis of the RE distribution in optical fibers have been borrowed from the analytical chemistry and materials science field. Such techniques include nano-secondary ion mass spectroscopy (NanoSIMS) [11], time of flight secondary ion mass spectroscopy (ToF-SIMS) [12], transmission electron microscopy [13] and Raman microscopy [14]. Although these techniques can offer adequate spatial imaging resolution for the direct investigation of optical fibers and they can offer information about the distribution of other dopants within the fiber core region, they normally require complex and time-consuming sample preparation and the use of high cost instrumentation. In an attempt to overcome such limitations, a number of optical imaging schemes have been applied over the years in order to provide information about the RE ion distribution in the core of optical fibers directly from the investigation of their cleaved endface [15–17]. With the exception of the work of Petreski et al. [15], where a fluorescence lifetime confocal optical microscopy technique was developed for the investigation of praseodymium-doped optical fibers, the other optical based systems were capable of providing information about the Er 3+ ion distribution in optical fibers with the application of a fluorescence intensity confocal optical microscopy scheme [16,17]. In the latter cases, the intensity of the backscattered fluorescence originating from the de-excitation of the 2H11/2 and 4S3/2 upper energy levels (Fig. 1) was monitored and taken as an indication of the local dopant concentration of the Er 3+ ions at a point. By scanning the excitation beam across the center of the fiber core, two-dimensional images or line-scans of the relative Er 3+ ion distribution were acquired. However, the acquired intensity profiles were assumed to be directly related to the relative Er 3+ ion distributions without establishing that the resulting backscattered fluorescence signal was indeed proportional to the erbium ion concentration. Care must be taken regarding the reliability of the intensity based measurements given that cooperative effects in RE doped fibers may yield incorrect values of dopant concentration [18]. In this work, the issue of the dynamics of different emission lines observed in Er 3+ doped optical fibers and their relationship to the total Er 3+ ion concentration is addressed by developing and solving a system of rate equations characteristic to the most likely electronic transitions observed in a Er 3+ doped silica glass. This exploration validates the use of direct pumping of the 2H11/2 (at 514 nm, as shown in some of our previous work [17]) or the 4F7/2 (at 488 nm in the work of [16]) levels with the subsequent detection of the backscattered fluorescence signal (around 550 nm) from the de-excitation of the 4S3/2 level as a measure of the relative Er3+ ion distribution in optical fibers. Furthermore, we extend
Γ Γ Γ α
Γ
α
Fig. 1. Partial energy level diagram for Er3+ in silica illustrating a six energy level system. Solid arrows represent radiative transitions, dashed arrows non-radiative decay and dotted arrows two ion interactions.
the relevance of this theoretical model as applied in our previous work with the use of a confocal optical microscope [17] to our more recent efforts in trying to improve spatial resolution using a Near-field Scanning Optical Microscope (NSOM). 2. Theoretical modeling The main focus of this article is to identify the specific electronic transitions that take place in Er 3+ doped silica glasses, which could be related directly to the concentration or distribution of the erbium ions within the core of optical fibers. The issue remains whether fluorescence intensity based measurements could possibly be related to the erbium dopant profile. As long as the process of de-excitation of an energy level is free from any possible inter-ionic cross-relaxation or upconversion effects then it can be assumed that the intensity of the emitted signal is proportional to the population of ions at that point. For radiative events during which the excited ions are returning to the ground state 4I15/2, there are three energy levels that have been identified as potential levels in which de-excitation can occur and whose intensity of emitting signal could be taken as an indication of the ion population. These levels are (Fig. 1): • the 4I13/2 state, where radiative decay back to the ground state from this level yields a fluorescence band centered at around 1530 nm, • the 4I11/2 state where de-excitation from this level occurs with the simultaneous emission of photons near 980 nm, • and the 4S3/2 state from which the ions can return back to the ground state with the concurrent emission of light at 550 nm. Depending on the associated energy level, excitation of the ions can be achieved with either resonant or near-resonant pumping and the collection of all energy transfers and radiative events can be explained by treating each system as a n-level system, where n the number of energy levels involved. In the case of the well-known metastable state 4I13/2, the ions could be excited using a near resonant pumping scheme with a 980 nm source, which excites the ions first to the 4I11/2 level before non-radiatively relaxing to the next lower level 4I13/2 from which they can then return back to the ground state by radiative emission centered on 1530 nm. It can be postulated that the intensity of this emission could be taken as an indication of the erbium dopant distribution while the excitation source is scanned across the endface of an optical fiber as long as there are no inter-ionic cross-relaxation or upconversion effects that can affect the relationship between the measured fluorescence signal and the investigated dopant concentration. However, the dynamics of the 4I13/2 state are strongly affected by various types of ion to ion interactions and in particular by excited state absorption (ESA) [19]. Therefore the emission intensity is not a direct measure of the relative erbium ion concentration. With the exception of the 1480 nm pumping scheme that does not suffer from ESA when used to pump Er 3+ doped optical fibers, ESA has been observed to originate from the 4I13/2 level for most of the other exhibited wavelengths used for pumping Er 3+ doped fibers [19]. If a scheme where the signal from the de-excitation of the 4I11/2 energy level is used as a tool to investigate the erbium ion distribution, crossrelaxation or upconversion effects can once again alter the dynamics of the system so that such an approach will not be accurate. Particularly in the most conventional case, where a 980 nm source is used to resonantly excite the ions at that level, ESA is known to take place in such a way that the already excited ions can be promoted to the 4F7/2 upper state. Furthermore, since most of the ions will instead relax to the metastable 4I13/2 level – by either non-radiative or radiative processes – before they return to the ground state, the ability to obtain concentration measurements based on the intensity of the detected backscattered fluorescence signal becomes even less realistic.
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On the other hand, if ion distribution measurements are carried out based on the detected signal from the de-excitation of the 4S3/2 state, non-radiative processes such as multiphonon decay are the predominant electronic transitions that have to be incorporated in theoretical models rather than any inter-ionic events as were encountered in the other cases. As it has been demonstrated by Desurvire et al. [20,21], the relaxation process between the 2H11/2 and 4 I13/2 (Fig. 1) energy levels is strongly dominated by fast non-radiative decay. In fact, the non-radiative transition processes for the upper energy levels are described by rates that on average are, almost by a factor of one thousand or larger, more probable than from those that describe radiative decay (Table 1). As a result, even in the presence of possible inter-ionic upconversion events in the lower energy levels, their contribution to the dynamics of the radiative transitions originating from the upper levels such as that of the 4S3/2 state is expected to be insignificant. The same argument can be drawn for the case where ESA promotes ions from the metastable 4I13/2 energy level to the 2H9/2 higher energy level. It is expected that the excited ions will return through a series of non-radiative processes to the lower energy states without affecting the dynamics of the radiative events originating from the de-excitation of the 4S3/2 state. Consequently it may be assumed that the intensity of the backscattered fluorescence signal originating from the de-excitation of the 4S3/2 energy level is proportional to the ion concentration. This statement is better demonstrated by investigating the rate equations that characterize all radiative and non-radiative events in the case of the six energy level diagram of Fig. 1. This diagram demonstrates the more realistic case where two ion interaction and ESA are also considered in the ensemble of events that describe the change in the ion populations for each level. A near-resonant pumping scheme at 514 nm is used as the source for the excitation of the ions from the ground state. Absorption of the pump (at a pump rate of RG) promotes the ground state ions to the 2H11/2 upper energy level. With the exception of a small fraction of excited ions that return back to the ground state through radiative decay (i.e. 2H11/2 → 4I15/2, 4S3/2 → 4I15/2 and 4S3/2 → 4I13/2), the vast majority of the ions will decay through a fast non-radiative process through lower energy levels until they reach the 4I13/2 metastable state. While at this energy level, most of the ions will return through radiative decay to the ground state. However, part of the ions can be promoted through ESA to the 2H9/2 upper energy level at a rate which according to the work of Laming et al. [22] is equal to: RE = 0:5RG :
ð1Þ
In addition, the ion population of the 4I13/2 level can be quenched due to an ion pair up-conversion process. Such a process involves two neighboring ions in the 4I13/2 state: one of the ions transfers its energy to the other, producing one up-converted 4I9/2 ion and one ground state ion. The up-converted ion quickly decays to the 4I13/2 state. The consequence is the loss of one excited erbium ion. This event has been modeled for erbium doped silicate fibers [23–26] and is described by a characteristic up-conversion coefficient (α), which is expressed in units of m 3/s. Table 1 Decay rates of electronic transitions encountered in erbium doped silica as depicted by the energy level diagram of Fig. 1. Transition
Type
Constant
Decay rate
Reference
2
Non-radiative decay Radiative decay Non-radiative decay Radiative decay Radiative decay Non-radiative decay Radiative decay
Γ43 P40 Γ32 P30 P31 Γ21 P10
~ 107 s− 1 ~ 3 × 103 s− 1 ~ 0.14 × 107 s− 1 ~ 103 s− 1 ~ 4 × 102 s− 1 ~ 2 × 108 s− 1 ~ 90 s− 1
[20] [20] [20] [20] [21] [20] [21]
H11/2 → 4S3/2 2 H11/2 → 4I15/2 4 S3/2 → 4I13/2 4 S3/2 → 4I15/2 4 S3/2 → 4I13/2 4 I9/2 → 4I13/2 4 I13/2 → 4I15/2
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The rates of change of the ion densities for each level of the energy level diagram of Fig. 1 take the form: dN5 = RE N1 −Γ54 N5 dt
ð2Þ
dN4 = RG N0 + Γ54 N5 −P40 N4 −Γ43 N4 dt
ð3Þ
dN3 = Γ43 N4 −P30 N3 −P31 N3 −Γ32 N3 dt
ð4Þ
dN2 1 2 = Γ32 N3 + αN1 −Γ21 N2 2 dt
ð5Þ
dN1 2 = Γ21 N2 + P31 N3 −P10 N1 −αN1 −RE N1 dt
ð6Þ
dN0 1 2 = P40 N4 + P30 N3 + P10 N1 + αN1 −RG N0 2 dt
ð7Þ
where Ni is the ion population for each level i (with i = 0, 1, 2, 3, 4 and 5), Γij represents the non-radiative decay between two energy levels i and j and Pij represents the radiative decay between two energy levels i and j. RG is the pump absorption rate for the ground state ions and can be determined as follows. RG = Pin
λ σabs hc Aspot
ð2:8Þ
where Pin and λ are the input power and wavelength of the excitation source, Aspot is the size of the beam spot, σabs is the absorption cross section for the ground state ions when an excitation at λ = 514 nm is used, while h and c are Plank's constant and the speed of light respectively. At each instant the overall ion density ρ is given by the sum of the ion densities at each level ρ = N0 + N 1 + N 2 + N 3 + N 4 + N 5
ð2:9Þ
which can be simplified to ð2:10Þ
ρ = N 0 + N1 + N2 + N3 + N4
,since at any instance the majority of the total ions exist between levels 4 I15/2 to 2H11/2. The units for the ion density at each level i is expressed in ions/m3, while the units for the pump rate as well as for the radiative and non-radiative decays are that of s− 1. By solving the above system of rate equations (Eqs (2)–(10)) for conditions of equilibrium (i.e. dNi/dt = 0), information regarding the relationship between the ion density at each level Ni and the total ion population ρ can be obtained (Appendix). Alternatively we can numerically solve the above set of equations (and those presented in the Appendix) using the values displayed in tables Table 1 and Table 2 and produce a number of graphical displays expressing the rate of change for the population of every energy state level (Ni) as a function of the overall ion density ρ and the characteristic up-conversion coefficient α. Table 1 summarizes the constant values that describe all ion transitions used in this 6 energy Table 2 Information about the experimental conditions used in solving the system of rate Eqs. (12)–(16). Parameter
Value
Λ Pin Aspot σabs Ρ Α
514 nm 2 mW 1 μm2 ~ 2 × 10− 25 m2 100–7500 ppm 4 × 10− 24–10− 22 m3/s
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F. Sidiroglou et al. / Journal of Non-Crystalline Solids 357 (2011) 3847–3852 Table 3 Fabrication characteristics of the erbium doped fiber samples investigated in this work.
Fig. 2. The ion population (N1) in the 4I13/2 state as a function of the total ion density (ρ) and the up-conversion coefficient. N1 and ρ are given in ions/m3.
level system, while Table 2 contains the parameters for which the above equations were solved. The ion density at each energy level can thus be evaluated and plotted as a function of the total ion population (for an erbium ion concentration between 100 and 7500 ppm) and the up-conversion coefficient (4 × 10 − 24 to 10 − 22 m 3/s). Fig. 2 and Fig. 3 represent the ion population in the 4I13/2 and 4S3/2 energy levels as a function of ρ and α. As the probability for two neighboring ions in the 4I13/2 level to interact increases, the ion density (N1) in 4I13/2 ceases to be linearly proportional to the total ion population of the system (Fig. 2). In contrast, the ion population N3 of the 4S3/2 energy state remains linearly dependent to the total ion population (Fig. 3) even in the presence of possible inter-ionic upconversion events in the lower energy levels. This demonstrates that the intensity of the backscattered fluorescence signal originating from the de-excitation of the 4S3/2 energy level will be proportional to the ion concentration, at a pump power and spot size typical of those used for measuring the erbium ion distribution in optical fibers as described in previous work [17] published by the authors.
Fiber
Sample code
Application type
Estimated Er3+ ion conc. (ppm)
Host composition
1 2 3 4 5
EDFA Er34 EDF-PCF Er10 Er35
Amplification fiber Experimental fiber Photonic crystal fiber laser Experimental fiber Experimental fiber
100 300 1000 2500 7600
Si, Si, Si, Si, Si,
Ge, Al, Ge, Al, Al, P Ge, Al, Ge, Al,
P P P P
and 7600 ppm were studied. In all cases the fiber preform was manufactured using the MCVD technique in conjunction with solution doping, with the silica core matrix also incorporating aluminum, phosphorous and germanium as network modifiers. Based on the above theoretical model, information about the way the Er 3+ ions are distributed in the core region of these fibers was obtained with the application of the FICOM technique [17]. The validity of this imaging scheme and its theoretical basis were then confirmed through a direct quantitative comparison of the data acquired using the FICOM [17] and NanoSIMS [10] techniques respectively. 4. Results Normalized transverse profiles displaying the relative Er 3+ ion distribution in the core region of the investigated fibers using the FICOM and NanoSIMS schemes are co-plotted in the series of graphs displayed in the following figures (Fig. 4 and Fig. 5). 5. Discussion A good agreement is evident between the extracted distribution profiles with the two different techniques. Any apparent discrepancies between the obtained data with the two different schemes are mainly a
3. Experimental procedures A total of five Er 3+ doped silica optical fibers (Table 3) having estimated erbium ion concentrations that ranged between 100 ppm
Fig. 3. The ion population (N3) in the 4S3/2 state as a function of the total ion density (ρ) and the up-conversion coefficient. N3 and ρ are given in ions/m3.
Fig. 4. Normalized transverse profiles of the relative Er3+ ion distribution in optical fibers, acquired from the application of the FICOM (continuous line) and NanoSIMS (dashed line) techniques.
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imaging set ups. A good quantitative agreement was evident, confirming the validity of the micro-fluorescence scheme and its associating theoretical model presented in this work. Appendix A At conditions of equilibrium (i.e. dNi/dt = 0), Eqs. (2)–(5) take the form, ð1Þ
Γ54 N5 = RE N1 ⇒ Γ54 N5 = 0:5RG N1
3+
Fig. 5. Normalized transverse profile of the relative Er ion distribution in Er35, acquired from the application of the FICOM (continuous line) and NanoSIMS (dashed line) techniques.
N4 = RG
ð11Þ
ðN0 + 0:5N1 Þ Γ43 ≫P40 ðN + 0:5N1 Þ ⇒ N4 = R G 0 Γ43 ðP40 + Γ43 Þ
N3 = RG Γ43
ðN0 + 0:5N1 Þ ðP40 + Γ43 ÞðP30 + P31 + Γ32 Þ
Γ43 ≫P40 &Γ32 ≫P30 + P31
⇒ N3 = RG
ð12Þ ðN0 + 0:5N1 Þ Γ32
ð13Þ direct consequence of the two following reasons. First, measurements with the two techniques were acquired from different sections along the length of each fiber, suggesting that possible elemental redistributions induced during the preform collapse process or the fiber draw process may result in slight variations in the distribution profiles along the length of the fiber. Second, from the obtained cross section analysis of each sample with the two techniques, it was found that most samples were to some extent non-symmetrical. Since there is no guarantee that the distribution profiles in each case have been extracted along the same coordinates, it is expected that such minor inconsistencies may exist between the two sets of data. Overall, information about the relative Er 3+ ion distribution in silica optical fibers was successfully obtained from the application of the FICOM technique according to the theoretical studies presented in this work. The good agreement that exists between the data acquired through the application of the optical and analytical systems constitutes a confirmation of the validity of the fluorescence based imaging technique and its associating theoretical model. From a spatial resolution point of view, both systems provided data at an estimated lateral resolution of the order of 0.4 μm and exhibited a repeatability for multiple measurements over the same region that deviated on average around 2.2% for the FICOM and 2.0% for the NanoSIMS. However, the major advantage of the FICOM method in comparison with the NanoSIMS configuration lies in its simplicity and ease of use. In contrast to the NanoSIMS arrangement, which requires a relatively expensive and not-widely available system, as well as time consuming and complex operation, the fluorescence based method represents a much cheaper and practical alternative. In addition, sample measurements can be obtained directly from the freshly cleaved endface of optical fibers without the need for special sample preparation techniques. 6. Conclusion Advances in the use of rare-earth doped silica optical fiber lasers and amplifiers in telecommunications and a range of industrial applications depend critically on an accurate knowledge of the rareearth dopant distribution within the core of the optical fiber or device. Here, a simple theoretical study of the spectroscopic characteristics of Er3+ doped silica optical fibers has established that information about the Er3+ ion distribution in optical fibers can be obtained by exciting the Er3+ ions using a 514 nm source and then monitoring the intensity of the backscattered fluorescence signal originating from the radiative transition of ions between the 4S3/2 upper energy level and the ground state. In order to test the practical validity of these theoretical studies, a FICOM was employed to analyze the relative Er3+ ion distribution in a range of optical fibers. The latter were also investigated with the use of a secondary ion probe and a direct quantitative comparison was performed between the information obtained with these two different
2
N2 =
1 N1 ðN + 0:5N1 Þ α + RG 0 2 Γ21 Γ21
ð14Þ
while from Eqs. (6) and (7) we get 2
ð1Þ& ð13Þ 1 N1 α + N1 ðRE + P10 Þ−N3 ðP31 + Γ32 Þ = 0 ⇒ 2 RG Γ32 ≫P31 1 N12 ðN + 0:5N1 Þ α + N1 ð0:5RG + P10 Þ−RG 0 ðP31 + Γ32 Þ = 0⇒ 2 RG Γ32 2
N0 =
1 N1 P N α + 10 1 : 2 RG RG ð15Þ
By substituting Eqs. (11)–(15) in the equation for the total ion density ρ and solve for N1 we get: 1 N12 P N 1 N2 ðN + 0:5N1 Þ α + 10 1 + N1 + α 1 + RG 0 2 RG 2 Γ21 Γ21 RG + RG
ðN0 + 0:5N1 Þ ðN + 0:5N1 Þ + RG 0 = ρ⇒ Γ32 Γ43
1 1 1 P 1 1 1 2 + + + αN1 + N1 10 + 1 + RG ðN0 + 0:5N1 Þ RG 2 RG Γ21 Γ21 Γ32 Γ43 1 1 1 + + ≅0 Γ21 Γ32 Γ43 ð16Þ −ρ = 0⇒ 1 1 1 P 2 + N1 10 + 1 −ρ = 0⇒ + αN1 2 RG Γ21 RG sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi P10 P10 1 1 − +1 + + 1 + 2αρ + RG Γ21 RG RG N1 = 1 1 α + RG Γ21
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