Modeling rare-earth doped microfiber ring lasers

doi:10.1364/OE.14.007073

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Modeling rare-earth doped microfiber ring lasers

Yuhang Li, Guillaume Vienne, Xiaoshun Jiang, Xinyun Pan, Xu Liu, Peifu Gu, and Limin Tong »View Author Affiliations

Optics Express, Vol. 14, Issue 16, pp. 7073-7086 (2006)
http://dx.doi.org/10.1364/OE.14.007073

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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical model
3. Discussion-Er and Yb doped microfiber ring lasers
4. Conclusions
– References and links

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Abstract

We propose a compact laser configuration based on resonating both the pump and signal light along a microfiber ring doped with active ions. We estimate the minimum Q-factor to obtain lasing and find that values already demonstrated in passive microfiber rings will be sufficient. We model the performance of this device in steady state using rate equations and show that pump resonance can significantly reduce the threshold and increase the quantum efficiency, especially for rings made of materials with weak active ion absorption. Numerical examples for erbium and ytterbium doped devices are presented. Taking into account scattering and coupling losses the optimum pump coupling factor is calculated. The dependences of the quantum efficiency and threshold power on the coupling losses are also investigated. We predict that efficient ytterbium-doped lasers can be obtained with a ring diameter down to a few tens of micrometers.

1. Introduction

Resonators have long been used in fields such as sensing [1

E. Udd, ed., Fiber Optic Sensors (Wiley, New York, 1991).

], telecommunications [2

C. K. Madsen and J. H. Zhao, Optical filter design and analysis: A signal processing approach (Wiley, New York, 1999).

], and metrology [3

P. Hariharan, Opital Interferometry , 2nd ed. (Academic, New York, 2003).

]. Very small size resonators down to submillimeter or even micrometer scale are particularly desired for photonics integration and biosensing [4

F. Vollmer, D. Braun, A. Libchaber, S. M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002). [CrossRef]

]. Several implementations of both passive and active microresonators have been demonstrated, including Si-SiO₂ microrings [5

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, and E. P. Ippen, “Ultra-compact Si-SiO₂ microring resonator optical channel dropping filters,” IEEE Photonics Tech. Lett. 10, 549–551 (1998). [CrossRef]

], photonic bandgap microcavities [6

K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

], microspheres [7

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000). [CrossRef]

], microdisks [8

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992). [CrossRef]

], and microtoroids [9

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett. 84, 1037–1039 (2004). [CrossRef]

]. Compared to these implementations micro- and nanofiber resonators offer simpler and/or more stable light coupling and can readily support single-mode operation. New fabrication methods for microfibers have recently been proposed [10

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef] [PubMed]

, 11

L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express 14, 82–87 (2006). [CrossRef] [PubMed]

], and microrings have been obtained by shaping a microfiber in the form of a loop [12

M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. 86, 161108 (2005). [CrossRef]

], or a knot [13

X. Jiang, L. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, and D. Yang, “Demonstration of optical microfiber knot resonators,” Appl. Phys. Lett. 88, 223501 (2006). [CrossRef]

]. The maximum value of the intrinsic Q factor obtained so far in a microfiber loop was 630 000 [12

M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. 86, 161108 (2005). [CrossRef]

], and a value as high as 10¹⁰ has been predicted [14

M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express 12, 2303–2316 (2004). [CrossRef] [PubMed]

]. When the microfiber is doped with active ions such as rare-earth ions, the microfiber ring can act both as resonator and as active medium for lasing. Resonance for the signal is necessary to obtain a laser but we show in this paper that resonating the pump as well provides an efficient method to reduce device size. Compact lasers are attractive not only for integration but also for operation in a single longitudinal mode. Short-length rare-earth doped fiber lasers have previously been reported [15

K. Hsu, C. M. Miller, J. T. Kringlebotn, and D. N. Payne, “Continuous and discrete wavelength tuning in Er:Yb fiber Fabry-Perot lasers,” Opt. Lett. 20, 377–379 (1995). [CrossRef] [PubMed]

], in which high reflectivity dielectric mirrors deposited on the fiber facets resulted in a Fabry-Perot cavity with a Q-factor sufficient to obtain low threshold powers (below 1 mW) but the quantum efficiency remained below 1‰ due to the short absorption length. Here we show that recirculating the pump near critical pump coupling in a microfiber ring can result in efficient ultracompact lasers.

The model is presented in section 2. In section 2.1, we start by analyzing how the pump intensity builds up at resonance. In section 2.2, we then combine the expression of the pump intensity at resonance with the rate equations of a three-level system to obtain the threshold power and the quantum efficiency in the proposed laser configuration. The physical meaning of these analytical expressions is highlighted. In section 3 we illustrate the effect of pump resonance by studying two different cases of rare-earth doping, namely erbium and ytterbium, and device optimization is discussed in details for both cases.

2. Theoretical model

Our model is based on two sets of equations. Coupling equations in the subsection 2.1 are used to analyze resonances for both pump and signal light, whereas the rate-equations in the subsection 2.2 describe the transitions in active ions. In the microfiber ring laser discussed in this paper, the pump light is either absorbed by the active ions (which leads to the population inversion) or lost by the scattering along the fiber or in the coupling region. Similarly, the signal light is either amplified as a result of population inversion or suffers loss by scattering. Here we consider the case of a three-level system. For completeness the case of four-level systems is treated in the appendix. Energy transfer effects which are detrimental to the quantum efficiency of lasers, such as concentration quenching, are not taken into account here but the numerical examples are chosen for concentrations where these effects are not prevalent. We do not consider the strong signal case where gain saturation may occur since high output power is not sought after here. On the other hand, our analysis focuses on obtaining compact and efficient devices. The diameter of the ring is chosen sufficiently small that absorption along a single path of the ring is low. Despite this low single-path absorption we will show that pump resonance results in high pump intensity and efficient pump absorption, paving the way to low threshold, high quantum efficiency compact fiber lasers.

2.1. Ring resonator equations

The microfiber ring operation for pump and signal is schematically illustrated in Fig. 1. Here we consider stationary operation. In this case, the relation between complex field amplitudes for the pump light in the fiber is [16

L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982). [CrossRef] [PubMed]

]

{\begin{matrix} E_{3, p} = {(1 - γ_{p})}^{\frac{1}{2}} [{(1 - K_{p})}^{\frac{1}{2}} E_{1, p} + j K_{p}^{\frac{1}{2}} E_{2, p}] \\ E_{4, p} = {(1 - γ_{p})}^{\frac{1}{2}} [j K_{p}^{\frac{1}{2}} E_{1, p} + {(1 - K_{p})}^{\frac{1}{2}} E_{2, p}] \end{matrix}} and E_{2, p} = E_{3, p} e^{\frac{- α_{p} π D}{2}} e^{jβπD},

(1)

where K_p , γ_p are the intensity coupling coefficient and coupling loss for the pump light, respectively. E_i,p (i=1,2,3,4) are the complex field amplitudes corresponding to I_i,p (i=1,2,3,4) shown in Fig. 1, D is the diameter of the ring, and β is the longitudinal propagation constant of the pump light in the microfiber.

Fig. 1. Schematic of a microfiber ring resonator. (a) The pump light I_1,p is partially coupled into the ring by a fiber taper at the coupling region of the ring, of intensity coupling coefficient K _p, and of coupling loss γ_p . The pump experiences loss of coefficient α_p , as well as dephasing while propagating along the ring, and is partially coupled outside at the coupling region, resulting in a transmitted intensity, I_4,p . Loss for the pump light is due to absorption and scattering, thus α_p =α_abs,p ,+α_sc,p . For a glass microfiber suspended in air the refractive index contrast is at least 30% and the bending loss is neglected here. (b) The signal light resonates in the ring in two directions, I _+,s and I _-,s, and is coupled out at the coupling region, resulting in transmitted intensities I_1,s and I_4,s . The loss coefficient, the gain coefficient, the coupling loss at the coupling region, and the signal coupling factor, are denoted as α_s , G, γ_s , and K _s, respectively. Loss is assumed to be solely due to scattering, so α_s =α_sc,s .

K_p =0 gives the nonresonant case for pump light,

{\begin{matrix} \frac{I_{3, p}}{I_{1, p}} = 1 - γ_{p} \\ \frac{I_{4, p}}{I_{1, p}} = {(1 - γ_{p})}^{2} e^{- α_{p} π D} \end{matrix}} .

(2)

When the resonance condition for β is satisfied [16

L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982). [CrossRef] [PubMed]

], the intensity relations are

{\begin{matrix} \frac{I_{3, p}}{I_{1, p}} = \frac{(1 - γ_{p}) (1 - K_{p})}{{(1 - {(1 - γ_{p})}^{\frac{1}{2}} K_{p}^{\frac{1}{2}} e^{\frac{- α_{p} π D}{2}})}^{2}} \\ \frac{I_{4, p}}{I_{1, p}} = \frac{(1 - γ_{p}) {(K_{p}^{\frac{1}{2}} - {(1 - γ_{p})}^{\frac{1}{2}} e^{\frac{- α_{p} π D}{2}})}^{2}}{{(1 - {(1 - γ_{p})}^{\frac{1}{2}} K_{p}^{\frac{1}{2}} e^{\frac{- α_{p} π D}{2}})}^{2}} \end{matrix}} .

(3)

As expected, setting K_p =0 in Eq. (3) gives the exact expression of Eq. (2).

Since the pump light experiences attenuation along the ring, the average intensity is

I_{p} = \frac{1}{π D} \int_{0}^{π D} I_{3, p} e^{- z α_{p}} dz = \frac{I_{3, p} (1 - e^{- α_{p} π D})}{α_{p} π D} .

(4)

The intensity enhancement factor E for resonant pump is given by

E = \frac{I_{p}}{I_{1, p}} = \frac{(1 - γ_{p}) (1 - K_{p})}{{(1 - {(1 - γ_{p})}^{\frac{1}{2}} K_{p}^{\frac{1}{2}} e^{\frac{- α_{p} π D}{2}})}^{2}} \frac{1 - e^{- α_{p} π D}}{α_{p} π D} .

(5)

The maximum value with regard to K_p is reached when dE/dK_p =0, giving

E_{max} = \frac{1 - γ_{p}}{1 - (1 - γ_{p}) e^{- α_{p} π D}} \frac{1 - e^{- α_{p} π D}}{α_{p} π D} = \frac{1}{\frac{1}{(1 - γ_{p})} + e^{- α_{p} π D}} \frac{1 - e^{- α_{p} π D}}{α_{p} π D} \approx \frac{1}{γ_{p} + 1 - e^{- α_{p} π D}} \frac{1 - e^{- α_{p} π D}}{α_{p} π D},

(6)

corresponding to the critical coupling condition K_p = (1-γ_p )e ^-α_pπD. In this case I_4,P is zero according to Eq. (3), which means that the pump light is totally consumed inside the ring [17

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whisper-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000). [CrossRef] [PubMed]

The enhancement factor E for microfiber rings with different α_pπD is plotted in Fig. 2(a). It can be seen that no enhancement occurs when K_p =0, as indicated by Eq. (2). When K_p reaches its optimum value, which corresponds to critical coupling, E increases largely, especially for rings of weak absorption materials. In Fig. 2(b), the values of E at critical coupling are plotted against α_pπD. It can be seen that E_max decreases dramatically as α_pπD becomes larger.

For the signal, light is coupled out in two directions, as is shown in Fig. 1(b), so that

\frac{I_{1, s}}{I_{-, s}} = \frac{I_{4, s}}{I_{+, s}} = (1 - γ_{s}) (1 - K_{s})

(7)

where K_s and γ_s are the intensity coupling coefficient and coupling loss for the signal light, respectively.

In practice, it should not be difficult to make the ring resonant for both the pump and the signal light. This is due to the fact that for practical diameters of the ring, a few tens of micrometers and above, the free spectral range does not exceed the emission linewidth of rare-earth ions in glasses, which is typically in the order of tens of nanometers. This means that at least one of the signal light resonance peaks will fall inside the gain curve. It is therefore sufficient to adjust the ring diameter so that a resonance peak coincides with the pump source spectrum.

Fig. 2. (a) Intensity enhancement factor E versus pump coupling coefficient K_p . Each curve corresponds to different α_pπD for the pump light, see labels. K_p =0 is the nonresonant case, whereas the peak values correspond to critical coupling. (b) Maximum of E versus α_pπD. The coupling loss is assumed to be 0.003 for both (a) and (b).

2.2. Rate equations

Atomic transitions in lasers can generally be treated as three-level transitions (TLT) or four-level transitions (FLT). The former case is schematically shown in Fig. 3, and will be discussed in details. The FLT case is treated in the appendix.

Assuming that the pump field and the dopant distribution are uniform across the fiber, that the degenerations of the upper and lower transition levels are equal, that τ₃₂ is several orders of magnitude smaller than τ, and that excited state absorptions and energy transfers between ions are not present, the population rate equations can be expressed as [18

Anthony E. Siegman, Laser (Mill Valley, California, 1986).

]

{\begin{matrix} \frac{d N_{1}}{dt} = - R N_{1} + \frac{N_{2}}{τ} + N_{2} W_{21} - N_{1} W_{12} \\ N_{1} + N_{2} = N \end{matrix}},

(8)

where

R = \frac{Γ_{p} I_{p}}{h ν_{p}} σ_{abs, p}

W_{21} = \frac{Γ_{s} (I_{+, s} + I_{-, s})}{h ν_{s}} σ_{em}

W_{12} = \frac{Γ_{s} (I_{+, s} + I_{-, s})}{h ν_{s}} σ_{abs}

. In these equations, N is the dopant concentration, N₁ , N₂ are the populations per unit volume for |E₁> and |E₂>, respectively. σ_abs,p is the absorption cross sectional area for the pump light, and σ_abs , σ_em are the cross sectional area for re-absorption and stimulated emission of the signal light, respectively. Γ_p and Γ_s stand for the fractional intensity inside the microfiber for the pump and signal light [19

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

]. τ denotes the spontaneous emission lifetime for |E₂>. I _+,s, I _-,s are the signal light intensities circulating in the ring in two directions and v_p , v_s are the frequencies of the pump and signal light.

Fig. 3. Schematic diagram for three-level transitions (TLT) with corresponding rates. σ_abs,p , σ_abs , σ_em stand for the cross sectional areas for the pump absorption, the signal re-absorption, and the stimulated emission, respectively. Since only a fraction of light propagates in the sub wavelength fiber [19

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

], the average pump and signal intensities in the fiber core are reduced by the overlap factors Γ_p and Γ_s . τ₃₂ and τ are the spontaneous radiation lifetimes from |E₃> to |E₂> and |E₂> to |E₁>, respectively.

The populations are dependent on the position along the fiber, but we are dealing with the case of weak pump absorption along a single path of the ring, so that an average pump value is considered within the ring and the spatial variation is neglected.

In the steady state (dN_i /dt=0), the gain compensates for the loss so that

K_{s} (1 - γ_{s}) e^{- α_{s} π D} e^{GπD} = e^{GπD - α_{tot, s} π D} = 1

(9)

where G=Γ_s (N₂σ_em -N₁σ_abs ) is the gain factor, and α_tot,s is the total loss of the signal light per round trip distributed over length,

α_{tot, s} = α_{sc, s} + \frac{1}{π D} \ln \frac{1}{K_{s}} + \frac{1}{π D} \ln \frac{1}{1 - γ_{s}} .

(10)

From Eqs. (4) and (8), the signal light intensity inside the ring is

I_{+, s} + I_{-, s} = \frac{1}{Γ_{s}} \frac{h v_{s}}{σ_{em}} \frac{(1 - \frac{α_{tot, s}}{Γ_{s} N σ_{em}}) \frac{Γ_{p} I_{3, p}}{h v_{p}} \frac{1 - e^{- α_{p} π D}}{α_{p} π D} σ_{abs, p} - (\frac{σ_{abs}}{σ_{em}} + \frac{σ_{tot, s}}{Γ_{s} N σ_{em}}) \frac{1}{τ}}{(1 + \frac{σ_{abs}}{σ_{em}}) \frac{α_{tot, s}}{Γ_{s} N σ_{em}}} .

(11)

2.3. Lasing conditions and quantum efficiency

The lasing threshold is reached when the signal loss equals the gain, which cannot exceed the value Γ_sσ_emN, corresponding to full inversion. It results in the condition for lasing

α_{tot, s} < Γ_{s} σ_{em} N .

(12)

The quality factor, Q, for a resonant cavity is given by Q=2πn/(λα), where n is the refractive index, λ is the free space wavelength, and α is the loss factor [20

B. E. A. Saleh and M. C. Teich, Fundamental of photonics (John Wiley & Sons, New York, 1991). [CrossRef]

]. According to expression (12) the condition for the quality factor of the signal light, Q_s , then becomes

Q_{s} > \frac{2 π n}{λ Γ_{s} σ_{em} N} .

(13)

Setting I _+,s+I _-,s=0 in Eq. (11), the threshold for the pump power is

P_{th} = \frac{1}{Γ_{p}} \frac{h v_{p}}{σ_{abs, p}} \frac{1}{τ} \frac{\frac{σ_{abs}}{σ_{em}} + \frac{α_{tot, s}}{(Γ_{s} σ_{em} N)}}{\frac{1 - α_{tot, s}}{(Γ_{s} σ_{em} N)}} \frac{1}{E} A

(14)

where A is the cross sectional area of the microfiber.

Using Eqs. (8) and (9) we have

α_{abs, p} = Γ_{p} N_{1} σ_{abs, p} = 1 - \frac{\frac{α_{tot, s}}{(Γ_{s} N σ_{em})}}{1 + \frac{σ_{abs}}{σ_{em}}} Γ_{p} N σ_{abs, p} .

(15)

The quantum efficiency for steady output can be obtained from Eqs. (3), (4) and (11) (assuming I _+,s=I _-,s):

η_{q} = \frac{\frac{(I_{3, s} + I_{4, s})}{(h υ_{s})}}{\frac{I_{p}}{(h υ_{p})}} = \frac{Γ_{p}}{Γ_{s}} \frac{[(1 - γ_{s}) (1 - K_{s})] [1 - \frac{α_{tot, s}}{(Γ_{s} N σ_{em})}]}{\frac{α_{tot, s} (1 + \frac{σ_{abs}}{σ_{em}})}{Γ_{s}}} N σ_{abs, p} E .

(16)

In order to get more physical insight into η_q , Eq. (16

L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982). [CrossRef] [PubMed]

) can be rewritten in the following form

η_{q} = \frac{α_{abs, p}}{\frac{1}{E}} \frac{(1 - K_{s}) (1 - γ_{s})}{α_{tot, s}},

(17)

thus

η_{q} \leq \frac{α_{abs, p} π D}{\frac{1}{E_{max}}} \frac{(1 - K_{s}) (1 - γ_{s})}{α_{tot, s} π D} = \frac{α_{abs, p} π D (1 - γ_{p})}{1 - (1 - γ_{p}) e^{- π D α_{p}}} \frac{(1 - K_{s}) (1 - γ_{s})}{α_{tot, s} π D} .

When πDα_p ≪1, 1-K_s ≪1, γ_s ≪1

η_{q} \leq \frac{α_{abs, p}}{\frac{γ_{p}}{1 - γ_{p}} \frac{1}{π D} + α_{p}} \frac{\frac{1 - K_{s}}{π D} (1 - γ_{s})}{α_{sc, s} + \frac{1 n \frac{1}{K_{s}}}{π D} + \frac{1 n \frac{1}{1 - γ_{s}}}{π D}}

\approx \frac{α_{abs, p}}{\frac{γ_{p}}{π D} + α_{abs, p} + α_{sc, p}} \frac{\frac{1 - K_{s}}{π D} (1 - γ_{s})}{α_{sc, s} + \frac{1 - K_{s}}{π D} + \frac{γ_{s}}{π D}} .

(18)

It can be seen that the first term in expression (18) is related to “useful” absorption, i.e. the fraction of pump attenuation used to excite the rare-earth ions to metastable state; the second term is the fraction of signal light extracted out of the cavity.

3. Discussion-Er³⁺ and Yb³⁺ doped microfiber ring lasers

Erbium and ytterbium ions are chosen as examples of dopants because of their technological importance. They also illustrate the difference between the weak and the high absorption cases. Erbium ions have the property of efficiently storing energy for light emission around 1.55 μm, the wavelength of choice for telecommunications, but present a relatively low absorption cross-section at practical pump wavelengths. Furthermore erbium ions in glasses are readily affected by detrimental concentration effects [21

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, New York, 1994).

]. Ytterbium ions offer high quantum efficiency with low heat dissipation as well as a high peak absorption cross-section and low concentration quenching. This makes them the ideal dopants for high power or compact fiber lasers [22

Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004). [CrossRef] [PubMed]

]. Here we consider the case of erbium in an aluminosilicate host, the most common host for erbium doped fibers [21

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, New York, 1994).

]. For the ytterbium ions the choice of the host glass is not as obvious but here we will consider a phosphate based composition (40P₂O₅-19SiO₂-40B₂O₃-1Yb₂O₃). This choice is motivated by the high rare-earth doping concentration possible in this matrix, as well as the presence of P=O bonds, which are known to be beneficial for energy transfer if erbium is added as a co-dopant [23

G. G. Vienne, W. S. Brocklesby, R. S. Brown, J. E. Caplen, Z. J. Chen, Z. E. Harutjunian, J. D. Minelly, J. E. Roman, and D. N. Payne, “Role of aluminum in Er³⁺: Yb³⁺ codoped aluminiphosphosilicate optical fibres,” Opt. Fiber Technol. 2, 387–393 (1996). [CrossRef]

3.1. Effect of pump resonance on Er³⁺ and Yb³⁺ doped microfiber ring lasers

Figure 4(a) and (b) show the threshold power (P_th ) and the quantum efficiency (η_q ) versus pump coupling factor K_p , for microfiber rings doped with Er³⁺ and Yb³⁺, respectively. The spectroscopic parameters are listed in Table 1, and the ring parameters are listed on the graphs. Various K_s have been tried and the optimum case, where the quantum efficiency is highest (corresponding to critical coupling), is shown here.

Table 1 Spectroscopic parameters for Er/Al/Si glass and Yb/P glass

Glass	N (10²⁰cm^-3)	σ_abs,p (pm²)	σ_abs,σ_em(pm²)	τ (ms)	Ref.
Er/Al/Si glass	0.5	0.19@980 nm	0.48@1535 nm, 0.58@1535 nm	10.2	[24 W. L. Barnes, R. L. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross section of Er³⁺ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004–1010 (1991). [CrossRef] ]
Yb/P glass	1.0	1.5@980 nm	0.1@1022 nm, 0.6@1022 nm	0.69	[25 X. Zou and H. Toratani, “Evaluation of spectroscopic properties of Yb³⁺-doped glasses,” Phys. Rev. B 52, 15889–15897 (1995). [CrossRef] ]

Fig. 4. Threshold power P_th (dotted line) and quantum efficiency η_q (solid line) for glass microfiber rings made of (a) Er³⁺ doped Al₂O₃-SiO₂ and (b) Yb³⁺ doped phosphate glass of spectroscopic parameters listed in Table 1. The diameters of the rings are assumed to be 1 mm for both (a) and (b). The diameters of the fibers are chosen to be 1.0 and 0.63 μm for (a) and (b) respectively, in order to maintain single-mode operation for the signal light and the same Γ_s in both cases [19

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

]. The coupling loss γ_p(γ_s) at the coupling region is assumed to be 0.3% [17

]. The intensity coupling coefficient for the signal light (K_s) is set to the optimum values, 0.979 for (a) and 0.937 for (b). The scattering loss is assumed to be the same for the pump and the signal light, and is set to 0.001 dB/mm (0.002 cm^-1) according to Ref [26

S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fiber waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef] [PubMed]

]. The fractions of pump and signal intensities (Γ_p and Γ_s ) inside the fiber are calculated according Ref. [19

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

], and are listed on the graphs.

3.2. Effect of coupler losses γ_s and γ_p on optimum threshold power and quantum efficiency

When K_p is adjusted, optimal values for the threshold power and the quantum efficiency are reached simultaneously at critical coupling. But the values of these optima are strongly affected by the coupling losses. This is illustrated in Fig. 5 (γ_p = γ_s assumed). In Ref. [17

], Cai et al. have reported that γ_p(γ_s) can be as low as 0.3%, and it is unknown at this stage whether γ_p(γ_s) can be further decreased. The optimum value of K_s depends both on D and γ_p(γ_s).

In Fig. 5, the red curve corresponds to the optimum cases for γ_s =0.003 and D=1 mm. Because there exist uncertainty on the achievable value of γ_p(γ_s), curves for values of K_s below and above this optimum are also shown.

Since γ_s contributes to α_tot,s there is an upper limit of γ_s in order to reach threshold, according to condition (12). This limit is indicated as γ_max in Fig. 5. For a given γ_s , a larger K_s results in a larger α_tot,s , and thus a larger P_th according to Eq. (14). But the situation is more complex for the quantum efficiency: with regard to η_q , there is an optimum of K_s , and the optimum value varies with γ_s . This explains the crossings observed in Fig. 5 between the quantum efficiency curves for different K_s .

In Fig. 5 it is observed that the optimal values for P_th and η_q depend more sharply on γ_p(γ_s) in the weak absorption case (Er³⁺ case) than in the strong absorption case (Yb³⁺ case). This is to be expected considering expressions (14) and (18): the coupling loss γ_p(γ_s) contributes much more to the total loss of both pump and signal light in the weak absorption case than in the strong absorption case. In the latter case, the rare-earth absorption is by definition larger, and larger γ_p(γ_s) can be tolerated. In order to obtain an efficient microfiber ring laser, γ_p(γ_s) should be as low as possible, and its value is particularly critical in the weak absorption case.

Fig. 5. Optimal threshold power P_th (dotted line) and quantum efficiency η_q (solid line) against the coupling loss γ_p(γ_s) for (a) Er³⁺ doped Al₂O₃-SiO₂ glass, and (b) Yb³⁺ doped phosphate glass microfiber rings. Curves of different colors stand for different K _s, as labeled; the maximum values of γ (labeled as γ_max ) are also shown. The parameters used for this simulation are listed on the graphs.

3.3. Effect of ring diameter on threshold power and quantum efficiency

For the purpose of miniaturization, we aim at evaluating the minimum possible ring size before the laser performance is significantly reduced. From expressions (10) and (12) a minimum possible diameter for lasing is deduced:

D_{min} = \frac{1 n \frac{1}{K_{s}} + 1 n \frac{1}{1 - γ_{s}}}{(Γ_{s} N σ_{em} - α_{sc, s}) π} .

(19)

Optimal threshold power P_th and quantum efficiency η_q against ring diameter D for rings made of Er³⁺ doped and Yb³⁺ doped glasses are shown in Fig. 6(a) and (b), respectively. It is clear in Eq. (19) that a high doping concentration N is desirable for miniaturization. Increasing erbium concentration chosen previously may result in concentration quenching. On the other hand, ytterbium ions are much more resilient to this effect and their concentration can be increased to 5.0×10²⁰ cm^-3, the value used to plot Fig. 6(b).

Like in Fig. 5, a set of curves for three different values of K_s has been plotted. It should be noted that the optimum value of K_s for ytterbium has changed compared to Fig. 5 because the ytterbium concentration has been increased. It should also be noted that for the ytterbium doped microfiber ring laser the optimum K_s corresponds to a diameter (D) of 0.2 mm, instead of 1 mm in Fig. 5. The curve for this optimum K_s is plotted in red. Since the diameter of 0.2 mm was chosen somewhat arbitrarily to define this optimum, plots for values of K_s below and above this optimum are also shown.

Figure 6 shows that η_q reaches a peak and then decreases slightly although the first term in expression (18) increases with D. This decrease is due to the fact that the second term in expression (18) slowly decreases as D increases, as more signal light is lost by scattering. This decrease in η_q is moderate, whereas P_th increases more rapidly after reaching its lowest value. Indeed, P_th depends on α_tot,s , and the term regarding to α_tot,s (see Eq. (14)), which changes much more dramatically when D increases.

It can be inferred from Eq. (19) that D_min can be much smaller when K_s approaches 1, and that for a given K_s , D_min is determined by γ_s. Moreover, it is anticipated that even higher ytterbium concentrations are possible before the onset of concentration quenching so that even smaller ring diameters are possible.

It has been estimated that for silica microspheres of diameter exceeding 20 μm, Q_rad , the contribution of purely radiative effects to the quality factor, is larger than 10²¹ and can therefore be neglected [27

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong Coupling,” Phys. Rev. A 67, 033806 (2003). [CrossRef]

]. The refractive indices of the glasses used here are close to the refractive index of pure silica. Both the surface of the microfibers and the microspheres are smoothened by surface tension. There may however be a little discrepancy between the values of Q_rad in the microspheres and in the microfibers. Therefore, we conservatively assume the upper limit of the ring diameter, where radiation losses may come into play, is 50 μm in our case. Below this value (shaded area in Fig. 6) η_q may be significantly reduced by radiation losses.

In Fig. 6(b) the quantum efficiency curve for K_s =0.990 indicates that an ytterbium doped microfiber ring laser with a ring diameter of only 50 μm can achieve a value of η_q as high as 70%. Comparing Fig. 6(a) to Fig. 6(b) it is clear that erbium doped microfiber ring lasers cannot be made as small as their ytterbium doped counterparts. However, we anticipate that it is possible to obtain the compactness of ytterbium doped microfiber ring lasers combined with emission around 1.5 μm by using an erbium: ytterbium co-doped glass [28

G. G. Vienne, J. E. Caplen, L. Dong, J. D. Minelly, J. Nilsson, and D. N. Payne, “Fabrication and Characterization of Yb³⁺: Er³⁺ Phosphosilicate Fibers for Lasers,” J. Lightwave Technol. 16, 1990–2001 (1998). [CrossRef]

]. In that case we also expect that the finite transfer time for the energy transfer will reduce the pump power for the onset of the output saturation effect, which should be the subject of a further study.

Fig. 6. Threshold power P_th (dotted line), and quantum efficiency η_q (solid line) v.s. ring diameter D for three different values of K_s, in (a) Er³⁺ doped Al₂O₃-SiO₂ glass and (b)Yb³⁺ doped phosphate glass microfiber rings. In (b), the concentration of Yb³⁺ ions is 5.0×10^-20 cm^-3 and K_s=0.939 is the optimum for a 0.2 mm diameter ring. Other parameters used for this simulation are listed in the figure and Table 1. The lowest limit of D (D_min) for the three different values of Ks is also indicated. The region below 50 μm is shaded to indicate that below this diameter radiation losses may need to be taken into account.

4. Conclusions

The above analysis provides a detailed description of continuous-wave microfiber ring lasers. Analytical expressions are derived for the lasing condition, threshold pump power, quantum efficiency, and optimum pump light coupling coefficient. Simulations of microfiber rings made of Er³⁺ doped and Yb³⁺ doped glasses are also presented. The main benefit of the microfiber ring laser configuration proposed here is the enhancement of the pump light intensity occurring at resonance. We show that compared to the non-resonant case, the pump intensity can be enhanced by a factor as large as 200 at critical coupling, leading to microwatt-level pump power thresholds and to a large increase in quantum efficiency, especially for compact rings made of materials with weak active ion absorption. We also show that the loss at the coupling region is very critical for small rings, and that strong absorption materials can tolerate larger coupling losses. Finally, we investigate how small microfiber ring lasers can be made while remaining efficient, and we anticipate that an ytterbium-doped microfiber ring laser of diameter as low as 50 μm can reach a quantum efficiency as high as 70%. To the best of our knowledge microfiber ring lasers have not yet been reported but the results presented here should provide useful guidelines for their implementation.

Appendices

Appendix

The four-level-transition (FLT) lasing process is illustrated in Fig. 7.

Fig. 7. Schematic of four-level-transition lasing process. The symbols used here are identical to the ones used for the three-level case, see caption of Fig. 3. In addition τ₄₃ and τ₂₁ stand for the spontaneous radiation lifetimes from |E₄> to |E₃> and from |E₂> to |E₁>, respectively.

Since τ₄₃, τ₂₁ are typically several orders of magnitude lower than τ, they are neglected here. The other assumptions are the same as for the TLT case. The rate equations are [18

Anthony E. Siegman, Laser (Mill Valley, California, 1986).

]

{\begin{matrix} \frac{d N_{3}}{dt} = R N_{1} - W N_{3} - \frac{N_{3}}{τ} \\ N_{1} + N_{3} = N \end{matrix}},

(20)

where

R = \frac{Γ_{p} I_{p}}{h ν_{p}} σ_{abs, p}

W = σ_{em} \frac{Γ_{s} (I_{+, s} + I_{-, s})}{h ν_{s}}

, symbols here have the same physical contents as in the TLT case.

The threshold power is

P_{th} = \frac{1}{Γ_{p}} \frac{h ν_{p}}{σ_{abs, p}} \frac{1}{τ} \frac{\frac{α_{tot, s}}{(Γ_{s} σ_{em} N)}}{\frac{1 - α_{tot, s}}{(Γ_{s} σ_{em} N)}} \frac{1}{E} A .

(21)

The quantum efficiency is

η_{q} = \frac{\frac{(I_{3, s} + I_{4, s})}{(h υ_{s})}}{\frac{I_{p}}{(h υ_{p})}} = \frac{Γ_{p}}{Γ_{s}} (1 - γ_{s}) (1 - K_{s}) (\frac{Γ_{s} N σ_{em}}{α_{tot, s}} - 1) \frac{α_{abs, p}}{σ_{em}} E .

(22)

The expression for quantum efficiency can be expressed in the same way as for the TLT case:

η_{q} = \frac{α_{abs, s}}{\frac{1}{E}} \frac{(1 - K_{s}) (1 - γ_{s})}{α_{tot, s}},

(23)

but here the steady state absorption coefficient for the pump light is

α_{abs, p} = Γ_{p} N_{1} σ_{abs, p} = (1 - \frac{α_{tot, s}}{Γ_{s} N σ_{em}}) Γ_{p} N σ_{abs, p .}

(24)

Acknowledgments

The work is supported by the National Natural Science Foundation of China (No. 60425517 and 60378036). We thank Z. Ma for helpful discussions.

References and links

1.	E. Udd, ed., Fiber Optic Sensors (Wiley, New York, 1991).
2.	C. K. Madsen and J. H. Zhao, Optical filter design and analysis: A signal processing approach (Wiley, New York, 1999).
3.	P. Hariharan, Opital Interferometry , 2nd ed. (Academic, New York, 2003).
4.	F. Vollmer, D. Braun, A. Libchaber, S. M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002). [CrossRef]
5.	B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, and E. P. Ippen, “Ultra-compact Si-SiO₂ microring resonator optical channel dropping filters,” IEEE Photonics Tech. Lett. 10, 549–551 (1998). [CrossRef]
6.	K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]
7.	M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000). [CrossRef]
8.	S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992). [CrossRef]
9.	A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett. 84, 1037–1039 (2004). [CrossRef]
10.	L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef] [PubMed]
11.	L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express 14, 82–87 (2006). [CrossRef] [PubMed]
12.	M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. 86, 161108 (2005). [CrossRef]
13.	X. Jiang, L. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, and D. Yang, “Demonstration of optical microfiber knot resonators,” Appl. Phys. Lett. 88, 223501 (2006). [CrossRef]
14.	M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express 12, 2303–2316 (2004). [CrossRef] [PubMed]
15.	K. Hsu, C. M. Miller, J. T. Kringlebotn, and D. N. Payne, “Continuous and discrete wavelength tuning in Er:Yb fiber Fabry-Perot lasers,” Opt. Lett. 20, 377–379 (1995). [CrossRef] [PubMed]
16.	L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982). [CrossRef] [PubMed]
17.	M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whisper-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000). [CrossRef] [PubMed]
18.	Anthony E. Siegman, Laser (Mill Valley, California, 1986).
19.	L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]
20.	B. E. A. Saleh and M. C. Teich, Fundamental of photonics (John Wiley & Sons, New York, 1991). [CrossRef]
21.	E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, New York, 1994).
22.	Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004). [CrossRef] [PubMed]
23.	G. G. Vienne, W. S. Brocklesby, R. S. Brown, J. E. Caplen, Z. J. Chen, Z. E. Harutjunian, J. D. Minelly, J. E. Roman, and D. N. Payne, “Role of aluminum in Er³⁺: Yb³⁺ codoped aluminiphosphosilicate optical fibres,” Opt. Fiber Technol. 2, 387–393 (1996). [CrossRef]
24.	W. L. Barnes, R. L. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross section of Er³⁺ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004–1010 (1991). [CrossRef]
25.	X. Zou and H. Toratani, “Evaluation of spectroscopic properties of Yb³⁺-doped glasses,” Phys. Rev. B 52, 15889–15897 (1995). [CrossRef]
26.	S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fiber waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef] [PubMed]
27.	J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong Coupling,” Phys. Rev. A 67, 033806 (2003). [CrossRef]
28.	G. G. Vienne, J. E. Caplen, L. Dong, J. D. Minelly, J. Nilsson, and D. N. Payne, “Fabrication and Characterization of Yb³⁺: Er³⁺ Phosphosilicate Fibers for Lasers,” J. Lightwave Technol. 16, 1990–2001 (1998). [CrossRef]

OCIS Codes
(140.3560) Lasers and laser optics : Lasers, ring
(140.4780) Lasers and laser optics : Optical resonators
(140.5680) Lasers and laser optics : Rare earth and transition metal solid-state lasers
(160.5690) Materials : Rare-earth-doped materials

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 5, 2006
Revised Manuscript: June 25, 2006
Manuscript Accepted: July 7, 2006
Published: August 7, 2006

Citation
Yuhang Li, Guillaume Vienne, Xiaoshun Jiang, Xinyun Pan, Xu Liu, Peifu Gu, and Limin Tong, "Modeling rare-earth doped microfiber ring lasers," Opt. Express 14, 7073-7086 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7073

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References

E. Udd, ed., Fiber Optic Sensors (Wiley, New York, 1991).
C. K. Madsen, and J. H. Zhao, Optical filter design and analysis: A signal processing approach (Wiley, New York, 1999).
P. Hariharan, Opital Interferometry, 2nd ed. (Academic, New York, 2003).
F. Vollmer, D. Braun, A. Libchaber, S. M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002). [CrossRef]
B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, and E. P. Ippen, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photonics Tech. Lett. 10, 549-551 (1998). [CrossRef]
K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915-1917 (2003). [CrossRef]
M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000). [CrossRef]
S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992). [CrossRef]
A. Polman, B. Min, J. Kalkman, T. J. Kippenberg and K. J. Vahala, "Ultralow-threshold erbium-implanted toroidal microlaser on silicon," Appl. Phys. Lett. 84, 1037-1039 (2004). [CrossRef]
L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003). [CrossRef] [PubMed]
L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, "Photonic nanowires directly drawn from bulk glasses," Opt. Express 14, 82-87 (2006). [CrossRef] [PubMed]
M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, "Optical microfiber loop resonator," Appl. Phys. Lett. 86, 161108 (2005). [CrossRef]
X. Jiang, L. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang and D. Yang, "Demonstration of optical microfiber knot resonators," Appl. Phys. Lett. 88, 223501 (2006). [CrossRef]
M. Sumetsky, "Optical fiber microcoil resonator," Opt. Express 12, 2303-2316 (2004). [CrossRef] [PubMed]
K. Hsu, C. M. Miller, J. T. Kringlebotn, and D. N. Payne, "Continuous and discrete wavelength tuning in Er:Yb fiber Fabry-Perot lasers," Opt. Lett. 20, 377-379 (1995). [CrossRef] [PubMed]
L. F. Stokes, M. Chodorow, and H. J. Shaw, "All-single-mode fiber resonator," Opt. Lett. 7, 288-290 (1982). [CrossRef] [PubMed]
M. Cai, O. Painter, and K. J. Vahala, "Observation of critical coupling in a fiber taper to a silica-microsphere whisper-gallery mode system," Phys. Rev. Lett. 85,74-77 (2000). [CrossRef] [PubMed]
AnthonyE. Siegman, Laser (Mill Valley, California, 1986).
L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2004). [CrossRef] [PubMed]
B. E. A. Saleh, and M. C. Teich, Fundamental of photonics (John Wiley & Sons, New York, 1991). [CrossRef]
E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, New York, 1994).
Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, "Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power," Opt. Express 12, 6088-6092 (2004). [CrossRef] [PubMed]
G. G. Vienne, W. S. Brocklesby, R. S. Brown, J. E. Caplen, Z. J. Chen, Z. E. Harutjunian, J. D. Minelly, J. E. Roman, D. N. Payne, "Role of aluminum in Er3+: Yb3+ codoped aluminiphosphosilicate optical fibres," Opt. Fiber Technol. 2, 387-393 (1996). [CrossRef]
W. L. Barnes, R. L. Laming, E. J. Tarbox, and P. R. Morkel, "Absorption and emission cross section of Er3+ doped silica fibers," IEEE J. Quantum Electron. 27, 1004-1010 (1991). [CrossRef]
X. Zou and H. Toratani, "Evaluation of spectroscopic properties of Yb3+-doped glasses," Phys. Rev. B 52,15889-15897 (1995). [CrossRef]
S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, "Supercontinuum generation in submicron fiber waveguides," Opt. Express 12, 2864-2869 (2004). [CrossRef] [PubMed]
J. R. Buck, and H. J. Kimble, "Optimal sizes of dielectric microspheres for cavity QED with strong Coupling," Phys. Rev. A 67, 033806 (2003). [CrossRef]
G. G. Vienne, J. E. Caplen, L. Dong, J. D. Minelly, J. Nilsson, and D. N. Payne, "Fabrication and Characterization of Yb3+: Er3+ Phosphosilicate Fibers for Lasers," J. Lightwave Technol. 16, 1990-2001 (1998). [CrossRef]

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