## Design of mid-infrared amplifiers based on fiber taper coupling to erbium-doped microspherical resonator |

Optics Express, Vol. 20, Issue 7, pp. 7616-7629 (2012)

http://dx.doi.org/10.1364/OE.20.007616

Acrobat PDF (1159 KB)

### Abstract

A dedicated 3D numerical model based on coupled mode theory and solving the rate equations has been developed to analyse, design and optimize an optical amplifier obtained by using a tapered fiber and a Er^{3+}-doped chalcogenide microsphere. The simulation model takes into account the main transitions among the erbium energy levels, the amplified spontaneous emission and the most important secondary transitions pertaining to the ion–ion interactions. The taper angle of the optical fiber and the fiber-microsphere gap have been designed to efficiently inject into the microsphere both the pump and the signal beams and to improve their spatial overlapping with the rare earth doped region. In order to reduce the computational time, a detailed investigation of the amplifier performance has been carried out by changing the number of sectors in which the doped area is partitioned. The simulation results highlight that this scheme could be useful to develop high efficiency and compact mid-infrared amplifiers.

© 2012 OSA

## 1. Introduction

^{9}) and small mode volume. These properties make WGMs interesting for many applications including the polarization transmission, coupled-resonator-induced transparency, biosensor analysis [1

1. H. C. Ren, F. Vollmer, S. Arnold, and A. Libchaber, “High-Q microsphere biosensor-analysis for adsorption of rodlike bacteria,” Opt. Express **15**, 17410–17423 (2007). [CrossRef] [PubMed]

2. F. Vollomer and A. Stephen, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods **5**, 591–596 (2008). [CrossRef]

3. L. Yang and K. J. Vahala, “Gain functionalization of silica microresonators,” Opt. Lett. **28**, 592–594 (2008). [CrossRef]

3. L. Yang and K. J. Vahala, “Gain functionalization of silica microresonators,” Opt. Lett. **28**, 592–594 (2008). [CrossRef]

7. G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, “Spectroscopic and lasing properties of Er^{3+} doped glass microspheres,” J. Non-Crystalline Sol. **352**, 2360–2363 (2006). [CrossRef]

9. L. Mescia, F. Prudenzano, M. De Sario, T. Palmisano, M. Ferrari, and G. C. Righini, “Design of Rare-Earth-Doped Microspheres,” IEEE Photon. Techonol. Lett. **22**, 422–424 (2010). [CrossRef]

11. Y. G. Boucher and P. Feron, “Generalized transfer function: A simple model applied to active single-mode microring resonators,” Opt. Commun. **282**, 3940–3947 (2009). [CrossRef]

^{3+}-doped chalcogenide microspheres. The design is realistically carried out, on the basis of the optical and spectroscopic parameters measured on chalcogenide glass [12

12. F. Prudenzano, L. Mescia, L. Allegretti, V. Moizan, V. Nazabal, and F. Smektala, “Theoretical study of cascade laser in erbium-doped chalcogenide glass fibers,” Opt. Mat. **33**, 241–245 (2010). [CrossRef]

^{3+}-doped chalcogenide glasses such as the radiative and nonradiative transition rates, at both pump and signal wavelengths, the stimulated emission of the signal, the amplified spontaneous emission noise (ASE), the lifetime and branching ratios of the considered energy levels, the ion–ion energy transfers and the excited state absorption (ESA). For the first time, WGM amplification in microspherical resonator at 2.7

*μ*m wavelength is accurately modeled by considering important design parameters such as the Q-factor, the spectral range, the loss effects, the mode volume, the coupling characteristics, the buildup factor and the spectroscopic properties of the Er

^{3+}ions. In particular, a number of simulations has been performed with the aim to identify the optimal geometric parameters leading to the best device performance. The paper is organized as follows: Section 2 gives the theoretical recalls of: i) WGMs propagation, ii) erbium-doped microspheres, iii) coupled mode theory describing the tapered fiber-microsphere system. The results and discussion are provided in Section 3 while conclusions are given in Section 4.

## 2. Theory

^{3+}-doped chalcogenide microsphere. The evanescent field of the optical fiber tunnels in the microsphere and the excitation of WGMs is achieved. Since the fundamental WGMs are localized near the microsphere surface, this coupling technique enables a very efficient excitation without affecting the high Q-factor characteristics. It allows the extraction of the cavity power through the same fiber and the control of the coupling characteristics. Moreover, the tapered fiber allows the focusing and the alignment of the input beam and it efficiently filters the high-order fiber modes. The design of this device requires the use of the electromagnetic mode analysis of both the fiber and the microsphere, the coupled mode theory and the rate equations formalism. In fact, the accurate calculation of the modal electromagnetic field distribution and of the propagation constants of the guided propagation modes in both the fiber and the microsphere is essential to evaluate the coupling coefficients, the quality factor, the mode volume, the transition rates for modeling the active behavior [13

13. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic Theory of Coupling from Tapered Fibers and Half-Blocks into Microsphere Resonators,” J. Lightwave Technol. **17**, 704–715 (1999). [CrossRef]

*k*= 2

*π*

*/*

*λ*is the wave vector in vacuum,

*λ*is the wavelength in vacuum and

*n*

*is the refractive index of the microsphere. In particular, the polarization direction of the electromagnetic field is considered constant along a fixed spherical coordinate. The solution of Eq. (1) is separable, i.e.*

_{s}*ψ*

*(*

_{l,m,n}*r,*

*θ*

*,φ*) =

*N*

_{s}*R*(

*r*) Θ (

*θ*) Φ (

*φ*); being

*ψ*

*=*

_{l,m,n}*E*

*for the transverse electric (TE) modes, and*

_{θ}*ψ*

*=*

_{l,m,n}*H*

*for the transverse magnetic (TM) modes.*

_{θ}*N*

*is the normalization constant calculated so that the volume integral of*

_{s}13. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic Theory of Coupling from Tapered Fibers and Half-Blocks into Microsphere Resonators,” J. Lightwave Technol. **17**, 704–715 (1999). [CrossRef]

*l,m,n*where

*l − m*+ 1 is the number of field maxima in the polar,

*θ*̂, direction and

*n*is the number of field maxima along the radial direction.

*φ*) can be expressed in terms of complex exponential functions depending on the mode number

*m*≤

*l*. If the sphere radius is larger than the effective wavelength, the WGMs are characterized by large values of

*m*and

*l*. Moreover, the approximation

*θ*≪ 1 for the polar angle has been considered because the corresponding modes are the most strongly coupled with the tapered fiber. In this case, the polar dependence Θ(

*θ*) can be represented by the Hermite polynomials of order

*N*=

*l − m*. Therefore, the radial solution

*R*(

*r*) can be expressed in terms of spherical Bessel functions depending on the mode orders

*l*and

*n*[13

13. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic Theory of Coupling from Tapered Fibers and Half-Blocks into Microsphere Resonators,” J. Lightwave Technol. **17**, 704–715 (1999). [CrossRef]

**17**, 704–715 (1999). [CrossRef]

*R*

_{0}is the sphere radius,

*β*

*is the propagation constant parallel to the surface of the microsphere,*

_{l}*α*

*is the constant describing the evanescent field decay from the microsphere along the radial direction,*

_{s}*n*

_{0}is the background refractive index,

*j*

*is the spherical Bessel function of the*

_{l}*l*-th order. Equation (2) enables the calculation of the resonance wavelengths and their corresponding modal field distributions.

14. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes” J. Opt. Soc. Am. B **16**, 147–154 (1999). [CrossRef]

15. C. L. Zou, Y. Yang, C. H. Dong, Y. F. Xiao, X. W. Wu, Z. F. Han, and G. C. Guo, “Taper-microsphere coupling with numerical calculation of coupled-mode theory,” J. Opt. Soc. Am. B **25**, 1895–1898 (2008). [CrossRef]

*A*of the internal cavity electromagnetic field can be obtained by solving the differential equation [14

_{l,m,n}14. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes” J. Opt. Soc. Am. B **16**, 147–154 (1999). [CrossRef]

*A*is the amplitude of the input electric field into the coupling region,

_{in}*τ*= 2

*πR*

_{0}

*n*is the circulation time inside the microsphere (round trip time),

_{eff}/c*n*is the WGM effective refractive index and

_{eff}*c*the speed of light in vacuum, Δ

*ω*=

*ω*

_{in}*−*

*ω*is the frequency detuning of the fiber input signal from the WGM frequency. The intrinsic lifetime

_{WGM}*Q*

_{0}is the intrinsic quality factor and

*κ*

_{0}is the intrinsic cavity decay rate which depends on the total losses due to the material absorption, surface scattering losses, radiative losses and whispering gallery losses [13

**17**, 704–715 (1999). [CrossRef]

*κ*is the cavity decay rate, denotes the coupling phenomenon between microsphere and optical fiber. In particular, the coupling coefficient K

*was calculated according to the overlap integral [13*

_{fs}**17**, 704–715 (1999). [CrossRef]

**E**

*is the field of the fundamental mode of the optical fiber and*

_{f}**E**

*is the WGM field of the microsphere. Both the electromagnetic fields are calculated by using the Eqs. (1)–(3) and normalizing on the*

_{s}*r̂*·

*θ*̂ plane such that In particular, for an adiabatic taper, the fiber radius,

*a*, can be approximated as where

*δ*is the tapered angle and

*a*

_{0}is the fiber radius at the waist of the tapered fiber, as shown in Fig. 1. As result, the z-depending propagation constant,

*β*, of the optical fiber mode can be calculated by solving the characteristic equation [13

_{f}**17**, 704–715 (1999). [CrossRef]

*J*and

_{υ}*K*,

_{υ}*υ*= 0,1, are the Bessel functions of the first kind and the modified Bessel functions of the second kind, respectively.

*i*〉

*i*= 1,2,...6), the electronic transitions and the energy transfers of the trivalent erbium ions in the chalcogenide glass. The state |6〉 is constituted by the

^{4}

*S*

_{3/2},

^{2}

*H*

_{11/2},

^{4}

*F*

_{7/2}manifolds because the nonradiative transition

^{4}

*F*

_{7/2}→

^{2}

*H*

_{11/2}is very fast, and the

^{4}

*S*

_{3/2}and

^{2}

*H*

_{11/2}are thermally distributed. However, the complex Er

^{3+}-ions system was justified by the fact that in the considered chalcogenide glass the lifetime of the

^{4}

*I*

_{13/2},

^{4}

*I*

_{11/2},

^{4}

*I*

_{9/2}energy manifolds are comparable and the lifetime of the

^{4}

*F*

_{9/2}energy manifolds is small but not negligible. In particular, the important energy exchange processes considered in the model are: 1) both 806 nm (

^{4}

*I*

_{15/2}→

^{4}

*I*

_{9/2}) and 980 nm (

^{4}

*I*

_{15/2}→

^{4}

*I*

_{11/2}) pump transitions, 2) 4.5

*μ*m (

^{4}

*I*

_{9/2}→

^{4}

*I*

_{11/2}), 2.7

*μ*m (

^{4}

*I*

_{11/2}→

^{4}

*I*

_{13/2}) and 1.55

*μ*m (

^{4}

*I*

_{13/2}→

^{4}

*I*

_{15/2}) stimulated emission transitions, 3) different spontaneous decays (point arrows), 4) various types of energy transfer mechanisms occurring in a pair of Er

^{3+}ions: (

^{4}

*I*

_{13/2},

^{4}

*I*

_{13/2}→

^{4}

*I*

_{15/2},

^{4}

*I*

_{9/2}), (

^{4}

*I*

_{11/2},

^{4}

*I*

_{11/2}→

^{4}

*I*

_{15/2},

^{4}

*S*

_{3/2}), (

^{4}

*I*

_{9/2},

^{4}

*I*

_{9/2}→

^{4}

*S*

_{3/2},

^{4}

*I*

_{13/2}), (

^{4}

*I*

_{15/2},

^{4}

*I*

_{9/2}→

^{4}

*I*

_{13/2}), (

^{4}

*I*

_{15/2},

^{4}

*S*

_{3/2}→

^{4}

*I*

_{13/2},

^{4}

*I*

_{9/2}), (

^{4}

*I*

_{13/2},

^{4}

*I*

_{9/2}→

^{4}

*I*

_{15/2},

^{4}

*S*

_{3/2}).

*r̂*·

*θ̂*is divided in

*q*sector, as showed in Fig. 3. In particular, Δ

*r*and Δ

*θ*are the radial and polar dimension of each sector,

*S*is the area of the

_{q}*q*-th sector,

*θ*is the maximum polar angle defining the doped area considered in the simulations.

_{max}*i*= 1,2,...6 is the concentration of

*i*-th energy manifold in the

*q*-th sector,

*q*-th sector defined as:

*ν*̃ =

*ν*is the WGM resonant frequency,

_{l,m,n}*σ*(

_{i,j}*ν*̃) is the emission cross section at the frequency

*ν*̃, when

*i*>

*j*, and the absorption cross section at the frequency

*ν*̃ when

*i*<

*j*, and (1/

*S*)

_{q}*S*, of the intensity profile. In particular, the intensity profile at pump,

_{q}15. C. L. Zou, Y. Yang, C. H. Dong, Y. F. Xiao, X. W. Wu, Z. F. Han, and G. C. Guo, “Taper-microsphere coupling with numerical calculation of coupled-mode theory,” J. Opt. Soc. Am. B **25**, 1895–1898 (2008). [CrossRef]

*r*̂ ·

*θ*̂. Thus, the Eqs. (12)–(13) can be written as where is the overlap factor of each WGM with the rare-earth profile in the

*q*-th sector. The condition

*φ*= 0 in Eqs. (15)–(16) is used by considering that the azimuthal dependence of the electromagnetic field takes the form exp(−

*jmφ*).

17. M. J. F. Digonnet, *Rare-Earth-Doper Fider Lasers and Amplifiers* (Marcel Dekker Inc., 2001). [CrossRef]

*dφ*= (

*c/n*)

_{eff}*dt*the differential equation governing the evolution of the amplitude of both the pump and the ASE within the doped microsphere can be expressed as:

*ν*[17

17. M. J. F. Digonnet, *Rare-Earth-Doper Fider Lasers and Amplifiers* (Marcel Dekker Inc., 2001). [CrossRef]

^{3+}ions. In particular, if the laser action occurs

18. K. Vahala, *Optical Microcavities* (World Scientific Publishing, 2004). [CrossRef]

*τ*,

*τ*,

_{ext}*ν*̃ is considered.

## 3. Numerical results

_{5}Ge

_{20}Sb

_{10}S

_{65}chalcogenide glass feasibility for the construction of microsphere amplifiers operating in the mid-infrared wavelength range. In order to carry out a realistic design, the simulations have been performed by taking into account the refractive index wavelength dispersion [12

12. F. Prudenzano, L. Mescia, L. Allegretti, V. Moizan, V. Nazabal, and F. Smektala, “Theoretical study of cascade laser in erbium-doped chalcogenide glass fibers,” Opt. Mat. **33**, 241–245 (2010). [CrossRef]

^{−6}s) of the state |6〉 and transition rates

*A*

_{61}=

*A*

_{62}=

*A*

_{63}=

*A*

_{64}= 0 have been considered in the calculations.

*λ*= 0.98

_{p}*μ*m and (b) signal wavelength

*λ*= 2.76

_{s}*μ*m, of the undoped microsphere coupled with the tapered fiber by considering sphere radius

*R*

_{0}= 25

*μ*m and taper-microsphere gap

*d*= 460 nm, at the pump wavelength, and

*d*= 1.8

*μ*m, at the signal wavelength. The different values of

*d*have been chosen to obtain the critical coupling condition at both wavelengths. The size of the fiber taper has been designed to ensure the fundamental mode propagation along the waveguide formed by the fiber waist surrounded by air, and to have a significant evanescent field into the space surrounding the taper. In particular, the considered waist radius and taper angle numerical values are

*a*

_{0}= 700 nm and

*δ*= 0.03 rad, respectively.

*V*includes only the sphere. In the calculation, the WGM electromagnetic field has been evaluated by solving Eq. (1) and considering the equations of radial, polar and azimuthal dependence reported in [13

**17**, 704–715 (1999). [CrossRef]

*λ*= 0.98

_{p}*μ*m and signal wavelength

*λ*= 2.76

_{s}*μ*m exciting the fundamental WGMs mode

*n*= 1,

*m*=

*l*= 352 and

*n*= 1,

*m*=

*l*= 118, respectively.

*q*ensuring a good compromise between the result accuracy and the computational time. Figure 5 summarizes the obtained results regarding the variation of the signal transmittance versus the sector number for different maximum polar angles,

*θ*. In the simulations, the thickness of the erbium-doped region and erbium concentration are

_{max}*s*= 3

*μ*m and

*N*= 0.5 w%, respectively. Other parameters are: input pump power

_{Er}*P*= 100 mW, input signal power

_{p}*P*= −50 dBm, taper angle

_{s}*δ*= 0.03 rad, taper-microsphere gap

*d*= 560 nm, waist radius

*a*

_{0}= 700 nm. It can be observed that for sectors number higher than 80, the transmittance is almost constant even by increasing

*θ*. This occurrence can be explained by considering that, for small

_{max}*θ*, only a part of the power distribution is taken into account in the calculation of the gain. By increasing

_{max}*θ*, the tails of the electromagnetic field along the polar direction are also considered for the signal amplification. For

_{max}*θ*>

_{max}*π*/10 and

*q*> 80, the transmittance slightly changes because the electromagnetic field distribution of the fundamental WGM on the

*r̂*·

*θ*̂ plane is almost completely considered in the calculations. Significant transmittance variation can be observed for

*θ*>

_{max}*π*/5 and

*q*< 80 because a poor sampling of the electromagnetic field occurs. However, we have selected

*θ*=

_{max}*π*/10 and

*q*= 30 since the obtained signal transmittance is practically equal to that calculated for

*θ*=

_{max}*π*/5 and

*q*> 80.

*P*= 100 mW, input signal power

_{p}*P*= −50 dBm, taper angle

_{s}*δ*= 0.03 rad, taper-microsphere gap

*d*= 560 nm, erbium concentration

*N*= 0.5 w%, waist radius

_{Er}*a*

_{0}= 700 nm. In all the investigated cases, the transmittance increases by increasing the thickness

*s*of the doped region because a better overlap between WGM electromagnetic field profile and the region containing the erbium ions occurs. For thicknesses of the doped region higher than 2

*μ*m the transmittance is quite constant because the WGM field intensity at both pump and signal wavelengths are bounded in a spatial region enclosed in

*s*< 2

*μ*m. Nevertheless, the transmittance increases by increasing the maximum polar angles

*θ*till

_{max}*θ*>

_{max}*π*/10. The curves

*θ*=

_{max}*π*/5 and

*θ*=

_{max}*π*/10 are practically unnoticeable.

*θ*for three different doped region thicknesses,

_{max}*s*=1,2,3

*μ*m. As expected, for

*θ*> 0.2 rad the signal transmittance slightly changes by changing the thickness of the doped region. Thus, according with the aforesaid remarks, the sector number

_{max}*q*= 30, maximum polar angle

*θ*=

_{max}*π*/10 rad, thickness of the doped region

*s*=3

*μ*m, sphere radius

*R*

_{0}= 25

*μ*m are fixed in the following simulations, allowing to obtain a good result precision without too much computational time consumption.

*δ*= 0.03 rad and input pump power

*P*= 100 mW. The transmittance (marked curves) at both pump and signal wavelengths for the undoped sphere are reported for a comparison. Good signal amplification is obtained for a fibre-microsphere gap close to 500 nm where the condition of critical coupling at the pump wavelength occurs. In fact, in this operational regime the most part of the pump signal is coupled into the microsphere. As result, a strong pump absorption rate occurs, the erbium ions are inverted and the signal stimulated emission rate takes place. It can be also observed that the amplifier performance depends on the erbium concentration. When the fiber is very close to the microsphere (d≅200 nm) both the pump and signal to be amplified are in a regime where the coupling is stronger than the loss (over-coupling). In this condition, the energy circulating in the microsphere poorly interacts with the active ions because a large portion of its is coupled out to the fiber. By increasing the gap up to about 550 nm the pump signal circulating in the microsphere is high enough to ensure a strong population inversion whose value mainly depends on erbium concentration. In under-coupling regime (d>550 nm) only a small part of input pump power is coupled in the microsphere. As result, the reduction of the population inversion due to the ion-ion interaction becomes stronger at higher erbium concentration causing a decreasing of the amplifier performance. Finally, for large gap values the signal amplifica-tion cannot be fulfilled because the pump signal in the microsphere is too much low. Figure 9 shows the signal transmittance versus the fibre-microsphere gap for three different taper angles, erbium concentration

_{p}*N*= 0.5 w%, input pump and signal power

_{Er}*P*(0) = 100 mW and

_{p}*P*(0) = −50 dBm, respectively. It can be observed that quite similar performance in terms of maximum signal amplification are obtained.

_{s}*P*= −30 dBm, ≈ 78 mW for

_{s}*P*= −40 dBm and

_{s}*P*= −50 dBm. In fact, for input pump power lower than the threshold one the absorption and the stimulated emission rates due to the internal cavity field at the pump and signal wavelength respectively, induce a weak population improvement of the

_{s}^{4}

*I*

_{11/2}energy manifold. The different values of the threshold pump power can be explained by taking into account the upconversion phenomena from

^{4}

*I*

_{13/2}and

^{4}

*I*

_{11/2}energy manifolds. In fact, with lower signal power the ion population of the

^{4}

*I*

_{11/2}energy manifold is quite high and the upconversion is more efficient. At the same time, the ion population of the

^{4}

*I*

_{13/2}energy manifold is not high enough to compensate the deleterious effect of the upconversion from the

^{4}

*I*

_{11/2}energy manifold. As result, in order to fully invert the active medium higher pump power is required. On the contrary, for input pump powers higher than the threshold one the population of the

^{4}

*I*

_{11/2}energy manifold steeply increases and a strong population inversion occurs. Finally, for input pump power higher than 85 mW the transmittance is, in all the three cases, quite constant because this regime corresponds to a fully inverted medium. Moreover, as expected, the transmittance value decreases by increasing the input signal power, because the gain saturation due to the amplified signal occurs.

*P*= −50 dBm, and for three different input pump powers. With respect to the steady operation an overshoot is observed which is narrowed by increasing the input pump power. The delay time of the peak corresponding to the lower input pump power is due to a slower dynamic of the absorption pump rate. Moreover, the absence of the spike for

_{s}*P*= 50 mW can be explained by considering that the pump absorption and stimulated emission rates grow in the same way during the time. The obtained numerical results show that the time constants of the transient gain dynamic depend on the input pump power and, the saturation and recovery times are typically less than few ms.

_{p}^{−6}°C

^{−1}) and thermal refraction coefficients (dn

*/dT=10*

_{s}^{−5}°C

^{−1}) of chalcogenide glasses [20

20. J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Crystalline Sol. **256–257**,6–16 (1999). [CrossRef]

## 4. Conclusion

^{3+}-doped chalcogenide microsphere amplifier evanescently coupled with a tapered optical fiber. The device performance has been investigated by changing the fiber-microsphere gap, the thickness of erbium doped region, the fiber taper angle, the erbium concentration and the operative parameters such as pump and signal power. A dedicated 3D model based on coupled mode theory and solving the rate equations has been developed and implemented in a home-made computer code. The electromagnetic analysis of the microsphere has been performed by finding the solution of the scalar Helmholtz equation in spherical coordinates. Moreover, peculiar parameters as quality factor and coupling coefficients have been evaluated by considering the calculated electromagnetic field profile of both microsphere and tapered optical fiber. A detailed analysis pertaining to the simulation accuracy has been performed by changing the amplitude of the computational domain and the number of the discretization points. In particular, sector number

*q*=30 and maximum polar angle

*θ*=

_{max}*π*/10 rad ensure good result precision and not too much computational time consuming. A significant signal amplification can be obtained for input pump powers higher than 80 mW and, in the small signal operation, maximum optical gain of about 7 dB has been calculated. The performed simulations indicate that the proposed Er

^{3+}-doped microspheres are good candidates for an efficient frequency-selective amplification, e.g. to restore signal attenuation and to be employed for compact integration of the active optical functionalities.

## References and links

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13. | B. E. Little, J. P. Laine, and H. A. Haus, “Analytic Theory of Coupling from Tapered Fibers and Half-Blocks into Microsphere Resonators,” J. Lightwave Technol. |

14. | M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes” J. Opt. Soc. Am. B |

15. | C. L. Zou, Y. Yang, C. H. Dong, Y. F. Xiao, X. W. Wu, Z. F. Han, and G. C. Guo, “Taper-microsphere coupling with numerical calculation of coupled-mode theory,” J. Opt. Soc. Am. B |

16. | H. A. Haus, |

17. | M. J. F. Digonnet, |

18. | K. Vahala, |

19. | A. W. Snyder and J. D. Love, |

20. | J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Crystalline Sol. |

**OCIS Codes**

(140.4480) Lasers and laser optics : Optical amplifiers

(230.5750) Optical devices : Resonators

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 22, 2011

Revised Manuscript: February 16, 2012

Manuscript Accepted: February 21, 2012

Published: March 20, 2012

**Citation**

Luciano Mescia, Pietro Bia, Marco De Sario, Annalisa Di Tommaso, and Francesco Prudenzano, "Design of mid-infrared amplifiers based on fiber taper coupling to erbium-doped microspherical resonator," Opt. Express **20**, 7616-7629 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7616

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### References

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- C. L. Zou, Y. Yang, C. H. Dong, Y. F. Xiao, X. W. Wu, Z. F. Han, G. C. Guo, “Taper-microsphere coupling with numerical calculation of coupled-mode theory,” J. Opt. Soc. Am. B 25, 1895–1898 (2008). [CrossRef]
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