Theoretical and experimental research on Er-doped and Yb-Er co-doped Al₂O₃ waveguide amplifiers

Abstract

The rate equations of Er-doped and Yb-Er co-doped systems pumped at 0.98 μm are presented, with consideration for the upconversion mechanisms such as cooperative upconversion, cross relaxation, and excited state absorption. A multi-theoretical model is founded to analyze the gain characteristics of Er-doped and Yb-Er co-doped Al₂O₃ waveguide amplifiers by using rate equations, a two-dimension waveguide finite element model and propagation equations with forward and backward amplified spontaneous emission. The dependence of the gain on amplifier length, pump power and doping concentration is obtained. The optimum design curve is given for designing waveguide amplifiers. The new theory is used to analyze the gain performance of a practical Yb-Er co-doped Al₂O₃ waveguide amplifier, and the analyzed results are in accordance with the experimental data.

Key words： integrated optics; Er-doped Al₂O₃ waveguide amplifier (EDAWA); Yb-Er co-doped Al₂O₃ waveguide amplifier (YEDAWA); multi-theoretical model, net gain

1 Introduction

The 1.53 μm Er-doped and Yb-Er co-doped waveguide amplifiers (EDWA, YEDWA) have attracted great attention in the research on active devices in all-optical communication because of their potential for having small sizes 1–323, low cost, stable and reliable performance and the possibility of integration with other components such as pump lasers, switches, filters, couplers and wavelength division multiplexers.

Currently, EDWAs with a variety of host material for Er have been studied, such as Si, SiO₂, phosphate glass, sodium calcium silicate glass, LiNbO₃ and Al₂O₃4^,5. Especially, Er-doped Al₂O₃ waveguide amplifier (EDAWA) and Yb-Er co-doped Al₂O₃ waveguide amplifier (YEDAWA) have attracted more attention 6. First, the high confinement, low-loss transmission and small bend radius can be realized because of the large refractive index changes between Al₂O₃ and the buffer layer (SiO₂), and the cover layer 4. Second, due to the similar crystal structures between Al₂O₃, Er₂O₃ and Yb₂O₃, high gain can be achieved in a small size with high concentration incorporation of Er and Yb. The absorption spectra of Er³⁺ and emission spectra of Yb³⁺ have greater overlap 7, the absorption cross-section Yb³⁺ at 0.98 μm is about an order of magnitude larger than that of Er³⁺, and the pumping efficiency by co-doping Yb can be greatly improved through energy transmission between Yb³⁺ and Er³⁺.

In 1996, Hoven et al. studied a 4-cm Er-doped Al₂O₃ waveguide amplifier integrated onto a silicon chip with an area of 15 mm²8. Recently, the research group developed an Er-Yb co-doped Al₂O₃ waveguide in the laboratory 9.

Chryssou et al. theoretically studied the gain of an Er-Yb co-doped Al₂O₃ channel waveguide amplifier with a 2 μm × 2 μm cross-section 10. However, for this waveguide structure, a signal light propagates with a non-single mode. In theoretical calculations, the result of whether the absorption and emission cross-sections of Er or Yb in an Al₂O₃ host material replaced by that in glass or other materials 10^,11 are limited and imprecise due to the lack of relevant experimental data on Er-doped Al₂O₃ film materials or amplified spontaneous emission and the upconversion mechanism are neglected 12.

In order to analyze the performance of the amplifier gain influenced by more factors so as to achieve a reasonable design optimization in technology, a multi-theoretical model is founded. The model consists of three interrelated parts: a) the components of the guiding mode and field distribution at signal and pumping wavelengths determined by waveguide geometry structure, physical parameters and coupled conditions between the fiber and waveguide by the finite element method (FEM); b) the rate equation with an eight-level system for Er³⁺ and Yb³⁺ to calculate the particle distribution of each level, considering the active particles' (Er³⁺ and Yb³⁺) linear and nonlinear mechanisms such as absorption, emission, upconversion and cross relaxation; c) the propagation equation to show the evolution of the signal light, pumping light and the forward or backward spontaneous emission along the waveguide. The optimum design curve of amplifier gain is obtained, which can be used to design EDAWA and YEDAWA including the doping concentration, etching size, waveguide length and pump power.

2 Multi-theoretical model

2.1 Guiding modes and field distributions

Waveguide design should assure single-mode operation at signal wavelength to enhance coupling efficiency between fiber and waveguide and to avoid mode dispersion in the amplification process. However, pump light, which can remain in a multi-mode state, is not like that. The mode field and normalized intensity distribution are calculated by FEM for a different rib height waveguide with a 1-μm thick active layer and a 3-μm etching width. By comparison, under the conditions of the single-mode, a high confinement of mode field and far away from the cut-off mode, the 0.8-μm rib height is more appropriate 13.

The parameters of the rib waveguide are shown in

Fig0 Rib waveguide(n₁ = 1.64, n₂ = 1.51, n₃ = 1, d = 1 μm, W = 3 μm, h = 0.8 μm)

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Fig. 1. Two fundamental modes \left(E_{11}^x\text{and}{E}_{11}^y\right) can propagate at 1.53 μm wavelength, whereas the other six modes \left(E_{11}^x,E_{11}^y,E_{21}^x,E_{21}^y,E_{31}^x,E_{31}^y\right) do so at 0.98 μm wavelength. The single-mode for 1.53 μm can be realized through polarized excitation. It is assumed that the signal and pumped light are directly coupled into the waveguide from the fiber. The input total electromagnetic field can be expressed in an expanded eigenmode field. The mode excitation fraction is{\alpha }_i=c_i{c}_i^{\ast }/{\displaystyle \sum _{i=1}^N{c}_i{c}_i^{\ast }}, where c_i is the expanded coefficient of theith guiding mode, c_i has a relation to the field distribution of the fiber output and position offset due to the misalignment of the fiber and waveguide in coupling 14; N is the number of guiding modes excited by the signal or pumped light. In the cross-section of the waveguide, the total normalized intensity distributions of signal and pump are as follows, respectively:

{\psi }_{\text{s}}(x,y)={\displaystyle \sum _{i=1}^{N_{\text{s}}}{\alpha }_{\text{s}i}}{\psi }_{\text{s}i},

{\psi }_{\text{p}}(x,y)={\displaystyle \sum _{i=1}^{N_{\text{p}}}{\alpha }_{\text{p}i}}{\psi }_{\text{p}i}.

In addition, mode power is also obtained from the mode excitation fraction and total power.

2.2 Rate equations

Fig0 Schematic representation of energy level and transition for Er³⁺-Yb³⁺ co-doped system pumped at 0.98 μm.

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Figure 2 shows the energy level structure and the transition schematic of a Yb³⁺-Er³⁺ co-doped system pumped at 0.98 μm. Considering the cooperative upconversion, excited state absorption, cross-relaxation and Yb-Er energy transfer, rate equations and conservation laws are founded as follows:

(1)\begin{array}{c}\frac{\text{d}{N}_1}{\text{d}t}=-W_{13}{N}_1-W_{12}{N}_1+W_{21}{N}_2+W_{31}{N}_3\\ +\frac{N_2}{{\tau }_2}+C_2{N}_2^2+C_3{N}_3^2-C_{14}{N}_1{N}_4\\ -K_{\text{tr}18}{N}_1{N}_8+K_{\text{tr}37}N,\end{array}

(2)\begin{array}{c}\frac{\text{d}{N}_2}{\text{d}t}=W_{12}{N}_1-W_{21}{N}_2-\frac{N_2}{{\tau }_2}+\frac{N_3}{{\tau }_3}\\ -2C_2{N}_2^2+2C_{14}{N}_{1}N,\end{array}

(3)\begin{array}{c}\frac{\text{d}{N}_3}{\text{d}t}=W_{13}{N}_1-W_{31}{N}_3-\frac{N_3}{{\tau }_3}+\frac{N_4}{{\tau }_4}\\ -2C_3{N}_3^2-W_{\text{ESA}}{N}_3+K_{\text{tr}18}{N}_1{N}_8\\ -K_{\text{tr}37}{N}_{3}N,\end{array}

(4)\frac{\text{d}{N}_4}{\text{d}t}=-\frac{N_4}{{\tau }_4}+\frac{N_5}{{\tau }_5}+C_2{N}_2^2-C_{14}{N}_{1}N,

(5)\frac{\text{d}{N}_5}{\text{d}t}=-\frac{N_5}{{\tau }_5}+\frac{N_6}{{\tau }_6},

(6)\frac{\text{d}{N}_6}{\text{d}t}=-\frac{N_6}{{\tau }_6}+C_3{N}_3^2+W_{\text{ESA}}{N}_3,

(7)\begin{array}{c}\frac{\text{d}{N}_7}{\text{d}t}=-W_{78}{N}_7+W_{87}{N}_8+\frac{N_8}{{\tau }_8}\\ +K_{\text{tr}18}{N}_1{N}_8-K_{\text{tr}37}{N}_3{N}_7,\end{array}

(8)\frac{\text{d}{N}_8}{\text{d}t}=-\frac{\text{d}{N}_7}{\text{d}t},

(9)N_{\text{Er}}=N_1+N_2+N_3+N_4+N_5+N_6,

(10)N_{\text{Yb}}=N_7+N_8,

where N_Er and N_Yb are the doping concentrations of Er and Yb, respectively, N_i and τ_i are the ion density and average lifetime of the i^th level, respectively, C₂ and C₃ are the upconversion coefficients of the ⁴I_13/2 and ⁴I_11/2 levels, accordingly, and C₁₄ is the cross-relaxation coefficient between ⁴I_15/2 and ⁴I_9/2. W₁₂, W₂₁ and W₁₃, W₃₁ are the stimulated absorption and stimulated emission transition probability for the signal and pump light, W_ESA is the excited state absorption transition probability (⁴I_11/2 → ⁴F_7/2). K_tr18 is the energy transfer coefficient from Yb³⁺ (²F_5/2) to Er³⁺ (⁴I_15/2). K_tr37 is the reverse energy transfer coefficient from Er³⁺(⁴I_11/2) to Yb³⁺(²F_7/2), but this reverse energy transfer can be neglected due to the lifetime of a μs and few particles existing in level ⁴I_11/2.

Amplified spontaneous emission should be considered to exactly analyze the gain of an Er or Yb-Er co-doped amplifier. Three kinds of light exist in a waveguide: signal light, pump light and spontaneous emission (forward and backward), so

(11)\begin{array}{c}{W}_{12(21)}(x,y,z)=\frac{{\sigma }_{\text{Er}-\text{a}12(\text{Er}-\text{e}21)}({\nu }_{\text{s}})}{h{\nu }_{\text{s}}}{I}_{\text{s}}(x,y,z)\\ +{\displaystyle \sum _{j=1}^M\frac{{\sigma }_{\text{Er}-\text{a}12(\text{Er}-\text{e}21)}({\nu }_j)}{h{\nu }_j}}\\ \times \left[I_{\text{ASE}+}\left(x,y,z,{\nu }_j\right)+I_{\text{ASE}-}\left(x,y,z,{\nu }_j\right)\right],\end{array}

(12)W_{13}(x,y,z)=\frac{{\sigma }_{\text{Er}-\text{a}13}\left({\nu }_{\text{p}}\right)}{h{\nu }_{\text{p}}}{I}_{\text{p}}(x,y,z),

(13)W_{\text{ESA}}(x,y,z)=\frac{{\sigma }_{\text{ESA}}\left({\nu }_{\text{p}}\right)}{h{\nu }_{\text{p}}}{I}_{\text{p}}(x,y,z),

(14)W_{78(87)}(x,y,z)=\frac{{\sigma }_{\text{Yb}-\text{a}78(\text{Yb}-\text{e}87)}\left({\nu }_{\text{p}}\right)}{h{\nu }_{\text{p}}}{I}_{\text{p}}(x,y,z),

where σ_Er-a12(ν_s) and σ_Er-e21(ν_s) are the absorption and emission cross-sections of Er³⁺ at 1.53 μm, respectively. σ_Er-a13(ν_p) is the absorption cross-section of Er³⁺ at 0.98 μm, σ_Yb-a78(ν_p) and σ_Yb-e87(ν_p) are the absorption and emission cross-sections of Yb³⁺ at 0.98 μm, respectively. σ_ESA is the Er³⁺ excited-state absorption cross-section. I is the light intensity, M is the sampling number of frequency slots of width Δν for a divided spontaneous emission spectrum (1.45–1.65 μm), and ν_j is the center frequency of each slot. σ_Er-a12(ν_j) and σ_Er-e21(ν_j) are the absorption and emission cross-sections for ν_j in the 1.45–1.65 μm band, respectively 6, which can be used to analyze the noise of the amplifier as well. The intensity distribution is expressed by normalized intensity distribution and actual power in a waveguide:

(15)I_{\text{s}(\text{p})}(x,y,z)={\psi }_{\text{s}(\text{p})}(x,y)P_{\text{s}(\text{p})}(z),

(16)I_{\text{ASE}\pm }\left(x,y,z,{\nu }_j\right)={\psi }_{\text{s}}(x,y)P_{\text{ASE}\pm }\left(z,{\nu }_j\right),

P_s(z), P_p(z) and P_ASE±(z, ν_j) represent the power of the signal, pump and spontaneous emissions going forward and backward, respectively.

Analyzing the gain of the Er-doped waveguide amplifier, we set N_Yb as 0 cm^-3 and the energy transfer coefficient between Yb and Er as 0. The optical parameters used in theoretical analysis are from the experimental data in related literature with Al₂O₃ as host material, which can be seen in

Tab0 Parameters of Yb³⁺-Er³⁺ co-doped Al₂O₃ film

parameter	value
pump cross-section (λp = 0.98 μm)	absorption	σEr-a13 = 1.7 × 10-21 cm2	σYb-a78 = 11.7 × 10-21 cm2
	emission	σEr-e31 = 0 cm2	σYb-e87 = 11.6 × 10-21 cm2
signal cross-section (λs = 1.53 μm)	absorption	σEr-a12 = 5.7 × 10-21 cm2
	emission	σEr-e21 = 5.7 × 10-21 cm2
excited state absorption cross-section	σESA = 0.8 × 10-21 cm2
excited state lifetime	τ2 = 7.8 ms	τ3 = 30 μs	τ4 = 1.0 ns	τ5 = 7.0 μs	τ6 = 20 ns	τ8 = 1.1 ms
upconversion coefficient (assuming a linearly increasing function of Er concentration)	NEr = 2.7 × 1020 cm-3 NEr = 4.4 × 1019 cm-3	C2 = C3 = C14 = 4.1 × 10-18 cm3/s C2 = C3 = C14 = 3.5 × 10-18 cm3/s
energy transfer coefficient	Ktr18 = 4.0 × 10-17 cm3/s

Table 1 1^,8^,9.

2.3 Propagation equations

The steady-state variation of the signal, pump and amplified spontaneous emission powers along the active waveguide are described by propagation equations 15:

(17)\frac{\text{d}{P}_{\text{p}}(z)}{\text{d}z}=-{\gamma }_{\text{p}}(z)P_{\text{p}}(z)-{\alpha }_{\text{p}}{P}_{\text{p}}(z),

(18)\frac{\text{d}{P}_{\text{s}}(z)}{\text{d}z}=\left[{\gamma }_{21}\left(z,{\nu }_{\text{s}}\right)-{\gamma }_{12}\left(z,{\nu }_{\text{s}}\right)\right]P_{\text{s}}(z)-{\alpha }_{\text{s}}{P}_{\text{s}}(z),

(19)\begin{array}{c}\frac{\text{d}{P}_{\text{ASE}\pm }\left(z,{\nu }_j\right)}{\text{d}z}=\pm \left[{\gamma }_{21}\left(z,{\nu }_j\right)-{\gamma }_{12}\left(z,{\nu }_j\right)\right]P_{\text{ASE}\pm }\left(z,{\nu }_j\right)\\ \pm mh{\nu }_j\text{Δ}\nu {\gamma }_{21}\left(z,{\nu }_j\right)\mp {\alpha }_s{P}_{\text{ASE}\pm }\left(z,{\nu }_j\right),\\ j=1,2,\dots ,M.\end{array}

Here

(20){\gamma }_{12(21)}\left(z,{\nu }_{\text{s}}\right)={\displaystyle {\iint }_A{\psi }_{\text{s}}(x,y){\sigma }_{\text{Er}-\text{a}12(\text{Er}-\text{e}21)}\left({\nu }_{\text{s}}\right)}{N}_{1(2)}(x,y,z)\text{d}x\text{d}y,

(21){\gamma }_{12(21)}\left(z,{\nu }_j\right)={\displaystyle {\iint }_A{\psi }_{\text{s}}(x,y){\sigma }_{\text{Er}-\text{a}12(\text{Er}-\text{e}21)}\left({\nu }_j\right)N_{1(2)}(x,y,z)\text{d}x\text{d}y},

(22)\begin{array}{l}{\gamma }_{\text{p}}(z)={\displaystyle {\iint }_A{\psi }_{\text{p}}(x,y)[{\sigma }_{\text{Er}-\text{a}13}\left({\nu }_p\right)N_1(x,y,z)}\\ +{\sigma }_{\text{ESA}}\left({\nu }_{\text{p}}\right)N_3(x,y,z)+{\sigma }_{\text{Yb}-\text{a}78}\left({\nu }_{\text{p}}\right)N_7(x,y,z)\\ -{\sigma }_{\text{Yb}-\text{e}87}\left({\nu }_{\text{p}}\right)N_8(x,y,z)]\text{d}x\text{d}y.\end{array}

α_s and α_p are waveguide losses for the signal and pump, A is the cross-section area of the active region, and m is the number of guiding modes at signal wavelength, which is decided by the waveguide film dielectric constant and the geometric structure. For example, for double fundamental modes, m = 2.

The boundary condition for waveguide amplifiers is

P_{\text{p}}(0)=P_{\text{P}},P_{\text{S}}(0)=P_{\text{S}},P_{\text{ASE}+}\left(0,{\nu }_j\right)=P_{\text{ASE}-}\left(L,{\nu }_j\right)=0,

where L is the waveguide length. The gain is given by

(23)G=10\lg \frac{P_{\text{s}}(z)}{P_{\text{s}}(0)}.

By FEM, the normalized intensity distributions ψ_s, ψ_p of the signal and pump are computed. The particle number N_i(x, y, z) of each level could be achieved by solving the rate equations under steady-state, then γ_p(z), γ₁₂(z,ν_j) and γ₂₁(z,ν_j) are calculated with ψ_s, ψ_p and N_i. Finally, 2M + 2 nonlinear differential equations, i.e., propagation equations, are solved by Runge-Kutta and a multi-step method. P_ASE±(z,ν_j), P_s(z) and P_p(z) are calculated step by step. The waveguide loss of the signal, spontaneous emission and pump light adopts a reference value of 0.35 dB/cm 8, and the step length is 20 μm.

By comparing the results of sampling number M = 50 (Δλ = 4 nm, Δν =512 GHz) with M = 200 (Δλ =1 nm, Δν =128 GHz), the relative gain difference (G₂₀₀ - G₅₀)/G₂₀₀ varies from 1 × 10^-6 to 1 × 10^-3 at N_Er = 2 × 10²⁰ cm^-3, P_p = 50 mW, L=15 cm, so it is enough to divide the spontaneous emission spectrum into 50 frequency slots.

3 Gain characteristics of waveguide amplifiers

3.1 Comparison of pump efficiency and gain between EDAWA and YEDAWA

For EDAWA and YEDAWA, the typical result of pump efficiency and gain of the amplifier as pump power changes is shown in

Fig0 Characteristic comparisons between YEDAWA and EDAWA.(a) Pumping efficiency comparison; (b) gain comparison

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Fig. 3. The input signal power is P_s(0) = 1 μW (-30 dBm), Er concentration N_Er = 2 × 10²⁰ cm^-3, Yb concentration N_Yb = 5 × 10²⁰ cm^-3, and the waveguide length is 2 cm. It is clear from Fig. 3 that the higher the pump power is, the higher the gain is, and finally the gain tends toward saturation. The gain of YEDAWA reaches saturation at the pump power of 5 mW while EDAWA needs a higher pump power to reach saturation. In addition, the gain of YEDAWA is higher than that of EDAWA.

As for pump efficiency, YEDAWA is obviously higher than that of EDAWA, especially at low pump power. The pump efficiency will drop as pump power increases, so co-doping with Yb can not improve an amplifier's pump efficiency at high pump power.

3.2 Parameter optimization of amplifiers

Fig0 Signal gain versus amplifier length for different input pump powers(N_Er = 2 × 10²⁰ cm^-3, N_Yb = 1 × 10²¹ cm^-3). (a) EDAWA; (b) YEDAWA

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Figure 4 illustrates that the net gains of EDAWA and YEDAWA vary with the amplifier length at different pump powers. As shown by the curves in Fig. 4, the gain increases at the beginning and then drops. In the initial stage, the signal light is so weak that the gain coefficient γ₂₁(z,ν_s) - γ₁₂(z,ν_s) in Eq. (18) can be considered as constant, and the gain increases linearly along the waveguide. And then the gain increases slowly, which can be explained by two reasons. On one hand, as the signal light is amplified, the saturation effect leads to a low gain coefficient. On the other hand, as pump power is wasted in the amplification process, the pump efficiency drops and the population inversion decreases, which also reduces the gain coefficient and the gain increases slightly. When it reaches a certain length where the gain coefficient equals the loss coefficient of the waveguide, the signal stops being amplified and the net gain comes to its maximum. If the length continues to increase, as the population inversion and the gain coefficient continue decreasing, even the loss and absorption surpass the gain, the signal light weakens, and the net gain begins to decrease. So there is an optimal waveguide length for the net gain of the amplifier which is relative to the pump power.

A linear characteristic is more obvious for YEDAWA in the region of increase and decrease of gain. A much longer waveguide and higher pump power are needed for higher gain. Under the conditions of Er concentration of 2 × 10²⁰ cm^-3 and pump power of 40 mW, the maximum gain for EDAWA is approximately 28 dB when the waveguide length is 16 cm, while approximately 32 dB can be achieved for YEDAWA with a waveguide length of only 10 cm under the same conditions.

Hoven et al. 8 theoretically predicted that the gain of a waveguide amplifier would achieve 20 dB for a 50 mW pump power at 1.48 μm, N_Er = 2.7 × 10²⁰ cm^-3 and 15 cm waveguide length. In this paper, using an optimized pump wavelength of 0.98 μm, an Er concentration of 2.0 × 10²⁰ cm^-3 and a waveguide length of 15 cm, we obtain a gain of 30 dB for 40 mW pump power.

Figure 4(b) also shows the theoretical results obtained by Chryssou 10, in which Er and Yb concentrations are identical and the pump power is 100 mW. However, Chryssou adopted a 2 μm × 2 μm channel waveguide so that the active region area is bigger and much higher pump power is needed. Furthermore, the signal propagates by way of multi-modes with E_{11}^x,E_{11}^y,E_{21}^x,E_{21}^y,E_{12}^x and E_{12}^y, and the amplification of the signal will bring mode dispersion, which is a disadvantage for the amplifier. So the pump efficiency of the rib waveguide amplifier introduced here is much higher than that of Chryssou. In addition, it can guarantee single mode propagation at signal wavelength.

Unlike the fiber amplifier, once a waveguide amplifier is fabricated, the length changes no more. For a certain pump power, the optimal length of the amplifier is such that the remaining pump power would make the net gain coefficient just equal to zero at the end of the amplifier; here the net gain reaches the maximum. The doping concentration indeed affects the optimal length as well. We conclude that an optimum matching relation exists among optimal length, doping concentration and pump power. For EDAWA, optimizing the three parameters synchronously, we get a practical optimization design curve as shown in

Fig0 Multi-parameter optimum of Er-doped Al₂O₃/SiO₂ waveguide amplifier

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Fig. 5. The horizontal axis denotes the pump power and the vertical axis denotes the optimized waveguide length; dashed lines mean Er concentration lines and the solid lines show contour lines of the maximum gain. The intersection point of the four parameters means that the corresponding maximum gain could be obtained under the condition of the corresponding pump power, Er concentration and optimal length. We can also choose suitable parameter values based on the experimental condition at hand to win the appropriate gain.

4 Experimental research of YEDAWA based on SiO₂/Si

Yb-Er co-doped Al₂O₃ thin film 16–19171819 based on SiO₂/Si is fabricated by medium frequency sputtering. The process includes: thermally oxidizing the Si (100) substrate; ultrasonic cleaning; depositing the Yb-Er co-doped Al₂O₃ film onto the surface of SiO₂ by twin targets medium frequency magnetron sputtering technology in a vacuum chamber; and, annealing the films at 850°C. The sputtering twin targets are two identical aluminum targets wherein a certain number of Er and Yb cylinders of Φ2 mm are embedded in the desired proportion. The twin targets are alternatively used as cathode and anode, which cannot only inhibit the arc discharge on the sputtering targets surfaces, but also eliminate “anode disappearing” in the usual DC magnetron sputtering and the instability of plasma discharge.

The Al₂O₃ film obtained is compact and homogeneous, with a layer thickness of 1.16 μm, refractive index of 1.65, Er concentration of 0.6 at.%, Yb concentration of 5.2 at.% and a size of 22.4 mm × 10.6 mm. The photoluminescence (PL) spectrum peak is around 1.53 μm, and FWHM is 40–60 nm. The film was etched by the Chuangweina Company in Beijing. Because of the difficulty in Al₂O₃ etching, the sample waveguide made at present is a nonstandard rib waveguide which is 10 μm wide, 1.04 μm high, 2.24 cm long; the waveguide space is 30 μm and there is air cladding.

The characteristics of the fabricated waveguide amplifier are theoretically simulated by the multi theory model. The finite element analysis shows that the multi modes propagate at both signal and pump wavelengths. Signal wavelength has six x-polarized modes and the pump wavelength has twenty-six modes. Because the big cross-section of the waveguide decreases the pump power density, the threshold pump power is bigger.

Pump and signal beams are combined into a single mode fiber using a wavelength division multiplexer. The end of the fiber was tapered from 8.2 μm to Φ0.96 μm. Pump and signal are directly coupled into the waveguide by means of a five-dimensional fiber regulator with precision in nanometers. Through an optical chopper (425 Hz) and a band-pass filter with a central wavelength of 1.54 μm and a bandwidth of 26 nm, the amplified signal received by an InGaAs detector, amplified by lock-in, is displayed and the data are processed by computer. In our experiment, the power supply of the signal is modulated at 1 Hz, and then the peak-to-peak value is measured. This not only separates the pump completely but also eliminates the amplified spontaneous emission and random scattered light. The measurement is done at room temperature.

Fig0 Dependence of net gain of YEDAWA on pump power

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Figure 6 gives the results of the experiment, showing that the amplifier gain varies with the pump power. The threshold pump power is around 18 mW, and the net gain can reach 8.44 dB at 70 mW pump power, which equals a gain of 3.77 dB/cm. The figure also shows the simulation results using the multi-theoretical model. The experimental data are in accordance with the numerical simulation results, and the two threshold pump powers are also very close.

The 2.24 cm YEDAWA gets a net gain of 10.6 dB (4.73 dB/cm) by theoretical calculation; the reason for the difference is that the optical parameters used here may be different from the actual parameters. Even though the host materials are identical, the preparation methods and techniques are different 20. In addition, the waveguide loss in calculation is 0.35 dB/cm, whereas the actual waveguide loss is perhaps a little larger, such as the scattering loss caused by interface flatness of the waveguide. It can be improved by enhancing film fabrication and etching techniques.

5 Conclusion

The gain characteristics of EDAWA and YEDAWA are analyzed by the multi-theoretical model, including rate equations, finite element field distribution, and propagating equations with forward and backward spontaneous emission. In the simulation, a more reasonable optimized program of parameters is established, which can achieve the synchronous optimization of pump power, doping concentration, waveguide length and gain. The optimum design curve is charted. The theoretical model is applied to a practical Yb-Er co-doped Al₂O₃ waveguide amplifier, and the results show that the calculations are in good agreement with the experimental data.

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