# Modelling of Output Statistics of Single andℳ-Mode Straight and Curved Er3+:Ti:LiNbO3 Waveguide Amplifiers

Authors: Niculae N. Puscas
Journal: Advances in OptoElectronics
 (2011)

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## Abstract

A theoretical analysis of some statistical parameters which characterize theEr3+-doped Ti:LiNbO3 single and ℳ-mode straight and curved waveguides is presented in this paper. In the derivation and the evaluation of the spectral optical quality factor, the power spectral density, the Fano factor, the statistical fluctuation, and the spontaneous emission factor we used the small gain approximation, and the photon statistics master equation of the linear amplifier (considering that the photon number distribution is determined by the normalized mode intensity profiles which are not uniform in the transversal section of the waveguide), transposed to the case of straight and curved amplifiers. The simulation results show the evolution of the above-mentioned parameters under various pump regimes and waveguide lengths.


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## Body

## 1. Introduction

Lithium niobate photonic circuits permit the generation, transmission, and processing of photons to be accommodated on a single chip. Compact photonic circuits with multiple components integrated on a single chip are crucial for efficiently implementing quantum information processing schemes.The study of the noise spectral distribution and the output statistics in a system where coherent and chaotic (thermal) fields are superimposed plays an important role in obtaining integrated amplifiers with low noise and high optical gain. This is the reason why, over the last decade, a great attention has been devoted to the analysis of the statistical properties ofEr3+-doped fibres and waveguide amplifiers [1–12].A novel technique for evaluating the output statistics of single andℳ-mode straight and curved Er3+-doped Ti:LiNbO3 amplifiers, like the spectral optical quality factor, the power spectral density, the output mean photon number, the statistical fluctuation, the Fano factor, and the spontaneous emission factor, is proposed in this paper. The reason of this analysis is related to the fact that often in the waveguides is excited not only the fundamental mode but also other high-order (ℳ) modes [11, 13], which influence the output gain, noise figure, signal-to-noise ratio, and the statistical properties of the waveguide.The paper is organized as follows. Section2 is devoted to the basic equations used to evaluate the population of the upper atomic energy level and the evolution of the pump, signal, power spectral density, Amplified Spontaneous Emission (ASE), the Fano factor, the statistical fluctuation, and the spontaneous emission factor which characterize the output statistics of the single and ℳ-mode waveguides. Also, in this section the model used for the calculation of optical field distribution in the bent waveguides is presented [6–11]. Section 3 deals with the discussion of the computed results, while the conclusion of this work is presented in Section 4.


## 2. Theoretical Considerations

The quantum theory of the statistical properties in the optical waveguide amplifiers is based on the more general quantum theories of coherent light, coherent light/matter interaction, noise, and laser oscillations [1, 8–11]. We used the two-level model to describe the above-mentioned processes [1, 8–11]. The transitions between the two lowest energetic level manifolds of the Er3+ ions incorporated into the LiNbO3 lattice determine absorption and optical amplification by stimulated emission in the range 1440 nm < λ < 1640 nm. The optical amplification of a signal wave  takes place if the population inversion with respect to the ground state I15/24 is achieved after the optical pumping to a pump band (an excited state) followed by a fast relaxation to the metastable level I13/24. Using a pump radiation having λ = 1484 nm it is possible to neglect the excited state absorption (ESA) and to consider a quasi-two-level model for the simulation of the Er3+-doped Ti:LiNbO3 waveguide amplifiers in the wavelength range 1440 nm < λ < 1640 nm.The interaction of pump/signal photons withEr3+ ions can be represented using the rate Equations [8–11]. In the case of the steady state regime by discretizing the considered frequency spectrum in a number of intervals Δν small enough so that the frequency-dependent quantities can be assumed constant over each of them, one obtains the equations for the pump, signal, and ASE evolution after integration over the waveguide transversal cross-section in the form [8–11]
(1)ddzPkm±(z)=±{[(σe,km+σa,km)∫AN2(x,y,z)im(x,y)dxdy-(σa,km∫AN2(x,y,z)im(x,y)dxdy+αkm)]×Pkm±(z)+hνkΔνσe,km∫AN2(x,y,z)im(x,y)dxdy},
where Pkm±(z) represent the power in the spectral interval Δν around frequency νk, for the polarization m in the forward (+) and backward (−) propagation directions, with x and y being the coordinates in the plane perpendicular to the waveguide axis z and N2 is the steady state solution of the rate equations for the upper populations of the two levels which is written in the form(2)N2(x,y,z)=NT0d(x,y)×∑k,m(τ/hνk)σa,km(Pkm+(z)+Pkm-(z))im(x,y)1+∑k,m(τ/hνk)(σa,km+σe,km)(Pkm+(z)+Pkm-(z))im(x,y).In (2) the Er3+ ions distribution in the waveguide has been written in the form NT(x,y)=NT0d(x,y), where NT0=NT(0,0) and d(x,y) is the normalized dopant distribution. Similar notations have been used for the cross-sections, σa,m(ν) and σe,m(ν) for the optical waveguide losses αkm. The term hνkΔν in (1) represents an equivalent spontaneous emission input noise power in the frequency slot Δν corresponding to the frequency νk [1].Spontaneous transitions from the excited states to the ground state are taken into account by the rateA21=1/τ,τ being the fluorescence lifetime of the Er3+ ions. The absorption and emission rates of the amplified spontaneous emission are W12ASEandW21ASE, respectively.The energy fine structure is taken into account by wavelength-dependent absorption and emission cross-sections. The cross-sections profiles constitute the basis for the computation of the gain, noise figure, and other parameters which characterize the amplifiers. The pump (signal) absorption, and emission rates areR12(W12) and R21(W21), respectively. To analyse the behaviour of the amplifier in terms of signal gain, pump absorption and ASE output power, it is convenient to distinguish in the optical power at frequency νk the coherent contribution due to injected signal or pump beams (stimulated emission or absorption) from the incoherent one due to the (amplified) spontaneous emission. We can do this by observing that the forcing term in (1) contributes only to the incoherent ASE power and by using different “k” indexes for the two kinds of contribution. The stimulated transition rates due to the pump, signal, and ASE fields can be written in terms of the corresponding field intensities Im(x,y,z,ν), the absorption and emission cross-sections, σa,m(ν) and σe,m(ν), and the photon energies hν [1, 8]:(3)R12,W12,W12ASE=∑m=TE,TM∫σa,m(ν)Im(x,y,z,ν)hνdν,R21,W21,W21ASE=∑m=TE,TM∫σe,m(ν)Im(x,y,z,ν)hνdν.
In (1)–(3) the index m allows to distinguish the two polarizations sides of the guided field. Moreover, the optical intensities can be expressed as the product of a power spectral density Pm(x,y) and the normalized field intensities im(x,y), which can be assumed constant in the frequency range of interest.The boundary conditions for (1) can be written in the form
(4)Pkm+(z=0)=(1-R0,km)Pin0,km+R0,kmPkm-(z=0),Pkm-(z=L)=(1-RL,km)PinL,km+RL,kmPkm+(z=L),
where R0,km,RL,km are the input and output reflectivities and Pin0,km,PinL,km are the optical powers injected at z=0 or at z=L. We assume that the signal is injected only from the section z=0 and that no noise power is entering the amplifier (i.e., Pin0,km=PinL,km=0 for the ASE beams). By properly choosing the values of R0 and RL we can account for the natural reflectivities of the LiNbO3-air interface, as well as analyse the case of a wavelength-selective coating; moreover, if we set RL to a high value (approx. 100%),our model can describe the behavior of a double-pass amplifier for the pump and the signal. For the numerical integration of the system of equations (1)-(2) in our model we took into account the overlap between the normalized field intensity and the (normalized) dopant distribution and the (normalized) upper laser level population by introducing the corresponding overlap integrals  ΓT,m and Γ2,m, respectively,
(5)ΓT,m(z)=∫Ad(x,y)im(x,y)dxdy,(6)Γ2,m(z)=1NT0∫AN2(x,y,z)im(x,y)dxdy.The effects of gain saturation due to the amplified signal or to the ASE power are accounted for in the expression for the populationN2: as the signal and ASE power grows the emission rate increases and N2 decreases, leading to a reduction of Γ2,m and consequently of  the gain factor akmΓ2,m in (1).Considering a collection of atoms with population density distributionN1(x,y,z) for the ground state and N2(x,y,z) for the excited state, the rate of change of the probability for having n photons at frequency ν in a single longitudinal mode (Pn) is given by the photon statistics master equation of the linear amplifier [1].Assuming that the normalized mode intensity profilesim(x,y) are not uniform in the transversal section of the waveguide the photon number distribution becomes n×im(x,y). Multiplying the photon statistics master equation by im(x,y) and integrating over the transversal section of the waveguide we obtain the following [9]:
(7)dPn(z,ν)dz=γe(z,ν)nPn-1(z,ν)+γa(z,ν)nPn+1(z,ν)-[γe(z,ν)(n+1)+γa(z,ν)n]Pn(z,ν),
where
(8)γe=γe(z,ν)=σe(ν)∫AN2(x,y,z)im(x,y)dxdy,γa=γa(z,ν)=σa(ν)∫AN1(x,y,z)im(x,y)dxdy.In (7) σa and σe represent the cross-sections for the absorption and emission processes. Furthermore, multiplying (7) by the number of photons, n, and summing over n we obtain the output expression for the photon mean value 〈n(z)〉:
(9)〈n(z)〉=G(z)〈n(0)〉+N(z),
where
(10)G(z,ν)=exp⁡{∫0z[γe(z′,ν)-γa(z′,ν)-α(ν)]dz′},(11)N(z,ν)=G(z,ν)∫0zγe(z′,ν)G(z′,ν)dz′
represent the spectral gain and the ASE photon number, respectively.For coherent input signals with Poisson’s statisticsPn(0)=(〈n(0)〉n/n!)e-〈n(0)〉 being well known that σ2(0)=〈n(0)〉. In the case of uniform inversion, when the population densities of the upper and lower levels (N1and N2, resp.) do not depend on the waveguide coordinate z, the variance takes the canonical form:
(12)σ2(z)=[G(z)〈n(0)〉+N(z)]+⌊2G(z)N(z)〈n(0)〉+N2(z)⌋.As a measure of the gain and the noise characteristics of the waveguide amplifiers, we adopted a quality factor(13)Q(z,ν)=G(z,ν)F(z,ν),
where (14)F(z,ν)=1+2N(z,ν)G(z,ν)
represents the noise figure which quantifies the noise properties of an optical amplifier contributing to the deterioration of the signal-to-noise ratio with a purely shot noise.As mentioned before spontaneous emission is also present in any amplifier. Small amount of this spontaneous emission gets amplified and comes out along with signal as amplified ASE noise.The ASE noise is generally modelled as white noise with the power spectral density [12]:
(15)S(z,ν)=nsp(z,ν)[G(z,ν)-1]⋅hν,
where nsp(z,ν) represents the spontaneous emission factor and h the Plank's constant. The amplifier output statistics can also be characterised by the Fano factor f(z)=σ2(z)/〈n(z)〉 and the statistical fluctuation e(z)=σ(z)/〈n(z)〉 [1]. For Poisson statistics (coherent light), the Fano factor and the statistical fluctuation correspond to f=1 and e=(〈n〉)-1/2, while for Bose-Einstein statistics (incoherent light), they are given by f=〈n〉+1 and e=(1+1/〈n〉)-1/2.Assuming an input signal characterised by Poisson statistics, and in the limit of high input signals and high gains, the Fano factor and the statistical fluctuation are given by(16)f(z,ν)≈1+2nsp(z,ν)[G(z,ν)-1],e(z,ν)≈1G(z,ν)〈n(0)〉{1+2nsp(z,ν)[G(z,ν)-1]}1/2,
where the spontaneous emission factor is given by(17)nsp(z,ν)=N(z,ν)G(z,ν)-1=G(z,ν)G(z,ν)-1∫0zγe(z′,ν)G(z′,ν)dz′.In (16) the factor [1+2G(L,ν)∫0L(γe(z′,ν)/G(z′,ν))dz′]1/2 characterizes the deviation of the output statistics from Poisson.Often in the waveguides is excited not only the fundamental mode but also other high-order (ℳ) modes, which influence the output gain, noise figure, and the statistical properties of the waveguide. The normalized field transversal intensity distribution can be written as [13]
(18)i(x,y)=∑j=1Mηjij(x,y)exp⁡[i(ωt-βjz)],
where βj are the propagation constants which are different for different modes because of their velocity [11] and
(19)ηj=∬Ep(x,y)⋅Ej(x,y)dxdy∬Ep2(x,y)dxdy∬Ej2(x,y)dxdy
are the overlap coefficients between the pump optical field, Ep(x,y), and the mode fields, Ej(x,y), excited in the waveguide, which satisfy a normalization condition ∑j=1ℳηj=1. In our model we considered that the waveguide is pumped by a radiation having a Gaussian distribution of the field using an optical fiber. The integrals in (19) are extended over the transversal section of the waveguide, with x representing the width and y the depth, respectively.In order to determine the optical field distribution in the bent waveguides we calculated first the refractive index profiles using the Fick’s diffusion law [12]. After that, the optical mode fields were calculated numerically for both TE and TM polarisation using the effective index method presented in papers [7, 14].


## 3. Discussion of the Simulation Results

The system of coupled first-order differential equations (1)-(2) for the optical power components and the upper population level can only be solved by numerical methods. Moreover, the presence of the boundary conditions (4) at both the extremities of the device requires an iterative procedure of integration: (1) integrate from z=0 to z=L the equations for Pkm+(z) and assuming Pkm-(z)=0 in the first iteration; (2) integrate from z=L to z=0 the equations for Pkm-(z), using the values of Pkm+(z) found previously; (3) restart from point (1), using for Pkm-(z) the value found at point (2). The iterations are stopped when the change in Pkm±(z) is smaller than a prescribed value.A particular care should be taken for the evaluation ofΓ2,m(z) because this quantity determines the incremental gain in (13) and depends on the actual value of the approximation for Pkm±(z); this implies that, at each step of the forward and backward integration along z, we have to evaluate the integral Γ2,m(z). Because of our choice of a Runge-Kutta formula (4th order, 4 stages) as the basic integration method, at each step we should evaluate four times (one for each stage) the overlap integral, within a large expense of computer time. Therefore, we decided to perform the transversal integration only at the first stage at each step, and to use the computed value also for the other three stages; this approximation appears not to have a great influence on the solution and represents a good compromise between accuracy and computation time. In our simulations, the spontaneous emission spectrum is divided into 100 slots which corresponds to a wavelength resolution Δλ = 2 nm in the region 1450–1650 nm.The intensity profilesim(x,y) used in the evaluation of Γ2,m(z) and ΓT,m are introduced in the model as a set of measured values, while the dopant distribution d(x,y) is approximated by Gaussian, erfc, or constant functions in depth (y) and in width (x). The simulation of the optical amplification in Er3+-doped LiNbO3 waveguide has been performed using parameters obtained from the literature [5–10]. Using (1), (2), (10), (11) and (13)–(17) we have calculated numerically the spectral dependence of the quality factor, the power spectral density, the Fano factor, and the statistical fluctuation for a single-pass configuration of the optical amplifier. We assumed a 1484 nm pump and a signal at λ = 1531 nm, having 1 μW input power.We used the following values for the absorption (a) and emission (e) cross-sections of the pump (p) and signal (s) for TE and TM polarizations: σTEa(1484 nm) = 5.61 × 10−25m2,σTMa(1484 nm) = 3.46 × 10−25m2, σTEe(1484 nm) = 1.92 × 10−25m2,σTMe(1484 nm) = 1.105 × 10−25m2,σTEa(1532 nm) = 17.24 × 10−25m2,σTMa(1532 nm)= 12.15 × 10−25m2, σTEe(1532 nm) = 16.36 × 10−25m2, σTMe(1532 nm) = 11.53 × 10−25m2. The Er-profile has been considered Gaussian in depth and constant in width, with a surface concentration of about 7 × 1025 m−3 and a diffusion depth of 20 μm and of 5.12 μm in width (defined at 1/e). We assumed the following values for the scattering loss and spontaneous emission lifetime: α = 3.7 dB·m−1) for TE, α = 4.8  dB·m−1 for TM in the case of straight waveguides, and α = 0.4 dB·cm−1 for TE and TM in the case of the curved ones and τ = 2.6 ms, respectively. The length of the waveguide in our simulations is L = 5.4 cm and the pump or signal assumed to be TE polarized if not explicitly stated.Figure1 presents the spectral evolution of the quality factor of the signal (13), the signal gain being defined  as G(z)=ln⌊Psignal(z)/Psignal(0)⌋) in the case of single-pass (signal output at z=L with R(L)=0) pass configuration, high pump regime (100 mW incident pump power) for single  and for the ℳ-mode operation.Figure 1
The spectral evolution of the quality factor pass configuration for single (S) and for theℳ-mode (M) operation.As can be seen from Figure1 the peak values of the quality factor are greater in the single mode operation because the noise figures in the case of ℳ-mode operation are greater than those corresponding to the single one. The ℳ-mode operation in comparison with the single one determines the diminution of the gain and the enhancement of the noise figure because the overlap integral between the population of the excited level and the normalized intensity field profile is smaller in the case of ℳ-mode operation than in the case of the single one (i.e., 1.21 times in the case of Gaussian profile of the dopant for a waveguide having 5.4 cm length and for an input pump power of 100 mW).Considering a radius of curvature of the waveguide is about 5 cm and its length is 2.5 cm (values commonly used in the laboratories when manufacturing integrated optics Mach-Zehnder interferometers) we obtained a quality factor of about 0.13 forλ = 1.531 μm.The power spectral density (Equation (15) is presented in Figure 2). For a high input pump power in the ℳ-mode operation the spontaneous emission factor (which determines the power spectral density) is less than 2 over most of the 60 nm spectral range considered in the figure (i.e., for Gaussian profile of the dopant in the spectral ranges 1.45 μm ÷ 1.48 μm and 1.62 μm ÷ 1.65 μm, which correspond to the amplification of an equivalent input noise of one photon per unit frequency).Figure 2
The power spectral density versus wavelength as in Figure1.In Figures3 and 4 the spectral dependences of the Fano factor (3) and the statistical fluctuation (4) in single pass configuration are presented.Figure 3
The Fano facto versus wavelength as in Figure1.Figure 4
The spectral dependence of the statistical fluctuation as in Figure1.The dependence of the Fano factor in the case of TE polarization for a signal atλ = 1531 nm versus the input pump power for a straight waveguide length L = 5.4 cm in the single and ℳ-mode operation is presented in Figure 5.Figure 5
The Fano factor versus the pump power in the single (S) andℳ-mode (M) operation.As can be seen form Figure5 the Fano factor increases rather low with the increasing of the pump power. In the same conditions the statistical fluctuation decreases when the pump power is augmented. This fact confirms that the output statistics is approximately Poissonian for pump powers about 100 mW.The Fano factor and the statistical fluctuation increase several orders of magnitude with the waveguide length over a range ~10 cm in both single andℳ-mode operation. In the case of straight waveguides, the output statistics is approximately Poissonian for waveguide lengths up to 6 cm.Computing the same parameters in the case of a curved waveguide having a length of about 2.5 cm for the above mentioned conditions we concluded that the photon statistics can be assumed to be Poissonian.Therefore, it seems that the amplifier length is responsible for the reduction of the Poissonian aspect of the amplified light. This is not the case for shorter waveguide lengths but higher pumping levels, for which the statistical properties of the light are roughly maintained. A consequence concerns the design of complex structures, when the coherence of the amplified light must be conserved. If the miniaturization of the integrated devices has to be considered, it is preferable to choose short lengths and high-level pumpings, rather than long arms and low pumping regimes.The above obtained simulated results show that the photon statistics which characterize theEr3+-doped Ti:LiNbO3  waveguide amplifiers (the Fano factor, the statistical fluctuation and the spontaneous emission factor) are correlated with the amplifying phenomena of light (gain, noise figure, signal-to-noise ratio, quality factor).


## 4. Conclusions

In the small gain approximation an original analysis of the output noise statistical properties of a single andℳ-mode Er3+-doped LiNbO3 straight and curved waveguide amplifiers has been presented. The simulations concern the quality factor, power spectral factor, Fano factor, and the statistical fluctuation. This analysis demonstrates that the Poissonian photon statistics are maintained for pump powers lower than 100 mW and waveguide lengths smaller than 5 cm for straight waveguides and 2.5 cm in the case of curved ones.Our simulation results concerning the optical gain, noise figure, and quality factor are in agreement with other experimental and theoretical results [15–17].The above-mentioned parameters are important when the coherence of the amplified light is an issue for the doped waveguides under investigation. In this paper we have shown that the coherence of the output amplified signal is more sensitive to the waveguide length than to the pumping power. The theoretical results of this simulation characterize theEr3+-doped LiNbO3 waveguide amplifiers from the point of view of noise statistical properties and can be used in the better understanding of the amplification process. Also, they can be used for the design of directional couplers, symmetrical and asymmetrical Mach-Zehnder interferometers, and other complex rare earth-doped integrated circuits.



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