Theoretical and experimental analysis of rare earth whispering gallery mode laser relative intensity noise

Jean-Baptiste Ceppe; Michel Mortier; Patrice Féron; Yannick Dumeige

doi:10.1364/OE.25.032732

1. Introduction

Solid state microcavity lasers have a lot of applications in integrated photonics or microwave photonics [1, 2]. Among available configurations, rare earth doped whispering-gallery-mode (WGM) micro-resonators can be used as narrow linewidth or low threshold miniaturized from visible to mid infrared lasers [3–7] due to their low volume and high quality (Q) factors. High purity microwave sources can be obtained using the beating of WGM lasers [8]. Solid-state micro-lasers can also serve as sensitive biological or chemical integrated sensors [9–13]. A lot of geometries (spheres, torus [14], disks [15]) and materials have been studied since the first demonstration of solid state WGM micro-resonator lasing [3]. These particular lasers have been extensively studied in terms of oscillation threshold [3,16], output power performances [17] and spectral purity [18]. Otherwise the phase or frequency noise of semiconductor or fiber lasers including a WGM resonator has been studied [19, 20]. Nevertheless, to date only few papers report on the dynamical properties of WGM miniaturized lasers: the auto-modulation regime [21] of a toroidal WGM laser have been demonstrated [22] and an extensive theoretical work dealing with intensity and frequency noise of micro-resonator Brillouin lasers have been published [23]. However, detailed study of the relative intensity noise (RIN) of WGM micro-lasers have not been reported. The intensity noise performance knowledge is of great importance for communication applications based on amplitude modulation [24]. In the purpose of sensing applications based on the use of micro-lasers, the RIN of the source is a crucial parameter since it will determine the sensitivity of the whole device. Moreover, microcavity laser noise properties investigation is of interest from a fundamental point of view [25–28] nevertheless, to our knowledge, neither theoretical nor experimental small mode volume WGM laser RIN analysis have been carried out so far.

In this paper we report on the RIN measurements of Erbium doped fluoride glass WGM micro-lasers for several pumping configurations. Due to the unique properties of WGM resonators, harmonics of the spiking frequency are observed in the RIN spectrum. Usual models of class-A or -B vertical cavity surface emitting laser RIN [29,30] have been extended to take into account WGM specificities in order to explain and exploit the experimental results. The paper is organized as follows. We first present our model detailing its specificity. Then we describe the experimental setup and the studied fluoride glass WGM micro-resonators. Finally we discuss some comparisons between theoretical and experimental results in the last section.

2. Model

We first describe the model used in this paper to analyse the RIN experimental results. The model is based on the coupled rate equations driven by noise sources. We will show that for WGM micro-lasers the linearisation usually made to solve the set of coupled differential equations in the frequency domain can not be used. In this section we propose an iterative method similar to the perturbation theory [31] to obtain an approximated solution to the coupled nonlinear differential equation system.

2.1. Notations

The Fourier transform of a signal f(t) will be denoted f̂(ω) = FT [f(t)] (ω). To lighten the calculation developments and to make easier the application of the Wiener-Khintchine theorem [32] we will use two different definitions of the operator 〈〉. In the time domain for a stationary signal f(t) it is defined as

〈 f (t) 〉 = lim_{T \to + \infty} \frac{1}{T} \int_{- T / 2}^{T / 2} f (t) d t .

In the frequency domain, 〈〉 will be related to the Fourier transform of two signals f(t) and g(t) by

〈 \hat{f} (ω) {\hat{g}}^{*} (ω) 〉 = lim_{T \to + \infty} \frac{1}{T} 𝔼 [{\hat{f}}_{T} (ω) {\hat{g}}_{T}^{*} (ω)]

where ĝ^*(ω) denotes the conjugate of ĝ(ω). f_T(t) = f(t) if t ∈ [−T/2, T/2] anf f_T (t) = 0 otherwise. 𝔼 represents here the mean expectation of a random variable.

2.2. Laser modelling

The Erbium ions (density 𝒩) will be modelled by an atomic 3-level system with an excited lifetime T₁ = 1/A (where A is the average probability of spontaneous emission) and an emission cross section σ_e. We consider here a single-mode laser characterized by a mode volume V_m and a photon lifetime τ_ph. n₀ is the refractive index of the cavity material. W_P represents the pumping rate. The dynamic evolution of a class-B laser is given by the following set of coupled differential equations where ΔN(t) is the inverted ion number and F(t) the intracavity photon number [33,34]

{\begin{cases} \frac{d Δ N}{d t} = \frac{Δ N_{0} - Δ N}{τ_{r}} - 2 κ F Δ N \\ \frac{d F}{d t} = - \frac{F}{τ_{ph}} + κ F Δ N \end{cases}

where we define the recovery time τ_r = 1/(W_P + A), the coupling between inverted ion and photon numbers

κ = \frac{c σ_{e}}{n_{0} V_{m}},

and the unsaturated inverted ion number

Δ N_{0} = N \frac{W_{P} - A}{W_{P} + A},

with N representing the total ion numbers N = 𝒩V_m.

2.3. Static solution above threshold

The first step of the calculation consists to solve Eqs. (3) and (4) in the stationary regime by setting $\frac{d Δ N}{d t} = 0$ an $\frac{d F}{d t} = 0$ . Above threshold, the inversion population value $\bar{Δ N}$ is clamped to the value

\bar{Δ N} = Δ N_{th} = \frac{1}{κ τ_{ph}},

where ΔN_th is the inversion population at threshold. Which gives the stationary value for the intracavity photon number

\bar{F} = \frac{η - 1}{2 κ τ_{r}},

where we have defined

η = \frac{Δ N_{0}}{Δ N_{th}}

. We can also evaluate the threshold pumping rate by

W_{P, th} = A \frac{N + Δ N_{th}}{N - Δ N_{th}},

and define the pumping rate

r = \frac{W_{P}}{W_{P, th}}

.

2.4. Small signal analysis

In this section we calculate the fluctuations of the photon number around its stationary value F̄. In this aim we define inversion population and photon number fluctuations δΔN(t) and δF(t) by

Δ N (t) = \bar{Δ N} + δ Δ N (t)

F (t) = \bar{F} + δ F (t) .

The usual approach consists to substitute Eqs. (10) and (11) in Eqs. (3) and (4) and to linearise the system.

\hat{δ Δ N} (ω)

and

\hat{δ F} (ω)

can be thus found algebraically by taking the Fourier transform of the whole linearized system. In the present case we want to take into account intracavity nonlinear effects and this method cannot be applied. Nevertheless the nonlinearities can be considered iteratively by applying a perturbation development of parameter λ of δΔN(t) and δF(t)

δ Δ N (t) = λ δ Δ N^{(1)} (t) + λ^{2} δ Δ N^{(2)} (t)

δ F (t) = λ δ F^{(1)} (t) + λ^{2} δ F^{(2)} (t),

where we assume 〈δΔN⁽¹⁾(t)〉 = 0, 〈δF⁽¹⁾(t)〉 = 0, 〈δΔN⁽²⁾(t)〉 = 0 and 〈δF⁽²⁾(t)〉 = 0. These two equations are injected in Eqs. (3), (4), (10), and (11), by identifying the first order terms in λ we obtain the following set of differential equations

{\begin{cases} \frac{d δ Δ N^{(1)}}{d t} = - \frac{η}{τ_{r}} δ Δ N^{(1)} - \frac{2}{τ_{ph}} δ F^{(1)} + ξ_{N} (t) \\ \frac{d δ F^{(1)}}{d t} = \frac{η - 1}{2 τ_{r}} δ Δ N^{(1)} + ξ_{F} (t), \end{cases}

where we have added driving Langevin forces ξ_N(t) and ξ_F(t) to take into account all of excess noise sources. We assume that these processes represent two white Gaussian noises with 〈ξ_N(t)〉 = 0 and 〈ξ_F(t)〉 = 0 which are characterized by their autocorrelation functions

〈 ξ_{N} (t) ξ_{N}^{*} (t + τ) 〉 = D_{NN} δ (τ)

〈 ξ_{F} (t) ξ_{F}^{*} (t + τ) 〉 = D_{FF} δ (τ),

where

(D_{NN}, D_{FF}) \in ℝ_{+}^{2}

are respectively the diffusion coefficients of N and F. Moreover we will assume no correlation between the two processes [31,35]

〈 ξ_{N} (t) ξ_{F}^{*} (t + τ) 〉 = 〈 ξ_{F} (t) ξ_{N}^{*} (t + τ) 〉 = 0 .

By identifying the terms in λ² we obtain

{\begin{cases} \frac{d δ Δ N^{(2)}}{d t} = - \frac{η}{τ_{r}} δ Δ N^{(2)} - \frac{2}{τ_{ph}} δ F^{(2)} - 2 κ δ Δ N^{(1)} (t) δ F^{(1)} (t) \\ \frac{d δ F^{(2)}}{d t} = \frac{η - 1}{2 τ_{r}} δ Δ N^{(2)} + κ δ Δ N^{(1)} (t) δ F^{(1)} (t), \end{cases}

which shows that the nonlinear coupling between δΔN⁽¹⁾(t) and δF⁽¹⁾(t), usually neglected, gives rise to the second order of the fluctuations of δΔN(t) and δΔF(t). This effect is weighted by κ which is inversely proportional to the laser mode volume. For highly confining low mode volume WGM microcavities this effect is enhanced.

2.5. Fluctuation calculation

By taking the Fourier transform of (14) and (15) we obtain

{\begin{cases} {\hat{δ Δ N}}^{(1)} (ω) = \frac{1}{D (ω)} [- \frac{2}{τ_{ph}} {\hat{ξ}}_{F} (ω) + j ω {\hat{ξ}}_{N} (ω)] \\ {\hat{δ F}}^{(1)} (ω) = \frac{1}{D (ω)} [(\frac{η}{τ_{r}} + j ω) {\hat{ξ}}_{F} (ω) + (\frac{η - 1}{2 τ_{r}}) {\hat{ξ}}_{N} (ω)] \end{cases}

where we have defined

D (ω) = \frac{η - 1}{τ_{r} τ_{ph}} - ω^{2} + j ω \frac{η}{τ_{r}} .

We use the same method to obtain the second order terms

{\begin{cases} {\hat{δ Δ N}}^{(2)} (ω) = - 2 κ \frac{\frac{1}{τ_{ph}} + j ω}{D (ω)} ({\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω)) = K_{N} (ω) ({\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω)) \\ {\hat{δ F}}^{(2)} (ω) = κ \frac{\frac{1}{τ_{r}} + j ω}{D (ω)} ({\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω)) = K_{F} (ω) ({\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω)), \end{cases}

where * represents the convolution product.

2.6. RIN expression

The RIN is defined as the normalized Fourier transform of the photon number fluctuation autocorrelation function

RIN (ω) = \frac{FT [〈 δ F (t) δ F^{*} (t + τ) 〉]}{{\bar{F}}^{2}} .

By setting λ = 1 in Eq. (13) we have to calculate four autocorrelation function Fourier transforms. Among them, using the Wiener-Khintchine theorem [32] we first evaluate

FT [〈 δ F^{(1)} (t) {(δ F^{(2)} (t + τ))}^{*} 〉] = 〈 {\hat{δ F}}^{(1)} (ω) {({\hat{δ F}}^{(2)} (ω))}^{*} 〉

= K_{F}^{*} (ω) 〈 {\hat{δ F}}^{(1)} (ω) {({\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω))}^{*} 〉 .

Noting that

{\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω) = FT [δ F^{(1)} (t) δ Δ N^{(1)} (t)]

and using one more time the Wiener-Khintchine theorem we obtain

FT [〈 δ F^{(1)} (t) {(δ F^{(2)} (t + τ))}^{*} 〉] = K_{F}^{*} (ω) FT [〈 δ F^{(1)} (t) {(δ F^{(1)} (t + τ) δ Δ N^{(1)} (t + τ))}^{*} 〉] .

Assuming that δF⁽¹⁾(t) ans δΔN⁽¹⁾(t) are Gaussian random processes, we can use the Wick-Isserlis theorem [32] which gives

〈 δ F^{(1)} (t) {(δ F^{(1)} (t + τ) δ Δ N^{(1)} (t + τ))}^{*} 〉 = 0,

leading to the following result

FT [〈 δ F^{(1)} (t) {(δ F^{(2)} (t + τ))}^{*} 〉] = 0 .

Using the same approach we can show that

FT [〈 δ F^{(2)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉] = 0 .

Finally we obtain

RIN (ω) = \frac{FT [〈 δ F^{(1)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉]}{{\bar{F}}^{2}} + \frac{FT [〈 δ F^{(2)} (t) {(δ F^{(2)} (t + τ))}^{*} 〉]}{{\bar{F}}^{2}}

= {RIN}^{(1)} (ω) + {RIN}^{(2)} (ω) .

RIN⁽²⁾(ω) consists of terms in λ⁴. To be exhaustive, we should have developed δΔN(t) and δF(t) up to the third order in Eqs. (12) and (13). Nevertheless, as we will see later, third order terms mainly contribute to a third harmonic spiking resonance which we deliberately do not consider in this work.

2.7. First order RIN

To obtain the expression of the RIN at first order we apply the Wiener-Khintchine theorem

{RIN}^{(1)} (ω) = \frac{FT [〈 δ F^{(1)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉]}{{\bar{F}}^{2}} = \frac{〈 {| {\hat{δ F}}^{(1)} (ω) |}^{2} 〉}{{\bar{F}}^{2}}

using Eq. (22) this gives

{RIN}^{(1)} (ω) = \frac{1}{{\bar{F}}^{2} {| D (ω) |}^{2}} [{(\frac{η - 1}{2 τ_{r}})}^{2} D_{FF} + [{(\frac{η}{τ_{r}})}^{2} + ω^{2}] D_{NN}],

where we used Eqs. (16), (17) and (18) and the generalized Wiener-Khintchine theorem [32] to write 〈|ξ̂_F(ω)|²〉 = D_FF, 〈|ξ̂_N(ω)|²〉 = D_NN and

〈 {\hat{ξ}}_{F} (ω) {\hat{ξ}}_{N}^{*} (ω) 〉 = 〈 {\hat{ξ}}_{N} (ω) {\hat{ξ}}_{F}^{*} (ω) 〉 = 0 .

Equation (37) is similar to the usual expression for the RIN of a class-B laser above threshold [29,30].

2.8. Second order RIN

Following the same method we obtain the second order RIN

{RIN}^{(2)} (ω) = \frac{FT [〈 δ F^{(2)} (t) {(δ F^{(2)} (t + τ))}^{*} 〉]}{{\bar{F}}^{2}} = \frac{〈 {| {\hat{δ F}}^{(2)} (ω) |}^{2} 〉}{{\bar{F}}^{2}},

using Eq. (25) this gives:

{RIN}^{(2)} (ω) = \frac{{| K_{F} (ω) |}^{2}}{{\bar{F}}^{2}} 〈 {| {\hat{δ F}}^{(1)} (ω) * {\hat{δ Δ N}}^{(1)} (ω) |}^{2} 〉,

which cannot be evaluated straightforwardly because it involves correlations of more complex products than those given in Eqs. (16), (17) and (18). We now introduce the function A₂(t) = δF⁽¹⁾(t)δN⁽¹⁾(t) and thus

{RIN}^{(2)} (ω) = \frac{{| K_{F} (ω) |}^{2}}{{\bar{F}}^{2}} 〈 {| {\hat{A}}_{2} (ω) |}^{2} 〉 = \frac{{| K_{F} (ω) |}^{2}}{{\bar{F}}^{2}} FT [〈 A_{2} (t) A_{2}^{*} (t + τ) 〉] .

where

〈 A_{2} (t) A_{2}^{*} (t + τ) 〉 = 〈 δ F^{(1)} (t) δ N^{(1)} (t) {(δ F^{(1)} (t + τ) δ N^{(1)} (t + τ))}^{*} 〉

which can be calculated using the Wick-Isserlis theorem

\begin{array}{l} 〈 A_{2} (t) A_{2}^{*} (t + τ) 〉 = 〈 δ F^{(1)} (t) δ N^{(1)} (t) 〉 〈 {(δ F^{(1)} (t + τ) δ N^{(1)} (t + τ))}^{*} 〉 + \\ 〈 δ F^{(1)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉 〈 {(δ N^{(1)} (t) δ N^{(1)} (t + τ))}^{*} 〉 + \\ 〈 δ F^{(1)} (t) {(δ N^{(1)} (t + τ))}^{*} 〉 〈 δ N^{(1)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉 \end{array}

The first term on the right of Eq. (43) is constant and thus we will note its Fourier transform Cδ(ω) with C ∈ ℝ₊. We now calculate the Fourier transform of

〈 A_{2} (t) A_{2}^{*} (t + τ) 〉

\begin{array}{l} 〈 {| {\hat{A}}_{2} (ω) |}^{2} 〉 = C δ (ω) + (FT [〈 δ F^{(1)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉]) * (FT [〈 δ N^{(1)} (t) {(δ N^{(1)} (t + τ))}^{*} 〉]) + \\ (FT [〈 δ F^{(1)} (t) {(δ N^{(1)} (t + τ))}^{*} 〉]) * (FT [〈 δ N^{(1)} (t) {(δ F^{(1)} (t + τ))}^{*} 〉]) \end{array}

which can also be written

\begin{array}{l} 〈 {| {\hat{A}}_{2} (ω) |}^{2} 〉 = C δ (ω) + 〈 {| {\hat{δ F}}^{(1)} (ω) |}^{2} 〉 * 〈 {| {\hat{δ N}}^{(1)} (ω) |}^{2} 〉 + \\ 〈 {\hat{δ F}}^{(1)} (ω) {({\hat{δ N}}^{(1)} (ω))}^{*} 〉 * 〈 {\hat{δ N}}^{(1)} (ω) {({\hat{δ F}}^{(1)} (ω))}^{*} 〉 . \end{array}

Using the same rules as those used to established Eq. (37) we have

\begin{array}{r} 〈 {| {\hat{A}}_{2} (ω) |}^{2} 〉 = C δ (ω) + U_{1} (ω) [{(\frac{η - 1}{τ_{r} τ_{ph}})}^{2} D_{NN} D_{FF} + 2 {(\frac{η}{τ_{r}})}^{2} {(\frac{2}{τ_{ph}})}^{2} D_{FF}^{2}] + \\ U_{2} (ω) {[(\frac{η - 1}{2 τ_{r}}) D_{NN} + (\frac{2}{τ_{ph}}) D_{FF}]}^{2} + \\ U_{3} (ω) [{(\frac{η - 1}{2 τ_{r}})}^{2} D_{NN}^{2} + {(\frac{2}{τ_{ph}})}^{2} D_{FF}^{2} + {(\frac{η}{τ_{r}})}^{2} D_{NN} D_{FF}] + U_{4} (ω) D_{NN} D_{FF}, \end{array}

where functions U_i(ω) with i ∈ {1, 2, 3, 4} are defined by

U_{1} (ω) = ({| D (ω) |}^{- 2}) * ({| D (ω) |}^{- 2})

U_{2} (ω) = (ω {| D (ω) |}^{- 2}) * (ω {| D (ω) |}^{- 2})

U_{3} (ω) = ({| D (ω) |}^{- 2}) * (ω^{2} {| D (ω) |}^{- 2})

U_{4} (ω) = (ω^{2} {| D (ω) |}^{- 2}) * (ω^{2} {| D (ω) |}^{- 2}) .

In conclusion, using Eqs. (47)–(50) and (25), the second order RIN can be evaluated which finally gives the total RIN including this contribution in Eq. (35). Note that in the experiments, the DC contribution giving Cδ(ω) will be filtered out and will not be taken into account in the modelling. We would like to underline that this model is not the juxtaposition of two Lorentzian fits. The second order calculations are directly derived from the first order calculations reducing the number of free parameters. In particular, when the level of the first order RIN is set via coefficients D_NN and D_FF, the mode volume through the value of κ unambiguously gives the height of the harmonic peak.

3. Experimental method

3.1. Experimental setup

The experimental part is devoted to the consistency checking of our model. For this purpose we used low mode volume WGM microcavities. The WGM lasers studied in this paper consist of ZBLALiP-fluoride glass microspheres doped with erbium ions. The microspheres are made by melting glass powders in a plasma torch, details on the manufacturing process can be found in [36]. For the present experiments, we used a doping rate of 0.1mol% and microspheres with diameters in the range 85 – 90 μm. The refractive index of ZBLALiP is n₀ = 1.49. The laser operation is obtained around 1.56 μm using the ⁴I_13/2 →⁴ I_15/2 transition of erbium. In our case, due to the melting process the excited state lifetime of this transition can be considered such as T₁ ≈ 10 ms [37] and the emission cross section will be taken equal to 3 × 10⁻²⁵ m² [38]. The ions are resonantly pumped using WGM around 1.48 μm. Figure 1 is a sketched of the two setups used in our experiments. In the first setup shown in Fig. 1(a), the pump is injected and the laser signal is extracted by a tapered fiber whose diameter is reduced to about 1 μm. In the second setup the pump is inserted using an half tapered fiber with a diameter reduced to less than 2 μm. The laser signal is collected in the counter-propagating mode using the same fiber as sketched in Fig. 1(b). The latter configuration is used to obtain dual emission operation for microwave applications [8]. We have tested both single-mode or multi-mode pumping regime using laser diodes emitting up to 100 mW around 1480 nm. The single-mode pumping is achieved thanks to a narrow linewidth (< 10 kHz) single-mode laser diode whereas for multi-mode pumping we use an usual broadband Fabry-Perot laser diode. The laser emission is measured using an InGaAs photodiode with a responsivity of 1 A/W at 1550 nm and a bandwidth of 1.2 GHz. The electrical signal is amplified using a transimpedance amplifier (TIA) with a maximal gain of 10⁶ V/A and a bandwidth of 1.2 MHz. The DC component giving the laser emitting power measured with a voltmeter (V) and filtered out using a DC block (cut-off frequency of 1 Hz).

Fig. 1 Experimental setups. a) tapered fiber setup, b) half tapered fiber setup. ISO : optical isolator, AV : variable attenuator, PC : polarization controller, WDM : wavelength multiplexer, PhD : InGaAs photodiode, TIA : transimpedance amplifier, V : voltmeter, ESA : electrical spectrum amplifier. i is the generated photo-current at the output of the photodiode.

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3.2. RIN measurement method

From an experimental point of view, the ESA enables us to measure the power spectral density (PSD) N_tot of the photo-current i after amplification which is characterized by a transfer function H(ω). This is related to the thermal noise N_kT, shot-noise N_sn and RIN N_RIN PSD by:

N_{tot} = N_{kT} + N_{sn} + N_{RIN} .

The experimental thermal noise PSD N_kT is measured with no signal at the photodiode (i = 0). The shot noise PSD reads

N_{sn} = 2 {| H (ω) |}^{2} Riq

where q is the elementary charge and R the load resistance. This value can also be experimentally obtained using a reference source without RIN giving the same photo-current i at the photodiode output. By noticing that

N_{RIN} = R {| H (ω) |}^{2} i^{2} \times RIN,

we obtain the experimental value of the RIN by

RIN = \frac{2 q}{i} (\frac{N_{tot} - N_{kT}}{N_{sn}} - 1),

where we deduce the photo-current i from the voltage V obtain thanks to the voltmeter V by i = V/G where G is the transimpedance gain.

4. Results

We used the same methodology for all the results presented in this section. We compare several experimental configurations and different microspheres with comparable diameters. The RIN spectra are all recorded for the maximal laser power obtained with a single-mode operation. This is checked using a spectral analyser (confocal cavity) with a spectral resolution of 100 MHz (not shown on Fig. 1). Note that the single-mode operation can also be checked by making sure that only one fundamental relaxation oscillation frequency is present in the RIN spectrum. In Fig. 2, RIN measurements are reported in the case of broadband pump laser a) in the half tapered fiber configuration and b) in the tapered fiber configuration. The RIN spectra show resonances (relaxation peak) which are the signature of a class-B laser operation. Harmonics of this relaxation oscillation frequency are also clearly visible on both spectra. Up to three harmonics (326, 488, 640 kHz) of the relaxation oscillation frequency (163 kHz) are present on Fig. 2(a). Note that on both spectra a broad weak peak at 34 kHz on Fig. 2(a) and 41 kHz on Fig. 2(b) can be attributed to a second laser mode near threshold which has no influence on our analysis. The experimental data are manually fitted using several free parameters: i) the pumping rate r and the photon lifetime τ_ph which give the position, the width and the amplitude of the fundamental resonance ii) the values of D_NN and D_FF influencing the global noise level and iii) the value of κ which unambiguously sets the ratio between the fundamental and first harmonic peaks. Our second order development is in good agreement for the RIN levels both at the spiking frequency and its first harmonic. To obtain a third peak in the simulation we would have developed δN(t) an δF(t) to the third order. In this case, the products of first and third order terms would give contributions at the first harmonic with amplitude similar to the third peak which should be neglected. Numerical values of the fitting parameters are given in Table 1. Error bars on the fitting parameters are set using the following method. Variations of r of about 1 % gives a change of 1 kHz. 10 % error on τ_ph induces variations of 1.5 dB on the relaxation peak height. Variations of D_NN up to 30 % gives only 1 dB variation in the low frequency white noise. D_FF influences the value of the amplitude of the relaxation peak, a 20 % variation induces 1 dB variation of the peak. Finally, a change in κ of 10 % gives a variation of 1 dB in the relative amplitude of the fundamental and harmonic resonances. The external Q-factor Q_e can be obtained from the operation wavelength λ₀ and the photon lifetime by Q_e = 2πcτ_ph/λ₀ and is also given in Table 1. These values are in the range of 10⁵ – 10⁹ which can be achieved with our experimental setup [37,39]. WGM are labelled using three quantum numbers (n, ℓ, m). n is the radial order of the mode, ℓ is the orbital number and the azimuthal number m is such that ℓ − |m| + 1 gives the number of maxima of the field in the polar direction. The volume of the WGM is deduced from κ (see Table 1) using Eq. (5). By comparing this value to numerical calculations carried out from the theoretical model [40]

V_{m} = \frac{∭_{ℝ^{3}} w (r) d^{3} r}{max [w (r)]}

where w(r) is the electromagnetic energy density deduced from the MIE theory [41], we can deduce the quantum numbers associated with the laser WGM. For the microsphere corresponding to Fig. 2(a), two sets of quantum numbers (n, ℓ − |m|) are found: (1, 5) or (2, 1), whereas for Fig. 2(b) we obtain the pair (1, 2). These quantum numbers are typical values of what can be characterized with our experimental setup [42]. Since the tapered fiber configuration seems to produce less noise at low frequencies, we tested it with a narrow linewidth pump laser. The results are given in Fig. 3. The fit gives an external Q-factor Q_e = 5.7 × 10⁷ and a WGM with n = 2 and ℓ − |m| = 1. Again, these values are totally consistent with what is usually observed with this kind of WGM microcavity. Finally, in this pumping configuration, the noise at low frequencies is reduced by one order of magnitude in comparison with the broadband pumping. This effect could come from the fact that, for a narrow pump laser, less non-efficient WGM pump modes are excited resulting in a lower excess noise. This assumption could also been done for the reduction of the low frequency noise when using a tapered fiber which allows a more transverse-mode selective pumping. As a remark, the RIN measured in WGM lasers at low frequency is 3 orders of magnitude higher than that of fiber lasers [43]; at the spiking frequency, the RIN can be even 6 orders of magnitude greater.

Fig. 2 RIN measurement for a broadband pumping. a) Half tapered fiber configuration and b) tapered fiber configuration. The physical and fit parameters are given in Table 1.

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Fig. 3 RIN measurement in the case of a narrow linewidth pump laser and a tapered fiber configuration. The physical and fit parameters are given in Table 1.

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Table 1. Physical and fit parameters for the WGM laser shown in the three previous figures. a : radius of the sphere, P_L : WGM laser power. We also give the deduced parameters Q_e and V_m. The mode volume is calculated in TE since the theoretical resonance wavelength values match well with experimental data in this polarization.

View Table

5. Conclusion

We developed a RIN model for class-B lasers including nonlinear coupling of population inversion and photon number fluctuations. This model is well suited for the description of low mode volume WGM lasers for which this coupling is not negligible. The model has been checked by characterizing noise properties of erbium-doped glass microspheres under several pumping configuration. We found that single-mode pumping using a tapered fiber is the most promising configuration for getting small low frequency intensity noise. The parameters obtained from the comparison between experiments and our model are consistent. In particular we have shown that the mode volume and the quantum numbers of the WGM laser could be deduced from the RIN spectrum analysis. This can be useful in the case of strongly multimode WGM microcavities for which dedicated experiments must be implemented to obtain a full spatial mode characterization. Furthermore, the fitting process could be improved by developing it to higher orders. Our approach could be extended to WGM laser frequency noise and exploited in the analysis of the whole noise properties of micro-lasers used in sensor applications.

Funding

Centre National d’Études Spatiales (CNES); Région Bretagne (ARED).

Acknowledgments

J.-B. Ceppe acknowledges support from the Centre National d’Études Spatiales (CNES) and Région Bretagne (ARED). Y. Dumeige is member of the Institut Universitaire de France. The authors acknowledge fruitful discussions with T. Chartier and S. Trebaol.

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	Fig. 2(a)	Fig. 2(b)	Fig. 3
2a (μm)	85	90	90
P_L (nW)	57.0	105	129
λ₀ (nm)	1563.6	1565.1	1565.7

r	9.5 ± 0.1	3.8 ± 0.1	6.5 ± 0.1
τ_ph (ns)	40 ± 4	(1.0 ± 0.1) × 10²	47 ± 5
D_NN (s⁻¹)	(2.0 ± 0.7) × 10¹⁹	(2.5 ± 0.8) × 10¹⁷	(1.5 ± 0.5) × 10¹⁶
D_FF (s⁻¹)	(1.5 ± 0.3) × 10¹⁴	(8.5 ± 1.7) × 10¹⁴	(3.0 ± 0.6) × 10¹⁴
κ (s⁻¹)	(2.0 ± 0.2) × 10⁻²	(2.0 ± 0.2) × 10⁻²	(2.2 ± 0.2) × 10⁻²

Q_e	(4.8 ± 0.5) × 10⁷	(1.2 ± 0.1) × 10⁸	(5.7 ± 0.6) × 10⁷
V_m (μm³)	(3.0 ± 0.6) × 10³	(3.0 ± 0.6) × 10³	(2.7 ± 0.5) × 10³

Theoretical and experimental analysis of rare earth whispering gallery mode laser relative intensity noise

Abstract

1. Introduction

2. Model

2.1. Notations

2.2. Laser modelling

2.3. Static solution above threshold

2.4. Small signal analysis

2.5. Fluctuation calculation

2.6. RIN expression

2.7. First order RIN

2.8. Second order RIN

3. Experimental method

3.1. Experimental setup

3.2. RIN measurement method

4. Results

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (3)

Tables (1)

Equations (50)

Optics Express