Analytical modeling of dual-frequency solid-state lasers including a buffer reservoir for noise cancellation

Kevin Audo; Abdelkrim El Amili; Mehdi Alouini

doi:10.1364/OE.26.008805

1. Introduction

The Buffer Reservoir (BR) approach for reducing intensity noise of solid-state lasers has opened new interesting perspectives. Indeed, this method which relies on the insertion of a very slight nonlinear absorption in the laser cavity offers a high bandwidth, passive, efficient and compact noise reduction solution overtaking all other known alternatives. This approach has been successfully implemented in single frequency lasers using Two Photon Absorption (TPA) [1,2] as well as Second Harmonic Generation Absorption (SHGA) [3] leading to an almost complete cancellation of the excess intensity noise at the relaxation oscillation frequency (RO), i.e., 50 dB reduction. A model based on rate equations of a single frequency laser including a BR has been developed to explain the origin of the intensity noise reduction [1]. In order to accurately describe the effects of the BR mechanism in a single mode laser, an additional rate equation ruled by the cross-section and time constant of the nonlinear absorber must be included in addition to the usual population and intensity rate equations. This additional rate equation suggested that the BR does not directly act as a “noise eater” but that the BR breaks the dynamical behavior of the laser damping its RO and consequently reducing the associated excess noise. This suggestion has been experimentally confirmed using SHGA in a Nd:YAG laser [3]. In this framework, it was shown that the time constant ruling the BR mechanism is essential for predicting both the noise reduction level but also the associated frequency shift of the resonant peak.

Dual mode solid-state lasers would highly benefit from the BR approach, intensity noise reduction of such lasers being a difficult task [4,5]. It was then tested experimentally but with less success. Indeed, in dual mode lasers, each mode exhibits two resonant frequencies associated to in-phase and anti-phase fluctuations that enhance the intensity noise of the considered laser mode [6,7]. While the BR mechanism was shown to be efficient in reducing the in-phase noise, it failed in reducing efficiently the anti-phase noise. Nevertheless a tradeoff between in-phase and anti-phase noise reduction was found by spatially separating the two modes in the BR. 25 dB of noise reduction was then obtained [8]. In order to go further, a predictive and accurate theoretical model is required. The theoretical description of the dynamical behavior, and then the intensity noise, of a dual-frequency laser including a BR must be conducted cautiously due to the existing interplay between the two modes in the active medium but also in the BR. In addition the time constant of the BR has to be taken into account since it was shown in single frequency lasers to play a detrimental role.

In this paper, we propose to derive a rigorous analytical model describing the dynamical behavior of a dual-frequency solid-state laser in presence of a BR. The predictive capability of this model will be tested and validated experimentally in the most general case using a dual mode codoped Er,Yb:Glass laser including TPA in different configurations.

2. THEORETICAL MODEL

2.1 Laser rate equations in the framework of Buffer Reservoir approach

In this section, we present a theoretical model which describes the dynamics of a CW dual-frequency solid-state laser including a nonlinear absorber (NLA). An Er,Yb:glass material is particularly chosen as gain medium because of its great interest in telecom applications. Moreover, an analytical description of codoped active medium is general enough so that the model can be easily transposed to any solid-state lasers.

2.1.1 Single mode operation

Before moving to the dual-frequency description, let us first start with a single-frequency laser including a NLA. In the most general case, the Er,Yb:glass active medium can be described with a six level energy diagram which couples the two dopants (see Fig. 1) [9,10].

Fig. 1 Energy levels of the Er:Yb active medium.

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Few approximations simplifying this level scheme can be made [11]. First, the lifetimes of the two levels ⁴I_11/2 and ⁴F_9/2 are supposed to be much shorter than the other lifetimes [12]. These levels can be thus adiabatically eliminated allowing to neglect the probabilities of inverse transfer $k_{T R, i n v}^{}$ and $k_{T R, i n v}^{'}$ coupling the Erbium excited levels ⁴F_9/2 and ⁴I_11/2 to the Ytterbium excited level ²F_5/2. Secondly, the transfer coefficients $k_{T R}^{}$ and $k_{T R}^{'}$ are considered to be equal. The insertion of the NLA inside the cavity is taken into account by including the intensity- and time-dependent nonlinear losses, $P_{N L}$ , through an additional rate equation. The single-frequency laser rate equations are thus expressed as:

\frac{d N_{2, Y b}}{d t} = σ_{Y b} I_{p} (N_{Y b} - 2 N_{2, Y b}) - γ_{Y b} N_{2, Y b} - k_{T R} N_{E r} N_{2, Y b},

\frac{d N}{d t} = - 2 σ I N - γ (N_{E r} + N) + k_{T R} N_{2, Y b} (N_{E r} - N),

\frac{d I}{d t} = \frac{c}{2 L} 2 e σ N I - γ_{c} I - γ_{P} I,

\frac{d P_{N L}}{d t} = ψ_{N L} I - γ_{N L} P_{N L},

where

N_{2, Y b}

,

N_{Y b} = N_{1, Y b} + N_{2, Y b}

,

N = N_{2, E r} - N_{1, E r}

,

N_{E r} = N_{1, E r} + N_{2, E r}

are respectively the populations of the ²F_5/2 level, the total Ytterbium population (Yb³⁺), the Erbium (Er³⁺) population inversion and the total Erbium population.

P_{N L}

is the time-dependent losses induced by the nonlinear mechanism, and

γ_{P} = \frac{c}{2 L} P_{N L}

is the nonlinear loss rate.

γ

and

γ_{Y b}

are the population inversion relaxation rate of Er³⁺ and Yb³⁺ respectively.

σ

and

σ_{Y b}

are respectively the emission cross-section of Er³⁺ and Yb³⁺.

I

is the laser photon flux and

I_{p}

is the photon flux of the pump laser.

e

and

L

are respectively the optical lengths of the active medium and of the resonator,

c

is the velocity of the light.

γ_{c}

is the photon decay rate which takes into account the linear losses due to the laser cold cavity as well as the residual linear losses coming from the nonlinear absorber.

γ_{N L}

is the inverse of the time response of the nonlinear mechanism, and is an essential parameter since it is shown to directly rules the noise reduction bandwidth [1]. Finally,

ψ_{N L}

is the cross-section of the nonlinear losses mechanism. Unlike the set of equations proposed in [1], we must emphasis that the rate equation of the Yb³⁺ population in the excited state must be taken into consideration. Indeed, as will be shown later, the dynamics of the Yb³⁺ excited state population is found to be very important in the overall behavior of the laser.

2.1.2 Dual mode operation

Starting from these equations, one can now extend them to dual mode operation. A dual-frequency laser sustains the oscillation of x- and y- orthogonally polarized modes. Each mode is thus ruled by two sets of rate equations, i.e., the inversion population and intensity (Eqs. (5)-(9)). A cross saturation term, $κ$ , is introduced in order to take into account the interaction between the two population inversions. This parameter is directly linked to the Lamb’s coupling constant by $C = κ^{2}$ [13]. It is must be noted that the value of the coupling constant, and consequently $κ$ , defines the Er³⁺ concentration which enters into picture for the x- and y-modes. Indeed, the total erbium population being shared between the two modes while constant, the more the two modes are coupled, the lower the erbium concentration associated to one mode is. Consequently, each mode interacts with an Er³⁺ population that reads $N_{E r, x} = N_{E r, y} = \frac{N_{E r}}{1 + κ}$ . Moreover, it is assumed that the Yb³⁺ excited state feeds in the same fashion the two excited states associated to the two Er³⁺ populations. In this framework, the population inversions for the two modes are equal. Finally, the results obtained in [8] suggest that the nonlinear losses have to be partially decoupled in order to model a dual-frequency laser including a BR without making any assumption on the NLA process implemented into the laser. Indeed, it has been shown that, while the in-phase noise is successfully reduced by the BR, the anti-phase noise is still present unless the two modes are partially decoupled in the nonlinear medium.

To describe this behavior, one has to consider two coupled rate equations for the nonlinear medium. We thus also introduce a coupling rate $η$ that is intended to describe the interaction of the two modes in the nonlinear medium. This way, if a polarization selection rule exists in the nonlinear medium as it is the case with second-harmonic generation (SHG), its effect can be taken into account. We finally end up with the following set of rate equations:

\frac{d N_{2, Y b}}{d t} = σ_{Y b} I_{p} (N_{Y b} - 2 N_{2, Y b}) - γ_{Y b} N_{2, Y b} - k_{T R} N_{E r} N_{2 Y b},

\frac{d N_{x}}{d t} = - 2 σ (I_{x} + κ I_{y}) N_{x} - γ (N_{E r, x} + N_{x}) + k_{T R} N_{2, Y b} (N_{E r, x} - N_{x}),

\frac{d N_{y}}{d t} = - 2 σ (I_{y} + κ I_{x}) N_{y} - γ (N_{E r, y} + N_{y}) + k_{T R} N_{2, Y b} (N_{E r, y} - N_{y}),

\frac{d I_{x}}{d t} = \frac{c}{2 L} 2 e σ N_{x} I_{x} - γ_{c, x} I_{x} - γ_{p, x} I_{x},

\frac{d I_{y}}{d t} = \frac{c}{2 L} 2 e σ N_{y} I_{y} - γ_{c, y} I_{y} - γ_{p, y} I_{y},

\frac{d P_{N L, x}}{d t} = ψ_{N L, x} (I_{x} + \frac{ψ_{N L, y}}{ψ_{N L, x}} η I_{y}) - γ_{N L} P_{N L, x},

\frac{d P_{N L, y}}{d t} = ψ_{N L, y} (I_{y} + \frac{ψ_{N L, x}}{ψ_{N L, y}} η I_{x}) - γ_{N L} P_{N L, y} .

The noise properties of the laser around the relaxation oscillations frequencies can be studied through the analysis of its frequency response. To this aim, we derive the transfer functions of the two modes. These transfer functions are derived from the rate equations by introducing a small perturbation starting from the equilibrium state of the laser.

2.2 Steady-state regime

The steady state equations are computed by setting to zero all the rate equations. At threshold, the laser intensity being null, the nonlinear absorber does not play any role. The population inversion for each mode is obtained when the unsaturated gain and losses are equal, leading to:

N_{i}^{t h} = \frac{γ_{c, i}}{2 e σ} \frac{2 L}{c},

where i,j = {x, y} stands for the x- polarized and y- polarized modes respectively. Above threshold, the stationary value of the Yb³⁺ excited state population and the Er³⁺ inversion population are given by:

N_{2, Y b}^{s t a t} = \frac{σ_{Y b} I_{p} N_{Y b}}{2 σ_{Y b} I_{p} + γ_{Y b} + k_{T R} N_{E r}},

N_{i}^{s t a t} = \frac{γ_{c, i}^{s t a t} + γ_{p, i}^{s t a t}}{2 e σ c / 2 L} .

The intensity-depend losses experienced by the x- and y- polarized modes can be expressed as:

P_{N L, i}^{s t a t} = \frac{ψ_{N L, i} I_{i}^{s t a t} + ψ_{N L, j} η I_{j}^{s t a t}}{γ_{N L}},

and the stationary values of the intensity of each modes read:

I_{i} = \frac{I_{s a t}}{1 - κ^{2}} [(\frac{N_{o, i}}{N_{i}^{s t a t}} - 1) - κ (\frac{N_{o, j}}{N_{j}^{s t a t}} - 1)],

with

I_{s a t} = \frac{w_{p} + γ}{2 σ}

,

w_{p} = k_{T R} N_{2, Y b}^{s t a t}

and

N_{o, i} = \frac{w_{p} - γ}{w_{p} + γ} N_{E r, i}

.

The intensities and nonlinear losses can be evaluated by calculating the population inversion $N_{x}$ and $N_{y}$ . For that purpose, we inject Eqs. (15) and (16) into Eq. (14) to end up with:

N_{x}^{s t a t} - N_{x}^{t h} - A_{x y} (\frac{N_{0, x}}{N_{x}^{s t a t}} - 1) - B_{x y} (\frac{N_{0, y}}{N_{y}^{s t a t}} - 1) = 0,

N_{y}^{s t a t} - N_{y}^{t h} - A_{y x} (\frac{N_{0, y}}{N_{y}^{s t a t}} - 1) - B_{y x} (\frac{N_{0, x}}{N_{x}^{s t a t}} - 1) = 0,

with

A_{i j} = \frac{N_{N L, i}^{s a t} - κ η N_{N L, j}^{s a t}}{1 - κ^{2}}

,

B_{i j} = \frac{η N_{N L, j}^{s a t} - κ N_{N L, i}^{s a t}}{1 - κ^{2}}

and

N_{N L, i}^{s a t} = \frac{ψ_{N L, i}}{2 e σ γ_{N L}} I_{s a t}

.

The derivation of the coupled Eqs. (17) and (18) leads to quartic equations which can be solved analytically by using Ferrari’s method [14].

2.3 Transfer functions

The knowledge of the steady state values allows us now to calculate the fluctuations of the intensities around their steady state values. In order to rigorously describe the RIN of the laser, both pump intensity fluctuations and intrinsic photon fluctuations coming from the amplified spontaneous emission have to be taken into account as noise sources. From a mathematical point of view and in the limit of semi-classical approximation, the photon fluctuations can be modeled as fluctuations of the photon decay rate. The laser RIN is thus given by [15]:

RIN(ω) = {| H_{pump} |}^{2} (\frac{S_{pump}}{I_{p}^{2}}) + {| H_{photon} |}^{2} (\frac{S_{photon}}{γ_{c}^{2}}),

where

H_{pump,i} = \frac{\frac{δ I_{i}}{I_{i}^{s t a t}}}{\frac{δ I_{p}}{I_{p}^{s t a t}}}

and

H_{photon,i} = \frac{\frac{δ I_{i}}{I_{i}^{s t a t}}}{\frac{δ γ_{c, i}}{γ_{c, i}^{s t a t}}}

are respectively the transfer functions related to the pump intensity fluctuations and to the cavity decay rate fluctuations.

\frac{S_{p u m p}}{I_{p}^{2}}

and

\frac{S_{p h o t o n}}{γ_{c}^{2}}

account for the spectral density of the fluctuation of pump intensity and intra-cavity photons. First, we analyze the intensity response of the two modes to small photon fluctuations arising from amplified spontaneous emission and possibly from residual loss fluctuations in the laser cavity. To take into account these fluctuations, we introduce small perturbations to

γ_{c, i}

at frequency

ω

around zero average value meaning

γ_{c, i} = γ_{c, i}^{s t a t} + δ γ_{c, i} e^{i ω t}

. This will in turn induce small fluctuations, at the same frequency around the stationary values, of the population inversions

N_{i} = N_{i}^{s t a t} + δ N_{i} e^{i ω t}

, the intensities

I_{i} = I_{i}^{s t a t} + δ I_{i} e^{i ω t}

, and the nonlinear losses

P_{N L, i} = P_{N L, i}^{s t a t} + δ P_{N L, i} e^{i ω t}

. By injecting these expressions in Eqs. (5)-(11), the fluctuations of

I_{x}

and

I_{y}

can be expressed as:

δ I_{x} D_{x} (ω) = T_{x} (ω) δ I_{y} + δ γ_{c, x} I_{x}^{s t a t},

δ I_{y} D_{y} (ω) = T_{y} (ω) δ I_{x} + δ γ_{c, y} I_{y}^{s t a t},

where:

\begin{matrix} D_{i} (ω) = i ω + γ_{c, i}^{s t a t} + \frac{ψ_{N L, i} I_{i}^{s t a t}}{γ_{N L}} \frac{c}{2 L} (1 + \frac{γ_{N L}}{i ω + γ_{N L}} + \frac{η ψ_{N L, j} I_{j}^{s t a t}}{ψ_{N L, i} I_{i}^{s t a t}}) \\ - 2 e σ \frac{c}{2 L} [N_{i}^{s t a t} + A_{i} I_{i}^{s t a t}], \end{matrix}

T_{i} (ω) = \frac{c}{2 L} [2 e σ κ A_{i} - \frac{η ψ_{N L, j}}{i ω + γ_{N L}}] I_{i}^{s t a t},

A_{i} (ω) = - 2 σ \frac{N_{i}^{s t a t}}{i ω + 2 σ (I_{i}^{s t a t} + κ I_{j}^{s t a t} + I_{s a t})} .

It is worth noticing that in Eqs. (20) and (21), the evolution of the intensity of two modes depends on each other.

With Eqs. (20) and (21), the evolution of each intensity can be written as a function of the photon fluctuations:

δ I_{i} = \frac{T_{i} δ γ_{c, j} I_{j}^{s t a t} + D_{j} δ γ_{c, i} I_{i}^{s t a t}}{D_{x} D_{y} - T_{x} T_{y}} .

One can see that the photon fluctuations and the nonlinear losses affecting the x- mode will influence the evolution of the intensity of the y- mode and vice versa. This appears through the parameters

T_{x}

and

T_{y}

which include the coupling between the two modes in the active medium and in the nonlinear absorber. For example, if the two modes are not coupled neither in the active medium nor in the nonlinear absorber, then

κ = 0

and

η = 0

, leading to

T_{x} = 0

and

T_{y} = 0

.

In order to compute the transfer function, one can now assume that the photon fluctuations arising from spontaneous emission are equal and not correlated for the two modes:

〈 {| δ I_{s p, x} |}^{2} 〉 = 〈 {| δ I_{s p, x} |}^{2} 〉 = 〈 {| δ I_{s p} |}^{2} 〉,

〈 δ I_{s p, x} δ I_{s p, y}^{*} 〉 = 0 .

Furthermore, we consider here that the stationary values of both mode intensities are equal, i.e.

I = I_{x} = I_{y}

. The transfer functions can then be deduced by solving Eqs. (20) and (21) and can be expressed as:

| H_{photon,x} (ω) |^{2} = {| \frac{\frac{δ I_{x}}{I_{x}^{s t a t}}}{\frac{δ γ_{c, x}}{γ_{c, x}^{s t a t}}} |}^{2} = {(γ_{c, x}^{s t a t})}^{2} \frac{| T_{x} |^{2} + | D_{y} |^{2}}{| D_{x} D_{y} - T_{x} T_{y} |^{2}},

| H_{photon,y} (ω) |^{2} = {| \frac{\frac{δ I_{y}}{I_{y}^{s t a t}}}{\frac{δ γ_{c, y}}{γ_{c, y}^{s t a t}}} |}^{2} = {(γ_{c, y}^{s t a t})}^{2} \frac{| T_{y} |^{2} + | D_{x} |^{2}}{| D_{x} D_{y} - T_{x} T_{y} |^{2}} .

Let us now focus on the transfer function due to the pump fluctuations. Similarly, we assume a small perturbation of the pump intensity at frequency ω,

I_{p} = I_{p}^{s t a t} + δ I_{p} e^{i ω t}

. This induces a variation at the same frequency around the stationary value of the Yb³⁺ excited state population

N_{2, Y b} = N_{2, Y b}^{s t a t} + δ N_{2, Y b} e^{i ω t}

, as well as around the stationary values of the Er³⁺ population inversion

N_{i} = N_{i}^{s t a t} + δ N_{i} e^{i ω t}

, the mode intensities

I_{i} = I_{i}^{s t a t} + δ I_{i} e^{i ω t}

, and the nonlinear losses

P_{N L, i} = P_{N L, i}^{s t a t} + δ P_{N L, i} e^{i ω t}

. The injection of these expressions in Eqs. (5)-(11) leads to two equations describing the fluctuations of the two orthogonal modes

I_{x}

and

I_{y}

:

δ I_{x} D_{x} (ω) = T_{x} (ω) δ I_{y} + J_{x} (ω) δ I_{p},

δ I_{y} D_{y} (ω) = T_{y} (ω) δ I_{x} + J_{y} (ω) δ I_{p},

where:

J_{i} (ω) = - \frac{c}{2 L} 2 e σ A_{i} k_{T R} \frac{β}{2 σ} (\frac{N_{E r, i}}{N_{i}^{s t a t}} - 1) I_{i},

and:

δ N_{2, Y b} = \frac{σ_{Y b} (N_{Y b} - 2 N_{2, Y b}^{s t a t})}{i ω + 2 σ_{Y b} I_{p}^{s t a t} + γ_{Y b} + k_{T R} N_{E r}} δ I_{p} = β δ I_{p} .

One can notice in Eq. (30) and (31) that the pump fluctuations

δ I_{p}

are weighted by the transfer function of the Yb³⁺ excited state population through

J_{i}

. As we will show in the following part, this transfer function is essential to describe accurately the conversion of the pump noise to the laser intensity noise.

So far, it has been assumed that the two modes experience the same pump fluctuations $δ I_{p}$ . To keep the model as general as possible, it has to reflect the fact that the influence of the pump intensity fluctuations on the two modes might be different depending on the laser configuration. Indeed, these two modes can be spatially separated in the active medium as done in [8,16] such that they may experience different pump fluctuations, especially when the pump intensity has a non-homogeneous spatial distribution. As a consequence, the pump fluctuations associated to x- and y- polarized mode, i.e. $δ I_{p, x}$ and $δ I_{p, y}$ , are taken with identical statistical average values, but are more or less correlated similarly to [17]. Under these conditions one can write:

〈 {| δ I_{p, x} |}^{2} 〉 = 〈 {| δ I_{p, y} |}^{2} 〉 = 〈 {| δ I_{p} |}^{2} 〉,

〈 δ I_{p, x} δ I_{p, y}^{*} 〉 = ξ 〈 {| δ I_{p} |}^{2} 〉 e^{i φ},

with

ξ \in [0, 1]

and

φ \in [0, 2 π [

are the amplitude and phase of correlation respectively. In the next, it is assumed that both pump fluctuations are in phase leading to

φ = 0

. Thus, the transfer function related to the pump fluctuations are given by:

| H_{p u m p, x} (ω) |^{2} = {| \frac{\frac{δ I_{x}}{I_{x}^{s t a t}}}{\frac{δ I_{p}}{I_{p}^{s t a t}}} |}^{2} = {(\frac{I_{p}^{s t a t}}{I_{x}^{s t a t}})}^{2} \frac{| T_{x} J_{y} |^{2} + | D_{y} J_{x} |^{2} + ξ (T_{x} J_{y} D_{y}^{*} J_{x}^{*} + T_{x}^{*} J_{y}^{*} D_{y} J_{x})}{| D_{x} D_{y} - T_{x} T_{y} |^{2}},

| H_{p u m p, y} (ω) |^{2} = {| \frac{\frac{δ I_{y}}{I_{y}^{s t a t}}}{\frac{δ I_{p}}{I_{p}^{s t a t}}} |}^{2} = {(\frac{I_{p}^{s t a t}}{I_{y}^{s t a t}})}^{2} \frac{| T_{y} J_{x} |^{2} + | D_{x} J_{y} |^{2} + ξ (T_{y} J_{x} D_{x}^{*} J_{y}^{*} + T_{y}^{*} J_{x}^{*} D_{x} J_{y})}{| D_{x} D_{y} - T_{x} T_{y} |^{2}},

3. Experimental analysis, model validation and discussion

3.1 Description of the experimental setup

Now, we present the dual-frequency laser that has been used to validate our model. We describe in this section the laser setup (see Fig. 2). The laser is composed of a 1.5-mm long Er,Yb:glass active medium pumped with a 976 nm diode laser providing ~450 mW. The plan-concave linear cavity is 4.9-cm long. One side of the active medium forms the input mirror coated for reflectivity at 1.55 µm (R>99.9%) and 95% transmission at the pump wavelength. The output coupler is a 5 cm curvature concave mirror with 99.5% reflectivity at 1.55 µm. In order to ensure a stable oscillation of two orthogonal polarized modes, a small birefringence is introduced using a 200-µm long YVO₄ plate separating the modes by 20-µm in the active medium. Depending on the laser configuration, a silica etalon can be inserted into the cavity to force single mode operation on both polarizations. The laser beam is coupled into an optical fiber and split in two parts using an optical coupler 99%-1%. One arm is dedicated to the RIN spectra measurements. It is composed by an InGaAs photodiode (Bandwidth: DC-1 GHz) followed by a low noise homemade amplifier (Gain: 50 dB and Bandwidth: 1 kHz-500 MHz). The electrical signal is finally recorded with a 3.6 GHz bandwidth Rohde&Schwarz electrical spectrum analyzer. The second arm is connected to a Fabry-Perot analyzer in order to insure that no mode hopping occurs during data acquisition. Each mode is analyzed independently using a polarizer placed at the output of the laser. In the following measurements, the optical power of both polarizations is carefully balanced by slightly adjusting the angle of the silica etalon or the NLA plate.

Fig. 2 Dual-frequency laser setup with the measurement bench. The laser is composed by an Er,Yb:glass laser. A polarizer at the output of the laser allows studying each mode. The measurement bench is composed by two arms. The first arm is dedicated for RIN spectrum measurement. It is composed by an InGaAs photodiode, an electrical amplifier and an electrical spectrum analyzer (ESA). The second arm is used to ensure that the laser remains single mode during measurements using a Fabry-Perot analyzer.

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3.2 Experimental analysis of the laser intensity noise without Buffer Reservoir

Let us consider first the dual-frequency laser without BR. This offers a simple way to quantify the contribution of the different noise sources on the intensity noise spectrum of the laser. In our model, the cross-sections $ψ_{N L, i}$ of the BR for the x- and y- polarized modes are thus set to zero.

The stationary values for both populations and intensities are deduced from Eqs. (14) and (16) respectively. Moreover, assuming that the two modes x- and y- experience equal intracavity losses $γ_{c} = γ_{c,}_{x} = γ_{c, y}$ the transfer functions in Eqs. (28)-(29) and (36)-(37) simplify further. The resonance frequencies, corresponding to in-phase and anti-phase noise, can be easily found by minimizing the denominator of the transfer function:

f_{I N} = \frac{1}{2 π} \sqrt{\frac{c}{2 L} 4 e σ^{2} N^{s t a t} I^{s t a t} (1 + κ)},

f_{A N} = \frac{1}{2 π} \sqrt{\frac{c}{2 L} 4 e σ^{2} N^{s t a t} I^{s t a t} (1 - κ)} .

It is worth mentioning that theses resonance frequencies are directly linked to the cross-saturation term

κ

which rules the coupling strength between the two modes in the active medium. The value of the Lamb coupling constant can be measured [18] or easily retrieved using these frequencies as already reported in [7]:

C = {[\frac{1 - {(\frac{f_{A N}}{f_{I N}})}^{2}}{1 + {(\frac{f_{A N}}{f_{I N}})}^{2}}]}^{2} .

In Fig. 3(a), we present the experimental RIN spectra obtained for both the x-polarized (1) and y- polarized (2) modes. By fitting the model to the experimental data, the laser parameters are extracted (see Table 1). By using Eq. (40), the two laser modes are found to be coupled in the active medium with

C = 0.52

. It corresponds to

κ = 0.72

. The correlation amplitude is found to be

ξ = 0.8

.

S_{pump} / I_{p}^{2}

is directly extracted from the measured RIN spectrum of the pump reported in Fig. 3(b).

S_{photon} / γ_{c}^{2}

is in the order of 0.8 0.10⁻¹⁸ Hz⁻¹.

Fig. 3 a) Experimental RIN spectra for the x- (1) and the y- polarized (2) mode (RBW: 100 Hz, I_phd = 1 mA). b) RIN spectrum of the pump laser (RBW: 100 Hz, I_phd = 1mA).

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Table 1. Values of the parameters used for the computation of the RIN spectra

View Table

Using the experimental parameters of Table 1, the theoretical RIN spectrum (Eq. (19)) is in a very good agreement with the experimental RIN spectra (see Fig. 4(a)). Specifically, the model is able to accurately predict both the resonance peaks as well as their amplitude levels. The theoretical RIN spectra of the x- polarized and y- polarized modes are the same since the photon decay time and the pumping rate are the same for the two modes.

Fig. 4 a) Theoretical RIN spectra for the two modes. b) Comparison of experimental and theoretical RIN spectra.

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Figure 4(b) is an illustration of how accurate is the model. The anti-phase noise peak is found at 31.8 kHz with amplitude of −90 dB/Hz for both experimental and theoretical plots. Similarly, the in-phase noise peak is found at 79.9 kHz with amplitude of −94 dB/Hz for both plots. Moreover, the noise levels aside the excess noise peaks are similar for the theoretical and the experimental RIN spectra. Such perfect agreement would have been not possible if pump and photon fluctuations were not considered together in the modelling. In particular, the level difference between the anti-phase and in-phase peaks can be explained only if the pump transfer function is taken into account. Indeed, the pump laser noise which is converted to laser noise experiences a first order low-pass filter during the energy transfer between Yb³⁺ and Er³⁺ ions. This justifies the fact that we have included in our model a rate equation for the Yb³⁺ populations in their excited state and that this rate equation is not eliminated adiabatically.

Accordingly our model enables to separate the two noise contributions sources in order to check their effect on the laser RIN spectrum (Eq. (19)). As shown in Fig. 5(a), this is done by vanishing the transfer function related to photon fluctuations ${| H_{photon} |}^{2} = 0$ or pump fluctuations ${| H_{pump} |}^{2} = 0$ . Below 25 kHz, the noise from the pump laser clearly dominates over the photon noise whereas this trend is inverted above 80 kHz. In the frequency range between the resonances, both noise sources contribute almost in the same way in the total RIN. As highlighted above, the pump laser noise which impacts the laser RIN is weighted by the Yb³⁺ response. The Yb³⁺ excited state fluctuations to the pump fluctuations are given by:

\frac{δ N_{2, Y b}}{δ I_{p}} = \frac{σ_{Y b} (N_{Y b} - 2 N_{2, Y b}^{s t a t})}{i ω + 2 σ_{Y b} I_{p} + γ_{Y b} + k_{T R} N_{E r}} .

As shown in Fig. 5(b), this transfer function behaves as a low pass filter discriminating the frequencies of the pump fluctuations higher than the inverse of the Yb³⁺ lifetime. This property is intrinsic to codoped active media [15].

Fig. 5 a) Theoretical RIN spectra where only the pump noise (1) and the photon fluctuations (2) are considered as a noise source. b) Transfer function of the Yb³⁺ excited state population.

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3.3 Experimental analysis of the laser noise including a Buffer Reservoir

Let us now focus on the laser including nonlinear losses. The nonlinear absorption can be obtained, for example, by two-photon absorption (TPA) through the insertion of a semiconductor plate in the cavity (see Fig. 6). In this case, we have already proved that the BR time response, i.e. $τ_{N L} = 1 / γ_{N L}$ , is related to the carrier recombination lifetime of the semi-conductor [1]. Thus, for carrier recombination lifetime several orders shorter than the inverse of the relaxation oscillations, the nonlinear absorption will be able to act instantaneously on the intensity fluctuations and thus to reduce resonant noise without any change of the resonant frequencies. When this condition is not fulfilled, the noise reduction is accompanied by a change of resonant frequencies. In order to test our model, we seek here to conduct our study in the most general framework. We thus choose to use a Si plate as BR. Indeed, in this case, the carrier recombination lifetime has been evaluated to be in the order of a few µs, which makes the BR action not instantaneous in regards of excess intensity noise lying on the range of a few hundred kHz. The Si plate has a thickness of 100 µm and acts also in the cavity as an étalon enabling single mode operation on both laser polarization modes.

Fig. 6 Dual-frequency laser with the Si plate bringing the TPA in the laser dynamics. a) In this first configuration, the Si plate is placed between the YVO₄ and the output coupler. b) In the second configuration the Si plate is placed between the active medium and the YVO₄.

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Moreover, we wish in this section to validate the necessity of considering two rate equations for the nonlinear losses. For that purpose, we study in the following two laser configurations with different coupling rates between the modes in the NLA. In order to modify the coupling rate, we change the spatial separation between the two modes in the Si plate. This is achieved by placing the Si plate at two different positions in the cavity with respect to the YVO₄ crystal. Thus, we are able to introduce successively common and selective nonlinear losses on the two laser modes. It is worthwhile to notice that the Si plate is placed close to the active medium for both configurations, i.e., in the Rayleigh range where the laser beam diameter is almost constant and where the photon density is maximum.

3.3.1 Strong mode coupling in the Buffer Reservoir

We first consider the case where the two laser modes are superimposed in the Si plate. The NLA is introduced between the birefringent element and the output mirror (see Fig. 6(a)). The corresponding experimental RIN spectra are represented in Fig. 7. Due to the non-efficient TPA, 10 dB reduction of the in-phase noise peak is achieved while only 4 dB reduction is reached on the anti-phase noise. This is consistent with the results reported in [8] which have been qualitatively explained by the fact that the TPA effect are ruled by the sum of the intensity fluctuations, i.e $δ I_{x} + δ I_{y}$ . However, the noise contribution from the two modes cancels each other at the anti-phase noise frequency. Thus, the NLA is not able to act on the anti-phase noise.

Fig. 7 a) Experimental RIN spectra for the x- (1) and the y- (2) polarized mode (RBW: 100 Hz, I_phd = 1 mA) and theoretical RIN spectrum (3). b) Evolution of the RIN spectrum for different TPA efficiencies.

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Let us now compare our experimental observations with the predictions of our model (Eqs. (5)-(11) and (19)). The laser transfer functions are calculated from the parameters in Table 1. The insertion of the Si plate adds small linear losses leading to $γ_{c} = 2.9 . 10^{8} s^{- 1}$ . Since the two modes are superimposed on the Si plate, the coupling rate $η$ in the NLA can be considered to be equal to 1. It must be noted that this coupling rate might be inferior to 1 if the NLA process is obtained through SHGA or if the TPA process is polarization dependent. The lifetime recombination of the carrier in our Si plate has been evaluated to be in the order of 3 µs [1]. Assuming that the two modes are equally affected by nonlinear losses, the TPA cross-section is thus estimated to be $ψ_{N L, i} = {9.10}^{- 23} {cm}^{2}$ . Under these circumstances, the computed RIN spectra of each mode are similar. As shown in Fig. 7(a), while the in-phase noise is affected by the nonlinear absorption, the anti-phase noise peak remains at the same amplitude. The model is able to predict with a very good agreement the effect of the NLA on the noise characteristics of the laser. Specifically, the reduction of the in-phase noise peak, reaching an amplitude of −104 dB/Hz, is determined with a very good accuracy. By contrast, the anti-phase noise peak does not experience significant reduction as expected with our RIN model (see Fig. 7(b)). This is explained by the fact that the two modes share a common BR whose effect depends on the total intensity fluctuations when this BR is realized through TPA. Besides, the model also takes into account the frequency shift of the noise peaks. Indeed, as the efficiency of the nonlinear mechanism increases, the in-phase noise peak shifts toward higher frequencies. On the contrary, no frequency shift occurs on the anti-phase noise because of its insensitivity to nonlinear absorption. This behavior which has been extensively discussed in [1] is due to the fact that the NLA process is ruled by the carrier recombination lifetime of the Si plate which is found to be at around 3 µs. This lifetime which can be relatively long, as in the present case, make it necessary to introduce a rate equation for nonlinear losses. Obviously, this rate equation can be adiabatically eliminated if the NLA process is fast with respect to the other time constants of the laser. We will see in the following that actually, two rate equations are required if the BR weakly couples the two laser modes.

3.3.2 Weak mode coupling in the Buffer Reservoir

We now consider the case of the weak coupling between the two modes in the NLA, i.e., $η < 1$ . To get experimentally this situation, the Si plate is placed between the active medium and the YVO₄ plate (see Fig. 6(b)). We have already shown in [8] that such spatial separation in the NLA, although weak, gives rise to the reduction of the anti-phase noise peak. As shown in Fig. 8(a) this reduction is about 14 dB for both resonant noise peaks as compared to the case where no TPA is inserted in the laser cavity (see Fig. 3(a)). This spatial separation, reduces the modes overlap in the Si plate and thus the coupling $η$ .

Fig. 8 a) Experimental RIN spectra for the x- (1) and the y- (2) polarized mode (RBW: 100 Hz, I_phd = 1 mA) and theoretical RIN spectrum (3). b) Evolution of the RIN spectrum for different TPA efficiencies.

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In order to estimate the coupling between the two modes in the NLA, we compute their spatial overlapping. Considering two Gaussian beams having the same waist and separated by a distance $d$ , the normalized overlap integral is given by:

R = \frac{\int I_{1} (x, y) I_{2} (x, y) d x d y}{\sqrt{\int I_{1}^{2} (x, y) d x d y \int I_{2}^{2} (x, y) d x d y}} = e^{\frac{- d^{2}}{w_{}^{2}}} .

The Si plate being inserted close to the active medium, we consider that the beam radii are equal in the NLA and in the active medium. The theoretical spectrum of Fig. 8(a) that accurately fit the experimental one (plot (3)), is obtained by adjusting the value of the waist to

w = 26 µm

. According to Eq. (42), this correspond to a mode coupling in the NLA of

η = 0.54

.

The photon decay rate and the cross-section of the nonlinear mechanism are set to $γ_{c} = 2.8 . 10^{8} s^{- 1}$ and $ψ_{N L, i} = 2 {.10}^{- 22} {cm}^{2}$ . All the other laser parameters are kept constant as for the previous configuration.

Here again, the model describes perfectly the noise behavior of the laser. In particular, the frequency and amplitude of the in-phase noise and anti-phase noise accurately fit the experimental measurements. More importantly, we confirm that the anti-phase noise can be reduced provided that the mode coupling in the BR is properly taken into account through the use of two coupled rate equation for nonlinear losses.

In Fig. 8(b), the evolution of the noise spectrum of the laser is computed for different nonlinear absorption efficiencies. It shows that the frequency of both in-phase and anti-phase noise peaks get shifted when the NLA cross-section is increased, meanwhile their amplitude decreases as expected. As already emphasized, this shift is due to the time constant of the NLA process which is in the present case limited by the carrier recombination lifetime in the Si.

The predictive capability of our fully analytical model, being assessed, it can be exploited to find the BR optimal characteristics for a given laser. This is illustrated for our laser in Fig. 9 where the predicted noise spectra are computed when (a) the coupling between the modes in the NLA is reduced to $η = 0.25$ and when (b) the recombination lifetime is set in the ns range. Here we kept the laser parameters constant as for the last configuration. As shown in Fig. 9(a), lowering the coupling strength in the NLA enables to reduce further the noise of the anti-phase peak as compared to Fig. 8(b). However, the in-phase noise peak remains unchanged as expected. In order to go further, we focus on the evolution of the theoretical RIN spectra when the recombination lifetime is reduced to 3 ns. Let us mention that in our modelling approach, and according to Eq. (15), the carrier recombination lifetime is connected to the stationary value of the nonlinear losses, $P_{N L, i}^{s t a t}$ . Consequently the cross-section of the NLA, $ψ_{N L, i}$ , must be adapted accordingly in order to keep a constant value of nonlinear losses. As shown in Fig. 9(b), by reducing the time constant of the BR to 3 ns (which is typical of GaAs), the in-phase and anti-phase resonant peaks are expected to be both efficiently reduced without any shift in frequency.

Fig. 9 Evolution of the RIN spectrum for different TPA efficiencies a) when the coupling strength in the NLA is reduced to 0.25 as compared to Fig. 8(b) and b) when the recombination lifetime is reduced to 3 ns.

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4. Conclusion

In this paper, we have presented a model describing the noise characteristics of a dual-frequency solid-state laser including a Buffer Reservoir mechanism. The development of this model led us to introduce two new rate equations which rule the nonlinear losses inserted on each mode. We have shown that these two equations have to be coupled in order to take into account both common and independent contributions of the nonlinear losses. Moreover, we have demonstrated that the noise contributions coming from the pump fluctuations and the photons fluctuations arising from the amplified spontaneous emission must be considered separately in particular when the active medium is codoped. More specifically, the energy transfer dynamics in the codoped Er,Yb:Glass active medium has to be taken into account to reflect accurately the effect of the pump fluctuations on the laser noise. This leads to an additional rate equation in the model. As a result, the dual-frequency laser with a BR must be described with a set of seven coupled rate equations whose linearization fortunately gives access to analytical expressions of the laser frequency response.

The development of this model has been guided by experimental observations that we conducted on a dual-frequency Er,Yb:Glass laser including a BR in different configurations. Accordingly, we ended up with a model that is able to accurately predict the experimental RIN spectra. In particular, it is shown that the theoretical intensity noise reduction induced by the BR as well as the associated frequency shift of the noise resonances are in very good agreement with the experimental results.

The model being validated, it has been used to predict possible further reduction of the intensity noise in dual-frequency solid-state laser by changing the mode coupling in the NLA and/or reducing the BR time constant. Finally, the set of rate equations describing the dual mode interaction with the BR is general enough to model different laser architectures and active media no matter the nature of the BR. In particular, this model which has been validated using a TPA mechanism as BR is compliant with SHG induced absorption [3].

Funding

Direction Générale de l’Armement (DGA) ; Région Bretagne ; Thales Research and Technology.

References and links

1. A. El Amili, G. Kervella, and M. Alouini, “Experimental evidence and theoretical modeling of two-photon absorption dynamics in the reduction of intensity noise of solid-state Er:Yb lasers,” Opt. Express 21(7), 8773–8780 (2013). [CrossRef] [PubMed]

2. A. El Amili, G. Loas, L. Pouget, and M. Alouini, “Buffer reservoir approach for cancellation of laser resonant noises,” Opt. Lett. 39(17), 5014–5017 (2014). [CrossRef] [PubMed]

3. A. El Amili and M. Alouini, “Noise reduction in solid-state lasers using a SHG-based buffer reservoir,” Opt. Lett. 40(7), 1149–1152 (2015). [CrossRef] [PubMed]

4. M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and P. Thony, “Offset phase locking of Er: Yb: Glass laser eigenstates for RF photonics applications,” IEEE Photonics Technol. Lett. 13(4), 367–369 (2001). [CrossRef]

5. M. Brunel, F. Bretenaker, S. Blanc, V. Crozatier, J. Brisset, T. Merlet, and A. Poezevara, “High-Spectral Purity RF Beat Note Generated by a Two-Frequency Solid-State Laser in a Dual Thermooptic and Electrooptic Phase-Locked Loop,” IEEE Photonics Technol. Lett. 16(3), 870–872 (2004). [CrossRef]

6. K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46(3), 1692–1695 (1992). [CrossRef] [PubMed]

7. M. Brunel, A. Amon, and M. Vallet, “Dual-polarization microchip laser at 1.53 µm,” Opt. Lett. 30(18), 2418–2420 (2005). [CrossRef] [PubMed]

8. A. E. Amili, K. Audo, and M. Alouini, “In-phase and antiphase self-intensity regulated dual-frequency laser using two-photon absorption,” Opt. Lett. 41(10), 2326–2329 (2016). [CrossRef] [PubMed]

9. E. Tanguy, C. Larat, and J.-P. Pocholle, “Modelling of the erbium-ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998). [CrossRef]

10. P. Laporta, S. Longhi, S. Taccheo, and O. Svelto, “Analysis and modelling of the erbium-ytterbium glass laser,” Opt. Commun. 100(1-4), 311–321 (1993). [CrossRef]

11. M. Alouini, “Theoretical and experimental study of Er³⁺ and Nd³⁺ solid-state lasers: application of dual-frequency lasers to optical and microwave telecommunications,” Ph.D. thesis, University of Rennes 1, Rennes (2001).

12. V. P. Gapontsev, S. M. Matitsin, A. A. Isineev, and V. B. Kravchenko, “Erbium glass lasers and their applications,” Opt. Laser Technol. 14(4), 189–196 (1982). [CrossRef]

13. W. E. Lamb, “Theory of an Optical Maser,” Phys. Rev. 134(6A), A1429–A1450 (1964). [CrossRef]

14. G. Cardano, The Great Art (Ars Magna) or The Rules of Algebra (Dovers, 1993).

15. S. Taccheo, P. Laporta, O. Svelto, and G. Geronimo, “Theoretical and experimental analysis of noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66(1), 19–26 (1998). [CrossRef]

16. M. Brunel, F. Bretenaker, and A. Le Floch, “Tunable optical microwave source using spatially resolved laser eigenstates,” Opt. Lett. 22(6), 384–386 (1997). [CrossRef] [PubMed]

17. S. De, G. Baili, S. Bouchoule, M. Alouini, and F. Bretenaker, “Intensity- and phase-noise correlations in a dual-frequency vertical-external-cavity surface-emitting laser operating at telecom wavelength,” Phys. Rev. A 91(5), 053828 (2015). [CrossRef]

18. M. Alouini, F. Bretenaker, M. Brunel, A. Le Floch, M. Vallet, and P. Thony, “Existence of two coupling constants in microchip lasers,” Opt. Lett. 25(12), 896–898 (2000). [CrossRef] [PubMed]

Laser parameters	Value	Laser parameters	Value	Laser parameters	Value
$κ$	$0.72$	$ξ$	$0.8$	$λ_{p}$	$976 nm$
$γ_{c}$	$2.8 {.10}^{8} s^{- 1}$	$N_{E r}$	$1 . 10^{20} {cm}^{-3}$	$N_{Y b}$	$20 {.10}^{20} {cm}^{-3}$
$L$	$4.9 cm$	$σ$	$6 {.10}^{- 21} {cm}^{2}$	$σ_{Y b}$	$10 {.10}^{- 21} {cm}^{2}$
$k_{T R}$	$0.8 {.10}^{- 15} {cm}^{3} {.s}^{-1}$	$γ$	$154 s^{- 1}$	$γ_{Y b}$	$1 {.10}^{3} s^{- 1}$
$e$	$1.5 mm$	$P_{p}$	$450 mW$	$r_{p}$	$40 µm$

Analytical modeling of dual-frequency solid-state lasers including a buffer reservoir for noise cancellation

Abstract

1. Introduction

2. THEORETICAL MODEL

2.1 Laser rate equations in the framework of Buffer Reservoir approach

2.1.1 Single mode operation

2.1.2 Dual mode operation

2.2 Steady-state regime

2.3 Transfer functions

3. Experimental analysis, model validation and discussion

3.1 Description of the experimental setup

3.2 Experimental analysis of the laser intensity noise without Buffer Reservoir

3.3 Experimental analysis of the laser noise including a Buffer Reservoir

3.3.1 Strong mode coupling in the Buffer Reservoir

3.3.2 Weak mode coupling in the Buffer Reservoir

4. Conclusion

Funding

References and links

Cited By

Figures (9)

Tables (1)

Equations (42)

Optics Express