JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL.?, NO.?, ?? 2013 … · 2013-11-14 · JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL.?, NO.?, ?? 2013 1 Phase Noise of the Radio Frequency (RF) Beatnote

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. ??, NO. ??, ?? 2013 1

Phase Noise of the Radio Frequency (RF) BeatnoteGenerated by a Dual-Frequency VECSEL

Syamsundar De, Abdelkrim El Amili,Ihsan Fsaifes, Gregoire Pillet, Ghaya Baili, Fabienne Goldfarb, Mehdi Alouini,

Isabelle Sagnes, and Fabien Bretenaker, Member, IEEE

Abstract—We analyze, both theoretically and experimentally,the phase noise of the radio frequency (RF) beatnote generatedby optical mixing of two orthogonally polarized modes in anoptically pumped dual-frequency Vertical External Cavity Sur-face Emitting Laser (VECSEL). The characteristics of the RFphase noise within the frequency range of 10 kHz - 50 MHzare investigated for three different nonlinear coupling strengthsbetween the two lasing modes. In the theoretical model, weconsider two different physical mechanisms responsible for theRF phase noise. In the low frequency domain (typically below500 kHz), the dominant contribution to the RF phase noise isshown to come from the thermal fluctuations of the semicondutoractive medium induced by pump intensity fluctuations. However,in the higher frequency domain (typically above 500 kHz), themain source of RF phase noise is shown to be the pump intensityfluctuations which are transfered to the intensity noises of thetwo lasing modes and then to the phase noise via the largeHenry factor of the semiconductor gain medium. For this lattermechanism, the nonlinear coupling strength between the twolasing modes is shown to play an important role in the valueof the RF phase noise. All experimental results are shown to bein good agreement with theory.

Index Terms—Dual-frequency laser, VECSELs, phase noise.

I. INTRODUCTION

THE increasing demand for spectrally pure optically-carried radio frequency (RF) signals comes from their

broad application areas such as wide-band signal processing[1], [2], long-range transmission of high purity RF references[3], [4], ultrastable atomic clocks [5], etc. The dual-frequencylaser, sustaining oscillation of two orthogonally polarizedmodes with frequency difference in the RF range, is a promis-ing source for direct generation of such optically-carried highpurity RF signals. The advantages of this technique for thedirect generation of optically-carried RF signals, comparedto other obvious techniques such as optical heterodyning oftwo independent lasers [6], [7], are the narrow linewidth,continuous frequency tunability, and 100 percent modulationdepth of the RF signal. Such tunable dual-frequency operationhas been realized for different solid-state lasers [8]–[10], but

Syamsundar De, Ihsan Fsaifes, Fabienne Goldfarb, and Fabien Bretenakerare with Laboratoire Aime Cotton, CNRS-Universite Paris Sud 11-ENSCachan, Orsay, France.

Abdelkrim El Amili and Mehdi Alouini are with Institut de Physique deRennes, CNRS-Universite de Rennes I, Rennes, France.

Gregoire Pillet and Ghaya Baili are with Thales Research and Technology,Palaiseau, France.

Isabelle Sagnes is with Laboratoire de Photonique et Nanostructures,CNRS, Marcoussis, France.

Manuscript received June ??, 2013.

the main limitation of these solid-state lasers comes from therelatively strong intensity noise due to relaxation oscillationsinherent to their class-B dynamical behavior [11], [12]. Thedual-frequency VECSEL does not suffer from relaxation os-cillations because of its class-A dynamical behavior, which isensured by achieving longer photon lifetime (∼10 ns) insidethe cm-long external cavity than the carriers’ lifetime (∼3 ns)of the semiconductor gain medium [13]. Moreover, similarlyto dual-frequency solid-state lasers, the RF beatnote generatedby the dual-frequency VECSEL is widely tunable by varyingthe intra-cavity phase anisotropy [13].

More importantly, the spectral purity of the beatnote, whichdepends on the correlation of the intensity and phase fluc-tuations of the two lasing modes, is very high since thetwo orthogonal polarization modes are oscillating inside thesame cavity. We have already investigated such correlationbehavior between the intensity noises of the two modes ofdual-frequency VECSEL for different coupling situations [14].Now, we know that the spectral purity of the RF beatnote isactually related to the correlation of the optical phase noisesof the two modes, since the positively correlated part of thephase fluctuations will cancel out in the beatnote generated byoptical mixing of the two laser modes. But the intensity noisecorrelation between the two laser modes is also expected tohave strong influence on the spectral purity of the RF beatnoteas phase fluctuations are coupled to intensity flcutuations dueto the large Henry factor of the semiconductor gain medium ofsuch a VECSEL [15]–[17]. Moreover, there are other reasonssuch as thermal or mechanical fluctuations of the laser cavity,which can create some additional phase noise.

In the present paper, we analyze, both experimentally andtheoretically, the behavior of the phase noise of the RFbeatnote for different coupling situations between the two lasermodes within the 10 kHz - 50 MHz frequency range. Indeed,the simultaneous oscillation of the two modes in the dual-frequency laser is ruled by the nonlinear coupling constantwhich must be the lowest possible for robust operation. Bycontrast, the spectral purity of the beat note is expected toincrease if the two modes share exactly the same opticalpath into the laser, which leads to a high nonlinear cou-pling constant. Thus, a thorough understanding of how thephase noise of the RF beatnote evolves with respect to themode coupling strength is useful. In the theoretical model,we suppose that the only source of noise is the intensitynoise of the pump diode laser, which is white within thefrequency range of our interest (10 kHz - 50 MHz). Moreover,

arX

iv:1

311.

3057

v1 [

phys

ics.

optic

s] 1

3 N

ov 2

013


e

o BC Output

Mirror Gain

Medium

LSC Lext

Lopt

Fig. 1. Schematic of the basic principle of the dual-frequency VECSEL. Thebirefringent crystal (BC) spatially separates the two modes polarized alongthe ordinary (o) and extraordinary (e) directions. They partially overlap inthe active medium. The length of the semiconductor 1/2-VCSEL is LSC .The length of the external cavity is Lext. The optical length of the cavity isLopt.

the pump fluctuations entering into the two laser modes arepartially correlated, but are in phase. Now in the model weconsider two different mechanisms through which the pumpintensity noise is affecting the phase noise of the RF beatnote.Firstly, the pump intensity noise is contributing to the higherfrequency (typically for frequencies larger than 500 kHz) partof the RF phase noise via phase-intensity coupling due tothe large Henry factor of the semiconductor active medium.The other contribution of the pump noise to the RF phasenoise (typically for frequencies lower than 500 kHz) is comingthrough the thermal fluctuations of the refractive index ofthe semiconductor structure and hence the fluctuations ofthe effective cavity length [18]. In Section II, we developa theoretical model aiming at describing the effects of thepump noise on the phase noise of the RF beatnote. Section IIIdescribes the experimental set-up for measuring the RF phasenoise. In Section IV, we compare the experimental results withthe predictions of our theoretical model for different couplingstrengths between the two modes, controlled by the spatialseparation of the two modes inside the active medium.

II. THEORETICAL MODEL

The basic working principle of our dual-frequency VEC-SEL, in which two orthogonally polarized modes are partiallyspatially separated in the active medium by an intra-cavitybirefringent element, is schematized in Fig. 1. To model theeffect of pump noise on the noise properties of this laser, wetake into account two different physical mechanisms. They aredescribed in the following subsections.

A. Propagation of intensity to phase noise via phase-intensitycoupling

In this subsection, we generalize the model developedearlier [14]. This model has proved to be successful to describethe correlations between the intensity noises of the two modesinduced by the intensity noise of the pump laser. We nowgeneralize this model to take into account the transfer ofintensity noise to the phase noise of the RF beatnote via phase-intensity coupling through the Henry factor. The rate equations

governing the laser behavior read [14]:

dF1(t)

dt= −F1(t)

τ1+ κN1(t)F1(t), (1)

dF2(t)

dt= −F2(t)

τ2+ κN2(t)F2(t), (2)

dN1(t)

dt=

1

τ[N01(t)−N1(t)]− κN1(t)[F1(t) + ξ12F2(t)],

(3)dN2(t)

dt=

1

τ[N02(t)−N2(t)]− κN2(t)[F2(t) + ξ21F1(t)],

(4)dφ(t)

dt=dφ1(t)

dt− dφ2(t)

dt=α

2κ[N1(t)−N2(t)]. (5)

Here F1 and F2 are the photon numbers in the two cross-polarized modes, N1 and N2 are the corresponding populationinversions, φ1 and φ2 are the optical phases of the two lasingmodes and φ is the phase of the RF beatnote, generatedby optical mixing of the two laser modes. τ1, τ2 are thephoton lifetimes inside the cavity for the two modes. Wechoose to consider two different photon lifetimes, since thetwo modes experience different losses inside the cavity. τ isthe population inversion lifetime, which is identical for thetwo modes. N01 and N02 are the two unsaturated populationinversions, since N01/τ and N02/τ denote the pumping ratesfor the two modes, respectively. The stimulated emissioncoefficient κ is proportional to the stimulated emission cross-section. α holds for the Henry factor, which is the ratio ofthe carrier induced change of real part of refractive indexto the corresponding change of imaginary part of refractiveindex [15]. The coefficients ξ12 and ξ21 are the ratios of thecross- to self-saturation coefficients, which take into accountthe effect of partial overlap between the two modes inside thegain medium. Therefore,

C = ξ12ξ21 (6)

defines the nonlinear coupling constant [19], on which dependsthe possibility for the two modes to oscillate simultaneously.This approach will allow us to vary the coupling constant (C)simply by changing the ratios of the cross- to self-saturationcoefficients, through adjustment of the spatial overlap betweenthe two modes in the active medium. This technique is verysimple, although it is not easy to describe in the framework ofthe usual spin-flip model [20]–[24]. The steady-state solutionsfor simultaneous oscillation of the two orthogonally polarizedmodes, obtained from Eqs. (1-4), are given as

F1 ≡ F10 =(r1 − 1)− ξ12(r2 − 1)

κτ(1− C), (7)

F2 ≡ F20 =(r2 − 1)− ξ21(r1 − 1)

κτ(1− C), (8)

N1 ≡ N1th =1

κτ1, (9)

N2 ≡ N2th =1

κτ2. (10)

Here r1 = N01/N1th and r2 = N02/N2th are the excitationratios for the two modes, where N01 and N02 are the steady-


state values of the corresponding unsaturated population in-versions. In the following, we suppose that the pump intensitynoise is the dominant source of noise in the consideredfrequency range (10 kHz - 50 MHz). The pump fluctuationsfor the two modes are modeled as:

N01(t) = N01 + δN01(t) , (11)

N02(t) = N02 + δN02(t) . (12)

The fluctutations in photon numbers and corresponding pop-ulation inversions around their steady-state values are definedas

F1(t) = F10 + δF1(t) , (13)F2(t) = F20 + δF2(t) , (14)N1(t) = N1th + δN1(t) , (15)N2(t) = N2th + δN2(t) . (16)

If now we substitute Eqs. (11-16) into Eqs. (1-4) and then per-form Fourier transformation, followed by linearization aroundthe steady-state, we obtain the following expression, relatingthe fluctuations of the photon numbers of the two laser modesto the pump fluctuations, in the frequency domain:[

δF 1(f)

δF 2(f)

]=

[M11(f) M12(f)M21(f) M22(f)

] [˜δN01(f)˜δN02(f)

], (17)

where the tilde symbol indicates Fourier transform and where

M11(f) =1

τ

[ 1τ2− 2iπf

κF20(r2/τ − 2iπf)]

∆(f), (18)

M12(f) =ξ12

ττ1∆(f), (19)

M21(f) =ξ21

ττ2∆(f), (20)

M22(f) =1

τ

[ 1τ1− 2iπf

κF10(r1/τ − 2iπf)]

∆(f), (21)

and

∆(f) = [1

τ1− 2iπf

κF10(r1/τ − 2iπf)]

× [1

τ2− 2iπf

κF20(r2/τ − 2iπf)]− C/τ1τ2 . (22)

Therefore, the fluctuations of the RF phase in frequencydomain can be easily obtained by Fourier transforming Eq. (5)and combining it with Eqs. (1) and (2), leading to:

δφ(f) =α

2[δF 1(f)

F10− δF 2(f)

F20] . (23)

Now in the experiment, the two modes, which are partiallyoverlapping in the gain medium, are pumped by the samepump laser. Therefore, we choose the following approxima-tions for the pump fluctuations [14]:

〈|δN01(f)|2〉 = 〈|δN02(f)|2〉 = 〈|δN0|2〉 , (24)

〈δN01(f) ˜δN∗02(f)〉 = η〈|δN0|2〉eiψ . (25)

The pump noises entering into the two laser modes are whitenoises of identical amplitudes, as implied by Eq. (24). Fur-thermore, Eq. (25) describes the correlation behavior between

the pump fluctuations for the two laser modes. For the theorywe have assumed that the pump intensity fluctuations for thetwo modes are partially correlated (0 < η < 1) and η isindependent of frequency within the frequency range of ourinterest (10 kHz - 50 kHz). Moreover, the pump fluctuationsare also assumed to be in phase (ψ = 0). Considering allthe above approximations, which have been shown to be valid[14], the power spectral density (PSD) of the phase noise forthe RF beatnote can be written as:

〈|δφ(f)|2〉 =α2

4[〈|δF 1(f)|2〉

F 210

+〈|δF 2(f)|2〉

F 220

−2〈Re(δF 1(f)δF∗2(f))〉

F10F20] . (26)

So Eq. (26) clearly shows the dependence of the RF phasenoise not only on the intensity noises (first 2 terms of righthand side) but also on the correlation between the intensitynoises of the two laser modes (last term on the right handside).

B. Thermal Fluctuations Induced by the Pump Fluctuations

Thermal fluctuations can have a deleterous effect on thephase noise of the laser modes. Generally, thermal fluctuationsintroduce fluctuations of cavity length either by varying theeffective refractive index inside the cavity or by the varying thephysical length of the cavity. These cavity length fluctuationsare translated into fluctuations of the optical phases of thelaser modes. In the present section, we start, following Refs.[25]–[27], by deriving the power spectral density (PSD) of theoptical phase noise for a single mode laser induced by suchthermal fluctuations. Then we extend this to the case of ourdual-frequency VECSEL.

It is worth mentioning that in this study we will notconsider the effect of temperature fluctuations either due toentropy change generated by the spontaneous emission usuallyin solid-state lasers [29], [30] or free electron generation-recombination associated with materials defects in the semi-conductor structure [31], which explain the so called 1/ffrequency noise.

1) Single mode laser: The instantaneous angular frequencyof the laser, which is an integer multiple of the free spectralrange of the cavity, is given by

ω(t) = qπc

Lopt(t), (27)

where q is an integer and c is the vacuum velocity of the light.The time varying quantity Lopt(t) denotes the total opticallength of the laser cavity, which can be expressed as

Lopt(t) = Lext + nSCLSC , (28)

where Lext is the length of the external cavity, and where nSCand LSC are the effective refractive index and the thickness ofthe semiconductor structure. Then the variation of the opticalfrequency due to total cavity length variation can be given as

δωth(t) = −ω(t)δLopt(t)

Lopt(t). (29)


The dominant contribution in the thermal fluctuations of theoptical length of the cavity comes from the fluctuations ofthe refractive index of the semiconductor structure [26], [27],leading to:

δωth(t) ' − ω0

Lextδn(t)LSC , (30)

where δn(t) is the temporal fluctuations of the refractive index,induced thermally and averaged over the volume of the opticalmode. We can describe it as

δωth(t) ' −ω0ΓthδT (t), (31)

where δT (t) denotes the fluctuation of the temperature, aver-aged over the optical mode volume and Γth is defined as

Γth =LSCLext

dn

dT. (32)

In general, the temperature fluctuation δT (t) of the semicon-ductor structure averaged over the optical mode volume is thesum of two terms [28]:

δT (t) = δTk(t) +RthδPp(t) ∗ θ(t), (33)

where δTk(t) defines the fundamental thermodynamic fluc-tuations of the semiconductor, when it is maintained at afixed temperature T and the last term of the right hand sidedescribes the effect of temperature fluctuations induced bypump power fluctuations.∗’ represents the convolution product.Rth is the thermal impedance of the semiconductor structure,averaged over the volume of the optical mode. δPp holdsfor the fluctuations of the pump power and θ(t) defines theimpulse temperature response of the semiconductor structureto the pump power. As usually performed [27], we model thetransfer function Θ(f), which is Fourier transform of θ(t), asa first-order low-pass filter, leading to:

|Θ(f)|2 =1

1 + (2πfτth)2. (34)

The order of magnitude of the response time (τth) of thermaldiffusion is given as [32]

τth 'w2p

2πDT, (35)

where wp is the waist of the pump beam on the semicon-ductor medium and DT is the thermal diffusion coefficient.To evaluate the order of magnitude of the first term in theright-hand side of Eq. (33), we obtain the variance of theintrinsic temperature fluctuations of the semiconductor due tothe fundamental thermodynamic fluctuations from Refs. [33]–[35]:

〈δT 2k 〉 =

kBT2

CvV. (36)

Here kB is Boltzman’s constant, Cv is density of specific heatand V ' πw2

0Lµc represents the volume of the semiconductor,occupied by the optical mode (w0 is the laser mode size).Therefore, the power spectral density (PSD) of the thermalfluctuations, induced by the fundamental thermodynamic fluc-tuations, is given as following

ST (f) =kBτthT

2

π2f2CvV|Θ(f)|2. (37)

Similarly the power spectral density of the thermal fluctua-tions, induced by the pump power fluctuations, is given by

SP (f) =Rth

2

4π2f2|Θ(f)|2RINP (f)P 2

P , (38)

where RINP is the pump relative intensity noise (RIN) andPP is the incident pump power. For our laser, if we comparethe PSD of the thermodynamic fluctuations (ST (f)) withthe PSD of the thermal fluctuations, induced by the pumppower fluctuations (SP (f)), we find that the effect of thefundamental thermodynamic fluctuations is very small (∼ 105

times smaller) compared with the effect of the pump powerfluctuations. Therefore, in the following, we neglect the effectof fundamental thermodynamical fluctuations, i. e., the firstterm in the right-hand side of Eq. (33). We can then Fouriertransform Eq. (33) and obtain the optical phase noise powerspectral density as

|δφopt(f)|2 =ω20Γ2

thRth2

4π2f2|Θ(f)|2RINP (f)P 2

P

= |Λ(f)|2|δPP (f)|2, (39)

where we have defined the transfer function

|Λ(f)|2 =ω20Γ2

thRth2

4π2f2|Θ(f)|2, (40)

and where|δPP (f)|2 = RINP (f)P 2

P (41)

holds for the power spectral density of the pump powerfluctuations.

2) Dual-frequency VECSEL: In the case of our dual-frequency VECSEL, the two orthogonally polarized modes,having a frequency difference in the RF range, are oscillatinginside the same cavity. Therefore, in the frequency domain,the phase fluctuations of the RF beatnote generated by opticalmixing of the two laser modes, are given by

δφRF (f) = δφ1(f)− δφ2(f)

= Λ1(f)δPP1(f)− Λ2(f)δPP2(f), (42)

where δφ1(f), δφ2(f) are fluctuations of the optical phasesof the two laser modes and δφRF (f) is the correspondingRF phase fluctuation in the frequency domain. Moreover,δPP1(f) and δPP2(f) are the pump fluctuations for the twocorresponding laser modes. Now, we make similar approxi-mations for the fluctuations of the pumping rates as in Eqs.(24-25), leading to:

|δPP1(f)|2 = |δPP2(f)|2 = |δPP |2, (43)

〈|δPP1(f)δP∗P2(f)|〉 = η|δPP |2 . (44)

Eqs. (43) and (44) describe white pumping noises of identicalamplitudes for the two laser modes and which are partiallyuncorrelated (0 < η < 1, η independent of frequency), but inphase (ψ = 0). Now we suppose that

|Λ1(f)|2 = |Λ2(f)|2 = |Λ(f)|2. (45)


Pump-diode λp=808 nm

BS

IO

λ/2

Fiber

½-VCSEL

R=25 mm T=0.5 %

SiC

Bra

gg M

irror

6 Q

W In

GaA

s

AR

Coa

ting

BC

d

PBS

λ/2

L

Etalon

FPI

D

Oscilloscope

D

Oscilloscope

RFA

ESA

Mixer

Local oscillator

Hea

t sin

k

Fig. 2. Schematic of the experimental set-up for measuring the phase noise of the RF beatnote. d: spatial separation between the two modes on the gainstructure, BC: birefringent crystal, L: lens, T : transmittance of the output coupler, R: radius of curvature of the output mirror, BS: beam splitter, λ/2:half-wave plate, PBS: polarization beam spliter, D: photodetector, RFA: radio frequency amplifier, ESA: electrical spectrum analyzer, OI: optical isolator, FPI:Fabry-Perot interferometer.

Therefore, the power spectral density (PSD) of the phase noiseof the RF beatnote in our dual-frequency VECSEL due tothermal noise, induced by pump power fluctuations, is foundto be

|δφRF (f)|2 = 2(1− η)|Λ(f)|2|δPP |2. (46)

Thus we can conclude that the phase noise of the RF beatnotedue to the thermal fluctuations of the refractive index ofthe semiconductor medium, induced by the pump intensityfluctuations, does not only depend on the PSD of the pumpnoise, but also on the correlation (η) between the pump fluctu-ations entering into the two laser modes. More specifically, thepump noise induced thermal fluctuations provide a significantcontribution to the RF phase noise as long as the pumpfluctuations for the two laser modes are not perfectly correlated(η < 1).

III. DESCRIPTION OF THE EXPERIMENTAL SET-UP

The experimental set-up designed for measuring the phasenoise of the RF beatnote, generated by optical mixing of thetwo orthogonally polarized laser modes, is schematized in Fig.2. The laser is operating at 1 µm and the pumping is done witha 808 nm diode laser, pigtailed with a multimode fiber anddelevering up to 3 W of optical power. The laser is based onthe 1/2-VCSEL structure, which is grown by Metal OrganicChemical Vapour Deposition (MOCVD) technique [26]. The1/2-VCSEL structure is initially grown on a GaAs substrate,then it is transferred to a SiC substrate. The high thermalconductivity (490 Wm−1K−1) of the SiC substrate enablesto dissipate the extra heat generated by the optical pumpand thus maintains a good efficiency of our laser. Moreover,the structure is placed on a Peltier thermoelectric cooler,whose temperature is maintained at 19◦C using a temperature

controller. There is a Bragg mirror inside the 1/2-VCSELstructure, which consists in 28 pairs of alternating GaAs/AlAslayers, providing 99.9 % reflectivity at 1 µm. The 1/2-VCSELstructure contains six strained balanced InGaAs/GaAsP quan-tum wells (QW), which are responsible for the gain at 1 µm.The structure possesses a linear gain dichroism. The anti-reflection coating of Si3N4 on top of the 1/2-VCSEL structurereduces the micro-cavity effect between the Bragg mirrorand the semiconductor-air interface. The planar-concave lasercavity is about 1.5 cm long. A concave mirror of reflectivity99.5 percent and radius of curvature 25 mm serves as theoutput coupler. Besides, this configuration ensures a class-Adynamical behavior of our VECSEL, since the photon lifetime(∼10 ns) is sufficiently longer than the population inversionlifetime (∼3 ns) [13], [14]. The intra-cavity YVO4 birefringentcrystal (BC) with anti-reflection coating at 1 µm introducesa polarization walk-off d proportional to its thickness. Thetwo laser modes are perfectly spatially overlapped betweenthe birefringent crystal BC and the output coupler as theBragg mirror is planar and the output mirror is concave. Thebirefringent crystal is oriented in such a way that the maximumgain axis of the semiconductor structure is aligned along thebisector of the ordinary and extraordinary eigenpolarizationsof the BC. Thanks to the intra-cavity BC, which spatiallyseparates the two orthogonally linearly polarized laser modeson the gain structure while enabling a balanced gain for thetwo orthogonal polarizations, simultaneous oscillation of thetwo modes is obtained by reducing the coupling constant valuebelow unity [13].

The waists of the two laser modes are identical and about62 µm. The position and the size of the pump beam is exper-imentally adjusted on the gain medium to provide maximum


and nearly identical gain for the two modes. We have usedthree different BCs of thicknesses 1 mm, 0.5 mm and 0.2mm, which correspond to spatial separations of 100 µm, 50µm and 20 µm respectively, hence different coupling strengthsbetween the laser modes [36]. The intra-cavity 150-µm-thickuncoated glass etalon forces both polarizations to oscillate insingle longitudinal mode.

The scanning Fabry-Perot interferometer with 10 GHz freespectral range enables us to analyze continuously the laserspectrum to check that the two orthogonal eigenpolarizationsare oscillating simultaneously in single-longitudinal-moderegime without any mode hopping during data acquisition.To generate the RF beatnote, the two orthogonally polarizedmodes are mixed by a half-wave plate (λ/2-plate), followed bya polarization beam splitter (PBS). The RF beatnote is detectedby a high-speed photodiode of 22 GHz bandwidth (DiscoveryDSC-30S). Then the RF signal is amplified using a low noiseRF amplifier (RFA). Now one part of the amplified RF signalis sent to an electrical spectrum analyzer (ESA) to monitor thebeatnote spectrum. Thereafter, the other part of the RF signalis downshifted to an intermediate frequency (IF) by mixing itwith a signal from a local oscillator (LO) (Rohde & SchwarzSMF100A) having very low phase noise compared to the RFphase noise of our laser. Finally the IF signal is recorded intime domain using a deep memory digital oscilloscope andprocessed numerically to obtain the phase noise spectrum ofthe RF beatnote.

Before measuring the phase noise of the RF beatnote gen-erated by our dual-frequency VECSEL, let us first recall that,in our model, we have approximated the pump intensity noiseas white, i.e., independent of frequency in the range of ourinterest (10 kHz to 50 MHz). Let us first check this assumptionby measuring the pump RIN spectrum. The correspondingexperimental result is shown in Fig. 3. One can see that theexperiment ensures that the pump noise is almost flat withinthe interesting frequency range (10 kHz− 50 MHz). The pumpRIN level is close to -138 dB/Hz, which we will take as thevalue that we will inject in our model in the following.

IV. RF PHASE NOISE: EXPERIMENTAL AND THEORETICAL

The phase noise measurements of the RF beatnote havebeen performed for three different BCs of thicknesses 1 mm,0.5 mm and 0.2 mm, which respectively correspond to spatialseparations of 100 µm, 50 µm, and 20 µm between the twolaser modes on the gain structure. For all the measurementsof different coupling situations between the laser modes, thepump power is fixed to 1.5 W and the power of the each lasermode is between 50 and 60 mW.

Fig. 4 shows the results for the 1-mm thick BC, whichspatially separates the two laser beams by 100 µm on thegain medium. In this case, the coupling constant (C) hasbeen measured to be equal to 0.1 [14], [36]. The powerspectral density (PSD) of the RF phase noise is shown inFig. 4(a), whereas the corresponding RF beatnote spectrum isreproduced in Fig. 4(b). In Fig. 4(a), the solid black and redcurves respectively correspond to the experimental and totaltheoretical PSD of the RF phase noise. The total theoretical

1 0 k 1 0 0 k 1 M 1 0 M- 1 6 0- 1 5 0- 1 4 0- 1 3 0- 1 2 0

RIN (d

B/Hz)

F r e q u e n c y ( k H z )Fig. 3. Experimentally measured RIN spectrum of the pump diode laser.

RF phase noise PSD [red curve in Fig. 4(a)] is obtained byadding two different physical mechanisms: (i) phase-intensitycoupling due to the large Henry factor of the semiconductorgain medium of our laser [see Eq. (26)] and (ii) the fluctuationsof the refractive index of the semiconductor structure dueto the temperature fluctuations induced by pump intensityfluctuations [see Eq. (46)]. The contribution of the phase-intensity coupling to the RF phase noise PSD, calculatedfrom Eq. (26), is shown by the dot-dashed blue curve in Fig.4(a), while the contribution of the thermal effect, calculatedfrom Eq. (46), is plotted as the dot-dashed green curve. Theparameters used for the simulations, which are given in thefigure captions, have been either obtained from the experimentor from preceding works performed with the same type ofstructure [28]. So in this low coupling condition (C = 0.1),the dominant contribution to the RF phase noise for offsetfrequencies between 700 kHz and 50 MHz comes from phase-intensity coupling. The RF phase noise coming from thermalfluctuations, induced by pump power fluctuations, is dominantat lower frequencies (between 10 and 700 kHz) but is filteredout for frequencies larger than a few hundred kilohertz by thethermal response time of the structure.

The total theoretical curve (solid red curve in Fig. 4(a))shows satisfactory matching with the experimentally measuredRF phase noise (solid black curve in Fig. 4(a)). The remainingsmall discrepancy probably comes from the crude approxi-mation of the thermal response of the structure by a simplefirst-order filter [see Eq. (34)] and also from the approximateexpression [see Eq. (35)] we chose for the thermal responsetime of the structure [32]. In the beatnote spectrum as shown inFig. 4(b), one can clearly see that the beat frequency, centeredaround 4.807 GHz, is sitting on a noise pedestal correspondingto the phase noise of Fig. 4(a).

Fig. 5 shows the results obtained with the 0.5-mm-thick BC,which corresponds to a spatial separation of 50 µm betweenthe two laser modes on the gain structure. For the theoreticalsimulations, we use the coupling constant (C) value of 0.35,to which we refer as a moderate coupling situation [14], [36].In Fig. 5(a), the solid black curve represents the experimental


Measured RF phase noise

Total theoretical RF phase noise

Noise from thermal effect only

Noise from phase-intensity

coupling only

(a)

RBW = 30 kHz

VBW = 30 kHz

(b)

Fig. 4. Results for the 1-mm-thick BC, which corresponds to a spatial separation (d) of 100 µm. (a) Power spectral density (PSD) of the phase noise of theRF beatnote. (b) Beatnote spectrum, measured by the ESA with resolution bandwidth (RBW) = 30 kHz and video bandwidth (VBW) = 30 kHz. Parametervalues used for simulation: C = 0.1, α = 12, r1 = 1.45, r2 = 1.5, τ1 = 5.8 ns, τ2 = 6 ns, τ = 3 ns, RINp = −138 dB/Hz, η = 0.85, DT = 0.22cm2s−1, dn/dt = 2.7× 10−4 K−1, LSC = 2.3 µm, Lext = 1.5 cm, wp = 50 µm, Rth = 6 K.W−1, Pp = 1.5 W.





coupling only

(a)

RBW = 30 kHz

VBW = 30 kHz

(b)

Fig. 5. Results for the 0.5-mm-thick BC, which corresponds to a spatial separation (d) of 50 µm. (a) Power spectral density (PSD) of the phase noise ofthe RF beatnote. (b) Beatnote spectrum, measured by ESA with resolution bandwidth (RBW) = 30 kHz and video bandwidth (VBW) = 30 kHz. Parametervalues used for simulation: C = 0.35, α = 12, r1 = 1.45, r2 = 1.5, τ1 = 5.8 ns, τ2 = 6 ns, τ = 3 ns, RINp = −138 dB/Hz, η = 0.85, DT = 0.22cm2s−1, dn/dt = 2.7× 10−4 K−1, LSC = 2.3 µm, Lext = 1.5 cm, wp = 50 µm, Rth = 6 K.W−1, Pp = 1.5 W.

PSD of the RF phase noise, whereas the solid red curve depictsthe corresponding theoretical one. The total theoretical curve(solid red curve) is obtained by considering the effects ofthermal noise as well as noise due to phase-intensity coupling.The PSD of the RF phase noise coming only from pump noiseinduced thermal noise, is shown as a dot-dashed green curveof Fig. 5(a), whereas the contribution of the phase-intensitycoupling due to the Henry factor is given by the dot-dashedblue curve of Fig. 5(a). As we can see from Fig. 5(a), the effectof the phase-intensity coupling is dominant for frequencieshigher than 500 kHz, whereas the thermal noise is dominantfor lower frequencies (10 kHz to 500 kHz). The theoreticalprediction (red curve of Fig. 5(a)) shows fairly good agreementwith the corresponding experimental results (black curve ofFig. 5(a)). The RF beatnote centered at about 4.227 GHz isshown in Fig. 5(b).

The results for the 0.2-mm-thick BC are shown in Fig. 6,which corresponds to a spatial separation between the twolaser modes of 20 µm. In this configuration, the couplingconstant C is shown to be equal to 0.65 [14], [36]. So this leadsto a relatively intense coupling between the laser modes. In this

case, the experimental and theoretical PSD of the RF phasenoise within the interesting frequency range 10 kHz − 50MHz, are respectively given by the solid black and red curvesin Fig. 6(a). Similarly to preceding results, the total theoreticalcurve (solid red curve of Fig. 6(a)) is obtained by taking intoaccount the two different mechanisms discussed above. We cansee that the lower frequency (10 kHz − 200kHz) RF phasenoise is mainly coming from pump noise induced thermalfluctuations, whereas the phase-intensity coupling effect iscontributing dominantly for frequencies higher than 200 kHz.The beatnote spectrum is given in Fig. 6(b), where we can seethat the beatnote is centered around 4.081 GHz and containsagain a few megahertz wide pedestal due to the phase noise.Again the theory (solid red curve) shows very good agreementwith experiment (solid black curve) for this strong couplingsituation. The little discrepancy for frequencies higher than20 MHz is coming from the fact that we are approaching themeasurement noise floor (∼ −130 dBc/Hz), which is limitedby our oscilloscope.

We can see from the result obtained in Sec. II that thethermally induced phase noise is independent of the value






coupling only

(a)

RBW = 30 kHz

VBW = 30 kHz

(b)

Fig. 6. Results for the 0.2-mm-thick BC, which corresponds to a spatial separation (d) of 20 µm. (a) Power spectral density (PSD) of the phase noise ofthe RF beatnote. (b) Beatnote spectrum, measured by ESA with resolution bandwidth (RBW) = 30 kHz and video bandwidth (VBW) = 30 kHz. Parametervalues used for simulation: C = 0.65, α = 12, r1 = 1.35, r2 = 1.4, τ1 = 6.3 ns, τ2 = 6.6 ns, τ = 3 ns, RINp = −138 dB/Hz, η = 0.85, DT = 0.22cm2s−1, dn/dt = 2.7× 10−4 K−1, LSC = 2.3 µm, Lext = 1.5 cm, wp = 50 µm, Rth = 6 K.W−1, Pp = 1.5 W.

of the coupling constant C, whereas the phase noise comingthrough phase-intensity coupling via Henry factor dependson C since the intensity noises of the two modes and theircorrelations depend on C [14]. So for very strong coupling(C = 0.65), the intensity noises of the two laser modesare increased for intermediate frequencies (between 200 kHzand 2 MHz) due to strong anti-phase relaxation mechanismcompared to in-phase relaxation mechanism of the coupledoscillator system [14]. As a result of that the RF phase noisecoming from the phase-intensity coupling is also increasedfor intermediate frequencies, as intuitively anticipated (Fig.6(a)). But for smaller values of coupling (C = 0.1 and0.35), the response of the antiphase relaxation mechanism islower (for C = 0.1) or comparable (C = 0.35) with theresponse of the inphase relaxation mechanism for intermediatefrequencies (between 200 kHz and 2 MHz). Moreover, the in-phase relaxation mechanism is independent of the strength ofcoupling. So the noise response of each laser mode, which isobtained by substracting the anti-phase response from the in-phase response, is relatively low at intermediate frequenciesfor smaller couplings. Therefore, the phase noise introducedvia phase-intensity coupling is relatively low at these fre-quencies in case of low (C = 0.1) and moderate coupling(C = 0.35), which explains the evolution of the experimentaland theoretical spectra with C. These results suggest that amoderate spatial separation of about 50 µm, corresponding toroughly 50 % mode spatial overlap, is a good tradeoff enablinggood robustness of the dual frequency oscillation as well asgood spectral purity of the beatnote.

V. CONCLUSION

In this study, we study both experimentally and theoreticallythe phase noise of the RF beatnote generated by opticalmixing of two orthogonally polarized modes of the dual-frequency VECSEL. We have investigated the RF phase noiseproperties for three different coupling strengths between thetwo laser modes. In the theory, we consider two differenteffects of pump intensity noise, which we suppose to be theonly source of noise in the frequency range of interest (10

kHz to 50 MHz). The first effect of the pump noise is togenerate intensity noises in the two laser modes, which thengive rise to phase noise via phase-intensity coupling due to thehigh Henry factor of the semiconductor gain medium. Nowthe intensity noises of the two laser modes are not perfectlycorrelated and the noise correlations depend on the strength ofthe nonlinear coupling between these modes [14]. The partialcorrelation between the intensity noises of the two laser modescomes from the fact that the pump noises for the two modesare not perfectly correlated (η < 1). Therefore, the opticalphase noises for the two laser modes, which are generatedby intensity noises via phase-intensity coupling, are also notperfectly correlated. As a result, the fluctuations of opticalphases of the two laser modes are not completely cancelledout, when we mix them to generate beatnote. This gives riseto the few megahertz wide noise pedestal of the RF beatnote.The other effect of the pump noise is to introduce thermalfluctuations of the semiconductor structure. As a result, therefractive index of the semiconductor and hence the effectivecavity length fluctuates, which creates the phase fluctuationsof the two modes oscillating inside the cavity. Now again,as the pump fluctuations for the two modes are not fullycorrelated, the phase fluctuations of the two modes originatingvia thermal fluctuations are not fully correlated. Therefore, inthe beatnote the optical phase noises of the two modes arenot completely cancelled out thus giving rise to the RF noisepedestal. In the RF phase noise spectra, the thermal noise isdominating for lower frequencies (typically below 500 kHz),whereas the noise coming from phase-intensity coupling isdominating for higher offset frequencies (typically within 500kHz − 50 MHz). Moreover, the thermal noise is independentof coupling between the two laser modes, whereas the noisecoming through phase-intensity coupling depends on couplingstrengths. This is due to the fact that intensity noises of thelaser modes, which are generating phase noises via phase-intensity coupling, as well as correlations between these noisesdepend on coupling strengths between the two laser modes.This study will help us to improve the purity of the opticallycarried RF signal, which is the very basic requirement for any


RADAR application.

ACKNOWLEDGMENTS

This work was partially supported by the Agence Nationalede la Recherche (Project NATIF No. ANR-09-NANO-012-01)and by the French RENATECH network.

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Syamsundar De was born in India in 1987. Hereceived his M.S. degree in Applied Physics (Op-tics, Matter, Plasma) from Ecole Polytechnique,Palaiseau, France, in 2012.

He joined Laboratore Aime Cotton, Orsay, Francein 2012 as Ph.D student. He is working on noise ofsolid-state and semiconductor lasers for microwavephotonics applications. His research interests arealso in non-linear and quantum optics.


Abdelkrim El Amili was born in 1981. He receivedthe M.Sc. from Universite de Bordeaux in 2006and the PhD degree from Universite Paris Sud 11in 2009, while working on cold atom manipula-tion using semiconductor waveguides at LaboratoireCharles Fabry de l’Institut d’Optique. He then joinedLaboratoire Aime Cotton to work on VECSELs. In2011 he joined the Institut de Physique de Renneswhere he is working on low noise dual-frequencylasers.

Ihsan Fsaifes Biography text here.

Gregoire Pillet received the Masters degree in laser-matter interactions physics from Ecole Polytech-nique, France, and the degree from the Ecole Na-tionale Superieure des Tele communication, France,in 2007. Since 2006, he has been pursuing thePh.D. degree at the Thales Research and Technology,Palaiseau, France.

Ghaya Baili was born in Tunisia in 1980. She re-ceived the M.S. degree in optics and photonics fromthe Ecole Superieure dOptique, University of ParisXI, Orsay, France, in 2004. She joined Thales Re-search and Technology France, Palaiseau, in 2005.She received the PhD degree from Universite ParisSud 11 in 2008. Her research interests are in thefield of lasers for microwave photonics applications.

Fabienne Goldfarb was born in Paris in 1976. Shereceived the PhD degree from Universite Paris Sud11 in 2003. She worked on molecular interferometryin Vienna from 2003 to 2005 and then joined Uni-versite Paris Sud 11 as assistant professor in 2005.Dr. Goldfarb’s research interests are in the field ofcoherent effects in atoms, slow and fast light, andmicrowave photonics.

Mehdi Alouini received the B.S. degree in fun-damental physics from the University of Rennes,Rennes, France, in 1995, the M.S. degree in opticsand photonics from the Ecole Suprieure dOptique,University of Paris XI, Orsay, France, in 1997, andthe Ph.D. degree in laser physics from the Universityof Rennes in 2001. He joined Thales Research andTechnology France, Palaiseau, in 2001 as ResearchStaff Member. His main activities cover the mod-eling and optimization of microwave photonics sys-tems as well as active polarimetric and multispectral

imaging. His research interests are also in solid-state and semiconductor laserphysics for microwave photonics applications.

Isabelle Sagnes was born in Tunisia in 1967. Shereceived the Ph.D. degree in physics from the Uni-versity Joseph Fourier, Grenoble, France, in 1994,for her work in electro-optical properties of epitaxialheterostructures on silicon. From 1994 to 1997, shewas with France Telecom/CNET-CNS, Grenoble,as a Process Engineer in silicon microelectronicstechnology, with a special interest in CVD growth ofepitaxial Si/SiGe heterostructures, rapid thermal pro-cessing (oxidation, nitridation, implant annealing),and bipolar and CMOS technologies. She joined the

CNET Laboratory, Bagneux, France, in 1998 as a Research Engineer insemiconductor III-V compounds and in 1999 the CNRS in the FT/CNRS jointlaboratory (LPN: Laboratoire de Photonique et Nanostructures), Marcoussis,France. The main fields of her research are MOCVD growth of vertical cavitysystems on GaAs and InP substrates, MOCVD growth of quantum dots onGaAs and InP substrates, heteroepitaxy of GaAs on Ge/Si virtual substrate,and III-V device technology as the processing of photonic crystals. She is anauthor or coauthor of about 200 papers in international journals.

Fabien Bretenaker (M06) was born in Metz,France, in 1966. After having graduated from EcolePolytechnique, Palaiseau, France, he received thePh.D. degree from the University of Rennes, Rennes,France, in 1992 while working on ring laser gyro-scopes for Sagem.

He joined the Centre National de la RechercheScientifique, Rennes, in 1994 and worked in Rennesuntil 2002 on laser physics and nonlinear optics.In 2003, he joined the Laboratoire Aime Cotton,Orsay, France, working on nonlinear optics, laser

physics, quantum optics, and microwave photonics. He teaches quantumphysics and laser physics in Ecole Polytechnique, Palaiseau, France. He isalso an Associate Editor of the European Physical Journal Applied Physics.Dr. Bretenaker is a member of the Societe Francaise de Physique, the SocieteFrancaise dOptique, the Optical Society of America, and IEEE Photonics.

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL.?, NO.?, ?? 2013 … · 2013-11-14 · JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL.?, NO.?, ?? 2013 1 Phase Noise of the Radio Frequency (RF) Beatnote