A classical corpuscular approach to optical noise Philippe Gallion Abstract

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1 A classical corpuscular approach to optical noise (invited paper) hilippe Gallion Département Communications et Electronique CNRS, URA 820, Ecole Nationale Supérieure des Télécommunications 46 Rue Barrault, aris Cedex 13 France gallion@com.enst.fr hone : (+33)(0) Fax : (+33)(0) Abstract An heuristic classical corpuscular approach to optical noise is presented. Optical signal generation and detection, including intensity squeezed-light, are described using only well-known electronics engineering concepts.

2 A classical corpuscular approach to optical noise (invited paper) hilippe Gallion Département Communications et Electronique CNRS, URA 820, Ecole Nationale Supérieure des Télécommunications 46 Rue Barrault, aris Cedex 13 France gallion@com.enst.fr hone : (+33)(0) Fax : (+33)(0) Introduction Quantum or semi-classical descriptions have been developed to study the optical intensity noise. A large number of noise sources are usually evoked in optical noise discussions: technical noises (non intrinsic vibrations, thermal fluctuations current fluctuations..) pump current shot-noise (vanishing out under quiet pumping conditions), fluctuations associated to non radiative recombinaisons (negligible far above the threshold current), stochastic nature of spontaneous, emission vacuum fluctuations, optical shot-noise related to absorption and to stimulated emission noise introduced by the internal and mirror(s) losses... Depending on the model all of them are not required. For instance, the quantum electrodynamics relate all these sources to zero point field fluctuations and momentum fluctuations at optical frequencies.. However electronics engineers are not usually familiar with its concepts and formalism. We propose an heuristic classical corpuscular theory using only well-known electronics engineering concepts.in this approach an optical beam is simply considered as a flow of classical particles without mutual interaction. It is valid as far as a large number of photon is observed which is the case.in usual electrical engineering situations The two only classical noise sources The intrinsic field fluctuations are described as the shot noise associated with the photon production or absorption including detection. The well-known shot noise result of the random times of arrival for particles (electrons or photons) without any correlation. For a rate ρ, with average value ρ of incident particles, the corresponding noise power spectrum density is: S ρρ (ω) = ρ. More usual forms of this relation are: S ii (ω) = ei or S pp (ω) = hν p, when i = ρe is the electrical current or when p = ρhν is the optical power. The noise linked to internal and to the mirror losses is taken into account in the form of partition noise Let us consider a beam, with photon rate ρ (t), reaching a beam splitter. For a given photon, there is a given probability p to be transmitted and a probability (1 p) to be reflected, independently of the other photons in the beam. In these conditions the transmitted (t) and reflected ρ 2 (t) photon rates obey the following relations: ρ(t) (1-R)ρ(t)+f ρ (t) ()+ t ρ 2 ()= t ρ(t) + ρ 2 = ρ 2 = ρ 2 2 = ρp(1 p) Rρ(t)-f ρ (t) R

3 The noises associated with the splitting are delta-correlated processes which can be represented by partition Langevin noise forces f ρi (t) The corresponding double-sided noise spectral densities are S ρ1 (ω) =S ρ2 (ω) = ρp(1 p) Noise in light generation For sake of simplicity, the laser studied here is a Fabry-erot laser. The internal photon number and carrier number N and the photon rate i.e. the number of photons emitted per unit of time by the laser facet ρ(t)are linked by the following standard rate equations : 1 N I N t () = G + F () t = G + FN () t ρ()= t + fm () t t t e where G = A(N N o ) is the gain per unit of time, p is the cold cavity photon lifetime inside the cavity, E the spontaneous carrier lifetime, M the time decay associated to mirror losses I the pumping current, e the charge of the electron, A the differential gain and N0 the transparency carrier number. The spontaneous recombinaison term N E, in the carrier number equation, become negligible under high level pumping conditions. The Langevin noise forces F and FN can be expanded as a function of f S (), t f A (), t f E () t and f I ()are t shot-noise Langevin forces linked to the stimulated emission, resonant absorption, spontaneous recombinations and the injected current noise. The partition noise forces f M () t and f D () t refer respectively to the laser facet reflection and the non-resonant absorption. The inversion population factor n sp is defined by n S = S (S A) = S G = S where S and A are the average stimulated emission and absorption rates respectively. When the laser is quietly pumped, for high power and low frequency, the output photon noise is determined by the ratio between the overall photon lifetime and the mirror loss photon lifetime. For low internal loss, it can be reduced below the shot noise limit. The photon flow is then an optical replica of the noiseless injected electron flow. Shot-noise normalized laser external photon noise of a high-reflection coated Fabry-erot at high emitted power for internal loss equal to 4,16,50 cm -1. The pumping level is r = 10. r = (I I th )/I th R= 07. L= 300µ m The squeezing level is : S ρρ ρ = = D Bruit externe normalisé E 10 3 α A =50 cm α =16 cm -1 Α α Α =4 cm M Fréquence (MHz) M

4 Linear optical attenuation To measure the intensity noise of an optical beam, partly collected light and attenuation problems have to be carefully considered. These phenomenons introduce a photon random killing and therefore corrupt the photon statistics. n 2 (z) n(z) z z + dz z n 2 (z + dz) n(z + dz) Because attenuation is a stochastic process, the transmission probability for a slice of thickness dz is defined as (1 αdz) and the absorption probability as αdz where α is the linear absorption coefficient of the medium (figure 2). Thus during a given observation time the average number of transmitted photons n(z + dz) is (1 αdz) n(z). In such conditions the mean square induced by this partition process is expressed as αdz(1 αdz)n(z). The mean square of the transmitted photon number follows: n 2 (z + dz) = (1 αdz) 2 n 2 (z) + αdz(1 αdz)n(z) The first term describes the attenuation of the incoming light fluctuations, while the second term arises from the partition noise introduced by the stochastic nature of attenuation. Let us now consider photons rate ρ(z,t) = n(z,t)/. The double-sided noise spectrum associated with ρ (z+dz,t) is written as: S ρ (ω,z + dz) = (1 αdz) 2 S ρ (ω,z) + αdz(1 αdz)ρ(z) The second term on the right hand side is an additional white noise and this equation. shows that propagation of fluctuation inside a linear attenuator induces noise whitening. The Relative Intensity Noise (R.I.N.) has been previously introduced to characterize laser noise and subsequently the correlation between the photons of the emitted beam. Several definitions have been already used [5,8,9]. The one used here correspond to that introduced by R. Schimpe and J. Arnaud : R.I.N. ( ω)= S ρρ( ω) ρ Taking into account that ρ ( z + dz)= ( 1 αdz)ρ () z the R.I.N. value in z+dz is equal to the one in z. Consequently, whatever the measuring conditions we obtain the same result. With such a definition, a coherent state would have a R.I.N. of zero and a amplitude squeezed state has a negative R.I.N. ρ 2 Intensity noise measurement δ (t)+δρ 2 (t) D2 +δ (t) ou δ (t) δρ 2 (t) Balanced detection ρ 2 arrangement ρ+δρ(t) D1 ρ 2 +δρ 2 (t) The R.I.N. cannot be obtained in a single manipulation and the dual balanced detection is certainly the more convenient tool for its measurement [23,24]. The laser output photon flow is divided into the two

5 arms by a linear and lossless beam splitter. hotons are converted into photoelectrons with two identical an ideal photodiodes. Under this condition the photoelectron statistics on a detector are a direct reproduction of the photon statistics. Both optical and electrical path lengths are supposed to be matched. On both arms, electrical fluctuation and DC components are splitted with a Tee bias. An hybrid junction subtracts or adds the fluctuations. Let T be the beam splitter power transmission coefficient. Then the mean reflected and transmitted photon rates are written as: = (1 T)ρ and ρ 2 = Tρ The noise in both arms is determined by the attenuated incident fluctuation and the partition noise introduced by beam partition. The spectral densities associated to (t) and ρ 2 (t) are linked to the spectral density associated to ρ(t) by : Sρ ( ω) = (( T) 2 Sρ( ω) + T( 1 T) ρ Sρ ( ω) = T 2 Sρ( ω) + T( 1 T) ρ reflection of initial fluctuation added partition noise transmission of initial fluctuation added partition noise The spectral densities associated with (t) ρ 2 (t) and (t) + ρ 2 (t) follow: S ρ1 +ρ 2 (ω) = S ρ (ω) S ρ1 ρ 2 (ω) = (2T 1) 2 S ρ (ω) + 4T(1 T)ρ S ρ1 +ρ 2 (ω) is always equal to the initial fluctuations while for a percent beam splitter S ρ1 ρ 2 (ω) gives the shot-noise level ρ corresponding to the laser output. Under these operating conditions, the experimental setup allows us to measure both laser noise and laser shot-noise level using the same setup and without any adjustment to the detectors or the laser. Then the R.I.N. is directly accessible. Conclusion The intrinsic field fluctuations are described as the shot-noise or partition noise associated with the photon production or absorption. hotons are considered as classical particles and a self-consistent model enables us to study the intensity noise of a semiconductor lasers and the generation of nonclassical states of light under normally and quietly pump condition. References 1. R. Loudon and. L. Knight, J. Mod. Opt, 34, p 709 (1987). 2. G.. Agrawal and N.K. Dutta. Van Nostrand, Rheinhold, New York (1986) Chap J. Arnaud, hys. Rev. A, 45, No 3, pp (1992). 4. Y. Yamamoto, S.Machida, O.Nilsson, hys. Rev. A, 34, No 5, pp (1986). 5.. Gallion, F. Jérémie and J.L. Vey. Opt. Quantum. Electron, 29, pp (1997). 6. R. Schimpe, Zeitschrift für hysik B, Condensated Matter, 52, pp (1983).