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Quantum Noise Properties of Non-Ideal Optical Amplifiers and Attenuators Home Search Collections Journals About Contact us My IOPscience Quantum noise properties of non-ideal optical amplifiers and attenuators This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Opt. 13 125201 (http://iopscience.iop.org/2040-8986/13/12/125201) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.151.240.150 The article was downloaded on 27/08/2012 at 22:19 Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF OPTICS J. Opt. 13 (2011) 125201 (7pp) doi:10.1088/2040-8978/13/12/125201 Quantum noise properties of non-ideal optical amplifiers and attenuators Zhimin Shi1, Ksenia Dolgaleva1,2 and Robert W Boyd1,3 1 The Institute of Optics, University of Rochester, Rochester, NY 14627, USA 2 Department of Electrical Engineering, University of Toronto, Toronto, Canada 3 Department of Physics and School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, K1N 6N5, Canada E-mail: [email protected] Received 17 August 2011, accepted for publication 10 October 2011 Published 24 November 2011 Online at stacks.iop.org/JOpt/13/125201 Abstract We generalize the standard quantum model (Caves 1982 Phys. Rev. D 26 1817) of the noise properties of ideal linear amplifiers to include the possibility of non-ideal behavior. We find that under many conditions the non-ideal behavior can be described simply by assuming that the internal noise source that describes the quantum noise of the amplifier is not in its ground state. We validate this model by showing that it reproduces the known predictions of two models of non-ideal amplifiers: a laser amplifier with incomplete inversion and a cascade of alternating ideal amplifiers and ideal attenuators. We also describe implications of this model for practical devices. Keywords: quantum noise, optical amplifiers (Some figures may appear in colour only in the online journal) 1. Introduction noise can be attributed mainly to the beating between the amplified signal field and the amplified vacuum fluctuations. Optical amplification is required for the operation of many A non-ideal amplifier or attenuator is a device that optical systems, including those used in telecommunica- introduces more noise than that required by the laws of tions, optical information processing, and quantum optics. quantum mechanics. Non-ideal amplifiers or attenuators have Examples of optical amplifiers include erbium doped fiber been treated theoretically by considering them to be comprised amplifiers [1, 2], semiconductor optical amplifiers [3], and of alternating pairs of ideal amplifiers and attenuators [8], by surface plasmon amplifiers [4]. Optical systems also often means of a three-dimensional quantum beam splitter model [9], contain loss elements, which are present either by design or by explicit consideration of the propagation of light through or because of the inability to fabricate perfectly transmitting a not-totally-inverted laser gain medium [10]. optical components. It is well known that both gain and loss introduce noise into an optical system, and this noise can be In this paper, we present a generalization of Caves’s [5] highly undesirable under many circumstances. treatment of the noise properties of linear amplifiers by In many cases of interest, the noise generated by an allowing for non-ideal behavior. We find that a broad class amplifier or attenuator can be ascribed to a purely quantum- of amplifiers and attenuators can be treated by making just mechanical origin. Indeed, the quantum-mechanical properties one additional assumption, namely that the internal quantum of the electromagnetic field pose a fundamental limit to the noise source of the device is not in its ground state. We noise level of an optical beam. An ideal phase-insensitive begin with a brief summary of the usual treatments for ideal optical amplifier can be treated by a quantum model that amplifiers or attenuators in section 2. Our new model is assumes the presence of two input ports, one for the signal introduced in section 3 and applied specifically to several types field and another that represents the internal noise source of of practical amplifiers. The paper concludes with a discussion the amplification process [5–7]. In these models, the amplifier and summary in section 4. 2040-8978/11/125201+07$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA J. Opt. 13 (2011) 125201 ZShiet al Here, the first term represents the amplification of the fluctuations present in the input field, and the second term represents the added noise. The noise properties of an amplifier are often character- ized by a noise figure [14], defined as Figure 1. Quantum model of an ideal linear amplifier (a) and an attenuator (b). Here and in figure 2, we use the symbol vˆ to designate (SNR) F ≡ in , an internal noise source cˆ that is in the vacuum state. (7) (SNR)out where (SNR)in and (SNR)out are the signal-to-noise ratios 2. Quantum model of an ideal linear amplifier or (SNRs) of the input and output fields, respectively, which are attenuator defined as follows: ˆn 2 Let us first summarize the quantum description of the noise (SNR) ≡ a , (8) in ˆ2 properties of an ideal amplifier for the case of a single-mode na input field propagating through an ideal linear amplifier with G2ˆn 2 intensity gain G. We adopt the standard model shown in (SNR) ≡ a . (9) out ˆ2 figure 1(a) which was introduced by earlier workers [5, 11–13]. nb This model ascribes the noise properties of the amplifier to an Note that in (SNR)out we use Gˆna for the amplified internal quantum noise source that represents the fluctuations signal field, not the total output field that includes the noise that must (by means of the fluctuation-dissipation theorem) contributions. accompany any gain or loss mechanism. The noise figure of an ideal optical amplifier is then found The strength of this noise source is adjusted to ensure that from the expressions given above to be the field operator of the output mode possesses standard boson commutation relations. Specifically, the photon annihilation 1 ˆn +1 F = 1 + 1 − a . (10) operators for the input and output fields are denoted by aˆ ˆ2 G na and bˆ, respectively, and are assumed to satisfy the standard commutation relations When the fluctuations of the input field follow Poisson statistics, so that nˆ2 =ˆn , one sees that the value of the [ˆ, ˆ †]=[ˆ, ˆ†]= . a a a a b b 1 (1) noise figure F of an ideal quantum amplifier approaches the ˆ The annihilation operator for the output field is expressed in value 2 (or 3 dB) when both G and na are much greater than terms of that of the input field and of a Langevin noise source unity [15]. as follows [5]: This same model can be used to describe an ideal attenuator of intensity transmission T . We again model this / / bˆ = G1 2aˆ + (G − 1)1 2cˆ†. (2) device in terms of an input signal field aˆ, an output field bˆ,and an internal noise field cˆ, as shown in figure 1(b). In order to ˆ Here c is a boson operator, denoting the internal noise field, preserve the commutation relations for the output field bˆ,we [ˆ, ˆ†]= which also must satisfy the relation c c 1. The internal express it as noise field is assumed to be uncorrelated from the input field, bˆ = T 1/2aˆ + (1 − T )1/2cˆ. (11) that is, [ˆa, cˆ†]=[ˆa, cˆ]=0. (3) Here, the Langevin operator is again assumed to be uncorrelated from the input operator: It is straightforward to verify that under these circumstances ˆ ˆ† the commutation relation [b, b ]=1 is satisfied. For the case [ˆa, cˆ†]=[ˆa, cˆ]=0. (12) of an ideal amplifier, one assumes that the internal noise field is in the vacuum state, that is, For an ideal attenuator, we again assume that cˆ denotes a † vacuum state input, i.e., ˆnc=ˆc cˆ=0. (4) ˆn =ˆc†cˆ=0. (13) Consequently, the expectation value of the photon number of c the output field is given by The expected photon number for the output field and its ˆ† ˆ variance are then given by ˆnb=b b=Gˆna+(G − 1), (5) ˆ† ˆ † ˆn =b b=T ˆn (14) where ˆna =ˆa aˆ is the average photon number of the b a input field. One sees that, in addition to the input field being and amplified by a factor of G, (G − 1) noise photons are added to ˆ2≡ˆ2−ˆ 2 the output field. The variance of the output photon number is nb nb nb similarly given by = 2ˆ2+ ( − )ˆ , T na T 1 T na (15) ˆ2≡ˆ2−ˆ 2 nb nb nb respectively, for such an ideal attenuator. Here, the first term = 2ˆ2+ ( − )(ˆ + ). G na G G 1 na 1 (6) represents the attenuation of the fluctuations present in the 2 J. Opt. 13 (2011) 125201 ZShiet al input beam, and the second term represents the noise added due to the random loss of photons from the signal field. The noise figure of the ideal attenuator is then found to be 1 ˆna F = 1 + − 1 . (16) ˆ2 T na Note that this noise figure can become arbitrarily large as the transmission of the attenuator approaches zero, which can be attributed to the random loss of photons from the signal field. Figure 2. One can describe a non-ideal amplifier using (a) a series of 3. Generalized model for a non-ideal linear amplifier alternating ideal amplifiers and attenuators, or (b) a generalized or attenuator quantum beam splitter model.
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