Abstract
Noise is one of the basic characteristics of optical amplifiers. Whereas there are various noise sources, the intrinsic one is quantum noise that originates from Heisenberg’s uncertainty principle. This chapter describes quantum noise in optical amplifiers, including population-inversion–based amplifiers such as an Erbium-doped fiber amplifier and a semiconductor optical amplifier, and optical parametric amplifiers. A full quantum mechanical treatment is developed based on Heisenberg equation of motion for quantum mechanical operators. The results provide the quantum mechanical basis for a classical picture of amplifier noise widely used in the optical communication field.
Keywords
- quantum noise
- noise figure
- quantum mechanics
- population-inversion–based amplifier
- optical parametric amplifier
1. Introduction
Noise is one of the important properties in optical amplifiers [1]. The intrinsic noise characteristic is determined by quantum mechanics, especially Heisenberg’s uncertainty principle. This chapter describes quantum noise in optical amplifiers in terms of quantum mechanics. After brief introduction of a classical treatment usually used in the optical communication field, properties of an optically amplified light, such as the mean amplitude, the mean photon number, and their variances, are derived based on first principles of quantum mechanics. Two kinds of optical amplifiers are treated: amplifiers based on two-level interaction in a population-inverted medium, i.e., an Erbium-doped fiber amplifier and a semiconductor optical amplifier, and those based on parametric interaction in an optical nonlinear medium. The results presented here provide the quantum mechanical basis to a phenomenological classical treatment conventionally employed for describing amplifier noise.
2. Classical treatment
A classical treatment of amplifier noise is widely employed in the optical communication field [1, 2], whereas it originates from quantum mechanics. Before presenting a quantum mechanical treatment, we briefly introduce the classical treatment. We first consider the light intensity or the photon number outputted from an amplifier. A photon-number rate equation for light propagating through a population-inverted medium can be expressed as
where
where
Eq. (2) shows that the output photon number is composed of amplified signal photons and ASE photons. Accordingly, the output amplitude is supposed to be a summation of amplitudes of amplified signal and ASE lights as
where
Intensity noise is evaluated using Eq. (3). The output intensity is given by
where the postulate of the ASE light phase being random is used in averaging. The first term represents 2 × <signal output intensity> × <ASE intensity>, which is called the “signal-spontaneous beat noise.” The second and third terms represent the intensity variance of the ASE light, which is called the “spontaneous-spontaneous beat noise.”
As an indicator for the amplifier noise performance, the “noise figure (NF)” is usually used. It is defined as the ratio of the signal-to-noise ratios (SNRs) at the input and output of an amplifier in terms of the optical intensity: NF ≡ (SNR)in/(SNR)out where SNR ≡ (mean intensity)2/(variance of the intensity) in the signal mode. The square of the mean intensity at the output is calculated from Eq. (3) as <|
On the other hand, the input SNR is evaluated for pure monochromatic light in the definition of the noise figure. In quantum mechanics, such a light is called “coherent state,” whose photon-number variance is equal to the mean photon number: <
where
The above-mentioned classical treatment is widely used for noise in optical amplifiers. However, it is based on phenomenological assumptions. (i) Eq. (3) is phenomenologically provided. Though the solution of the photon-number rate equation indicates that the output photon number is composed of amplified signal photons and ASE photons (Eq. (2)), this result does not logically conclude that the output amplitude is a linear summation of the amplified signal and the ASE amplitudes as Eq. (3). (ii) The phase of ASE light is assumed to be random, which is a phenomenological postulate, not logically derived from first principles. Although the above classical treatment is correct and useful in fact, we need quantum mechanics for theoretically confirming its validity, which is presented in the following sections.
3. Quantum mechanics
In this section, we briefly review quantum mechanics, especially the Heisenberg picture [4]. The basic concept of quantum mechanics is that a physical state is probabilistic and the theory only provides mean values of physical quantities, which is given by a quantum mechanical inner product of a physical quantity operator
where
The most important operator in discussing quantum mechanical properties of light is the “annihilation operator,”
We discuss quantum noise of optical amplifiers in the following sections, using the above-mentioned framework of quantum mechanics. Note that the above operator
4. Population-inversion–based amplifiers
Erbium-doped fiber amplifiers (EDFAs) are widely used in optical communications. Optical semiconductor amplifiers are also being developed for compact and integrated amplifying devices. They amplify signal light through interaction between light and a two-level atomic system with population inversion. This section discusses quantum noise in population-inversion-based amplifiers [5].
4.1. Heisenberg equation
The Hamiltonian for a light-atom interacting system can be expressed as [4]
The first and second terms are the Hamiltonians of light and atoms without interaction, respectively, where
Applying the above Hamiltonian to the Heisenberg equations for
Employing the variable translations
where Δ
We solve Eq. (10) by an iterative approximation. First, the first-order solutions are derived by substituting the initial values {
The solutions of these equations are
Next, these first-order solutions are substituted into the right-hand side of Eq. (10a), and the second-order solution is calculated as
We regard Eq. (13) as the time evolution of the field operator during a short time
where
Eq. (14) is the basic expression for discussing quantum properties of light that travels through an amplifier. For the discussion, we also need an initial state of the system at
where |
4.2. Mean amplitude
We first discuss the mean amplitude. The mean amplitude, denoted as
The average of the transition operator
Assuming that the energy levels of each atom are densely distributed, the summation in this equation can be replaced by an integral in the frequency domain, and the real part of Eq. (18) is further calculated as
where Ω is the frequency detuning;
In this expression, the contents of the integral is an odd function around the resonant frequency Ω = 0. Thus, the imaginary part equals 0. Regarding the average of
Eq. (21) describes the time evolution of the mean amplitude of light traveling through an amplifying medium, i.e., the time evolution in a frame moving along with the light, during a short time. This expression can be translated to the spatial evolution along the medium length as
where
Eq. (23) includes (
where
4.3. Mean photon number
We next discuss the mean photon number. The short-time evolution of the photon-number operator is expressed from Eq. (14) as
from which the short-time evolution of the mean photon number is obtained as
In deriving Eq. (26), higher-order interaction terms are neglected, because the short-time evolution is considered here. This short-time evolution is translated to the short-length evolution along the medium length as
from which the following spatial differential equation is obtained:
This equation is equivalent to the photon-number rate equation given in Eq. (1). Therefore, similar to Eq. (1), the output of the mean photon number is calculated as
The first and second terms represent amplified signal photons and ASE photons, respectively. It is noted that ASE photons appear at the output even though there is no such light in the mean amplitude as shown in Eq. (24).
4.4. Amplitude fluctuation
We next discuss amplitude fluctuations or noise. The light amplitude has two quadratures, i.e., the real and imaginary components. The operators representing each component are
From Eq. (14), the short-time evolution of the mean square of the real component is expressed as
where
From Eqs. (30) and (31), the short-time evolution of the variance of the real component is obtained as
This equation is translated to the short-length evolution as
from which the following differential equation is obtained:
From Eq. (34), the variance of the real component at the output is calculated as
The variance of the imaginary component is similarly calculated as
For a coherent incident state, i.e., whose amplitude variances is
The first term 1/4 corresponds to the inherent quantum noise of a coherent state, and the second term represents amplitude fluctuations at the amplifier output in a classical picture. Recalling that the mean amplitude at the amplifier output is that amplified from the incident light with no addition mean field, as indicated in Eq. (24), Eq. (36) suggests that the amplifier output can be regarded as a summation of a clean signal light (i.e., coherent state), displaced from the initial mean amplitude position, and fluctuating light, whose mean value and variance are 0 and
4.5. Photon-number fluctuation
We next discuss photon-number fluctuations. These fluctuations are evaluated employing the variance of the photon number as
where higher-order interaction terms are neglected as before. This expression can be translated to the short-length evolution as
from which the following differential equation is obtained:
with
Here,
From this equation and Eq. (29), the photon-number variance at the amplifier output is expressed as
Recalling that the mean photon numbers of the amplified signal and the spontaneous emission are
The first and second terms in Eq. (42) correspond to the classical intensity noise represented by Eq. (4), as described above, supporting the classical treatment. In addition, the inherent quantum noises are included in Eq. (42), owing to the full quantum mechanical treatment, and the amplified excess noise is simultaneously included as well. Sometimes in the classical treatment, the inherent quantum noise is phenomenologically added as the shot noise arising at the electrical stage after direct detection [2]. In fact, however, it exists in the optical stage as derived above. Therefore, the inherent quantum noise is sometimes called “optical shot noise.”
4.6. Noise figure
The noise figure, defined as the ratio of the signal-to-noise ratios (SNRs) at the input and output of an amplifier in terms of the light intensity or the photon number, is usually used as an indicator for the noise performance of an amplifier. Based on the above results, we describe the noise figure of population-inversion-based amplifiers in this subsection. The output SNR is obtained from Eqs. (29) and (42) as
where only the signal power and the signal-spontaneous beat noise are taken into account, assuming that the amplified signal is sufficiently larger than the spontaneous emission. On the other hand, the input SNR is evaluated for a coherent state, according to the definition of the noise figure, which is
This expression equals the classical result given by Eq. (6).
The noise figure is proportional to the population inversion parameter
The fact that the noise performance is determined by the population inversion parameter can be intuitively understood as follows. The source of amplifier noise is spontaneous emission. A small amount of spontaneous emission suggests a good noise performance. However, spontaneous emission is roughly proportional to the signal gain (Eq. (29)), which is desired to be high as an amplifier. Thus, the amount of spontaneous emission normalized to the signal gain, (ASE power)/(signal gain), can be an indicator for the noise performance. The spontaneous emission rate is proportional to the number of atoms in the upper energy level
5. Optical parametric amplifiers
Whereas population-inversion–based amplifiers are widely used, there is another type of optical amplifiers, that is an optical parametric amplifier (OPA) based on optical nonlinearity [8]. When signal light is incident onto a nonlinear medium along with intense pump light, a signal and idler photons are created from one pump photon in case of second-order nonlinearity, satisfying the energy conservation of ℏ
5.1. Heisenberg equation
The Hamiltonian for parametric interaction between signal and idler via pump light(s) can be expressed as [11]
The first and second terms are the Hamiltonians of signal and idler lights without interaction, respectively, where
From the Heisenberg equation with the above Hamiltonian, temporal differential equations for the field operators are obtained as
These temporal differential equations can be translated to spatial ones as
where
Here, we consider the propagation phase of the right-hand term in the above equations. The coefficient
where
From Eq. (49), the signal field operator at the output is calculated as
where
As shown later, degenerate and nondegenerate OPAs have definitely different characteristics. For simplifying mathematical expressions, hereafter, we rewrite Eq. (50) as
with
The mean values of the physical quantities after amplification can be evaluated using Eq. (51). In the evaluation, we need the initial state in addition. Here, we assume that only signal light is incident to an OPA, and express the initial state as
5.2. Mean amplitude, photon number, and signal gain
The mean amplitude and photon number at the output are evaluated by
Eq. (53) indicates that the signal field is simply amplified while preserving the phase state, with no additional field on average. On the other hand, Eq. (54) shows that the output photons consist of two components. The first term is proportional to the incident photon number, which corresponds to the amplified signal photons with a gain of
It is noted in this expression that parameter
where Eq. (55) is applied. This expression is equivalent to the spontaneous photon number in population-inversion-based amplifiers indicated in Eq. (29) with
Regarding degenerate OPA, on the other hand, its mean output amplitude is calculated as
where
where Δ ≡
The mean photon number in degenerate OPA is calculated from Eq. (51) as
Unfortunately, this equation cannot be further developed, because we cannot readily calculate
In this expression, the first term represents amplified signal photons, and the second term represents spontaneous emission whose mean amplitude is zero as indicated in Eq. (57).
From the first term in Eq. (60), the signal gain is expressed as
which is dependent on the relative phase Δ =
5.3. Amplitude fluctuation
Next, we evaluate the amplitude noise in OPAs. For the evaluation, the light amplitude is decomposed into two quadratures and the variance of each quadrature is calculated, as in Section 4.3. In case of OPAs, the output field operator is phase-shifted by (
The calculation result for nondegenerate OPA is expressed as
where Eqs. (55) and (56) are applied. The first term represents noise amplified from the incident light, and the second term represents additional noise superimposed via OPA. Note that Eq. (63) is equivalent to the amplitude variance of population-inversion-based amplifiers shown by Eq. (35) with
The first term is the inherent quantum noise of a coherent state, and the second term represents noise superimposed via OPA in a classical picture. The sum of the second terms of the two quadratures equals the photon number of spontaneous emission light indicated in Eq. (56). This consideration supports the classical noise treatment, described in Section 2, where spontaneous emission with random phase is superimposed onto signal light at the amplifier output.
For degenerate OPA, on the other hand, the amplitude variances are calculated as
where
5.4. Photon-number fluctuation and noise figure
Next, we discusses photon-number fluctuations in OPAs, which are evaluated through the photon-number variance as
Subsequently, the photon-number variance is obtained as
where Eqs. (55) and (56) are applied. Recalling that the mean photon number of the amplified signal is
Next, we consider degenerate OPA. As indicated by Eq. (59), properties of the photon number in degenerate OPA are hard to evaluate for an arbitrary initial state. Thus, we assume a coherent incident state here. From Eq. (51), the average of the square of the photon-number operator for the initial state |Ψ0> = |
Subsequently, the photon-number variance at the output is
where Eq. (61) is applied. This expression cannot be decomposed and interpreted as that of nondegenerate OPA indicated by Eq. (68), which could be because the amplitude distribution is not simply isotropic in two quadratures, unlike nondegenerate OPA.
The noise figure can be evaluated from the results obtained above. For nondegenerate OPA, it is obtained as
where only the signal-spontaneous beat noise is considered for the output SNR, according to the definition on the noise figure. This noise figure equals that of ideal population-inversion–based amplifiers indicated by Eq. (44) with
For Δ = 0, NF = 1 (0 dB), suggesting no SNR degradation in phase-synchronized degenerate OPA. In fact, a noise figure of less than 3 dB in a phase-sensitive amplifier has been experimentally demonstrated [12, 13].
6. Conclusion
This chapter describes quantum noise of optical amplifiers. Full quantum mechanical treatment based on the Heisenberg equation for physical quantity operators was presented, by which quantum properties of optical amplifiers were derived from first principles. The obtained results are consistent with a conventional classical treatment, except for the inherent quantum noise or the zero-point fluctuation, providing the theoretical base to the conventional phenomenological treatment.
References
- 1.
Yamamoto Y, Inoue K. Noise in amplifiers. Journal of Lightwave Technology. 2003; 21 :2895-2915. DOI: 10.1109/JLT.2003.816887 - 2.
Olsson N. Lightwave system with optical amplifiers. Journal of Lightwave Technology. 1989; 7 :1071-1082 - 3.
Yariv A. Quantum Electronics. 3rd ed. USA: Wiley; 1989. 676 p - 4.
Loudon R. The Quantum Theory of Light. 3rd ed. New York: Oxford; 2000. 438 p - 5.
Inoue K. Quantum mechanical treatment of optical amplifiers based on population inversion. IEEE Journal of Quantum Electronics. 2014; 50 :563-567. DOI: 10.1109/JQE.2014.2325906 - 6.
Smart RG, Zyskind JL, Sulhoff JW, DiGiovanni DJ. An investigation of the noise figure and conversion efficiency of 0.98 μm pumped erbium-doped fiber amplifiers under saturated conditions. IEEE Photonics Technology Letters. 1992; 4 :1261-1264 - 7.
Laming RI, Zechael MN, Payne DN. Erbium-doped fiber amplifier with 54 dB gain and 3.1 dB noise figure. IEEE Photonics Technology Letters. 1992; 4 :1345-1347 - 8.
Marhic M. Fiber Optical Parametric Amplifiers, Oscillators and Related Devices. 1st ed. New York: Cambridge; 2008. 366 p - 9.
Hansryd J, Andrekson PA, Westlund M, Li J, Hedekvist P. Fiber-based optical parametric amplifiers and their applications. IEEE Journal of Selected Topics in Quantum Electronics. 2002; 8 :506-520 - 10.
Inoue K. Quantum noise in parametric amplification under phase-mismatched conditions. Optics Communication. 2016; 366 :71-76. DOI: 10.1016/j.optcom.2015.12.034 - 11.
Walls D, Milburn G. Quantum Optics. 2nd ed. Berlin: Springer; 2008. 425 p - 12.
Imajuku W, Takada A, Yamabayashi Y. Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit. Electronics Letters. 2000; 36 :63-64 - 13.
Tong Z, Bogris A, Lundström C, McKinstrie CJ, Vasilyev M, Karosson M, Andrekson A. Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier. Optics Express. 2010; 18 :14820-14835