Stochastic pulse models of a partially-coherent elementary field representation of pulse coherence

Carlos R. Fernández-Pousa

doi:10.1364/OE.21.009390

1. Introduction

A recurrent topic in coherence theory is the search for representations of genuine (i.e., nonnegative) and tractable correlation functions for modeling the coherence properties of random fields [1–9]. This trend has evolved in parallel in both spatial and temporal domains, mainly due to the equivalence between the properties (Hermiticity and nonnegativity) of the two-point cross-spectral density W(x₁,x₂) across an one-dimensional quasi-monochromatic source and the two-time mutual coherence function (MCF) Γ(t₁,t₂) of a random light pulse.

Focusing our discussion on the temporal domain, a number of representations of Γ(t₁,t₂) with different degrees of generality has been described in the literature. In addition to the broad class of phase-space distributions [1], the list includes the matrix representation [2], the coherent-mode expansion [3] and, more recently, the elementary-field representation [4–6], the reproducing kernel Hilbert space representation [7, 8] and the complex Gaussian representation [9]. Among them the coherent-mode expansion plays a preferential role. This representation is based on the observation that, if the MCF is square integrable in addition to Hermitian and nonnegative, Mercer’s theorem leads to a diagonal expansion of Γ(t₁,t₂) in terms of a (possibly infinite) number of coherent modes [3]. Its construction entails the solution of a Fredholm integral equation with kernel Γ(t₁,t₂) but, as pointed out in [7], this procedure yields analytical models only in some cases. Several of the aforementioned families of genuine kernels have been proposed to overcome this limitation. Although more restrictive than the coherent-mode expansion, a particularly intuitive family is the elementary-field representation [4, 5], where the MCF of an optical pulse is modeled as an incoherent sum of shifted MCFs of fully coherent elementary pulses:

Γ (t_{1}, t_{2}) = \int d u λ (u) e_{0}^{*} (t_{1} - u) e_{0} (t_{2} - u)

with λ(u) a nonnegative and integrable function and e₀(t) analytic signals describing elementary pulses with central optical frequency ν₀ equal to that of the pulse represented by the MCF.

A different approach, motivated from the theory of the reproducing kernel Hilbert space, describes the MCF as a continuous generalization of the coherent-mode expansion [7, 8]:

Γ (t_{1}, t_{2}) = \int d x p (x) e_{0}^{*} (t_{1} | x) e_{0} (t_{2} | x)

where p(x) is a nonnegative function that may contain impulses and e₀(t | x) a collection of functions depending on parameters x. The class of MCFs (2) includes the coherent mode expansion, which is recovered for single-parameter, discrete weighting function p(x) = Σ_n c_nδ (x−n), with c_n > 0 the eingenvalues and e₀(t |n) the corresponding eigenfunctions of kernel Γ(t₁,t₂). Class (2) also includes (1) for p(x) = λ(x) and time shifts e₀(t | x) = e₀(t − x), and the dual model of (1) [6] for frequency shifts e₀(t |x) = e₀(t) exp(j2πxt).

The question addressed in this paper is the connection of these rather abstract representations with stochastic models of pulse emission, where pulses are emitted according to a certain law, or of pulse fluctuations, where internal pulse parameters such as amplitude or width are random variables (rv) that fluctuate in a certain range according to a probability density function (pdf). Such a connection requires the construction of a pulse model e(t) whose second-order correlation function 〈e*(t₁)e(t₂)〉 coincides with MCF (1) or (2). If λ(u) and p(x) are integrable it is clear that we can always associate a pulse fluctuation model, since then (1) and (2) can be interpreted as probabilistic averages over pdfs ~λ(u), p(x) describing the distribution of emission instants u in the elementary field e₀(t) or other internal pulse parameters x in e₀(t |x).

This is not the only possibility, of course. MCF (1) was reproduced in [5] as the second-order correlation function of a continuous emission of elementary pulses, e(t) = e₀(t) ⊗ a(t), with ⊗ standing for convolution, driven by a random function a(t) that describes the amplitude temporal density, i. e., the emission amplitude per unit time. Absorbing amplitude normalization in e₀(t), a(t) may be alternatively interpreted as the number of emissions per unit time, so that λ(u) in (1) is an emission rate. The complex driving function a(t) is taken as modulated complex white noise, a(t) = $\sqrt{λ (t)}$ n(t), so that its correlation is

〈 a^{*} (t_{1}) a (t_{2}) 〉 = \sqrt{λ (t_{1}) λ (t_{2})} 〈 n^{*} (t_{1}) n (t_{2}) 〉 = λ (t_{1}) δ (t_{2} - t_{1})

and using e(t) = e₀(t) ⊗ a(t) MCF (1) is straightforwardly recovered. This second approach is substantially different from the former, however, as it entails the random emission of deterministic pulses instead of the emission of a single, random pulse.

In this regard, we show in this paper that there exists a class of generalized elementary-field models, which can be derived as a subclass of (2), where it is possible (a) to separate the MCF in mutually incoherent constituent MCFs (section 2); (b) to describe the random constituents by at least three different pulse models, all having the same MCF but differing a higher-order, the first two being those described above (section 3); and (c) to show that the fourth-order correlation function of the third model includes those of the first two in the appropriate limits (section 4). These models not only provide a description of MCFs as pulse emissions or fluctuations but also a means of extending elementary-field representations to higher order.

2. A partially-coherent elementary-field representation of pulse coherence

The MCFs we consider here are based on partially-coherent elementary pulses. This class is obtained by setting x = (u,z), e₀(t |x) = e₀(t −u| z) and p(x) = λ(u) p(z) in (2):

\begin{matrix} Γ (t_{1}, t_{2}) = \int d u λ (u) \int d z p (z) e_{0}^{*} (t_{1} - u | z) e_{0} (t_{2} - u | z) \\ = \int d u λ (u) {〈 e_{0}^{*} (t_{1} - u | Z) e_{0} (t_{2} - u | Z) 〉}_{Z} \\ = \int d u λ (u) Γ_{0} (t_{1} - u, t_{2} - u) \end{matrix}

Nonnegative functions λ(u), p(z) are assumed integrable and may contain δ functions. Function p(z) is normalized as ∫ dz p(z) = 1 to interpret it as a pdf. In the first line of (4) λ(u) has again the interpretation of emission rate of elementary pulses e₀(t |z) and so N₀ = ∫ du λ(u) is its total number. We stress that this interpretation of N₀ relies on the fact that λ(u) can be understood as an emission rate. As will be shown below, in models involving the emission of a single pulse λ(u) cannot be an emission rate, and thus the number of emitted pulses is not N₀. Pulses e₀(t |z) are assumed random with internal parameters described by a rv Z with pdf p(z), so that the integral over z in (4) becomes a probabilistic average over Z, as shown in the second line. The third line describes Γ(t₁,t₂) in terms of the MCF Γ₀(t₁,t₂) of elementary fields e₀(t |z) and shows that (4) generalizes (1) by representing the MCF as an incoherent sum of shifted MCFs of, in general, partially-coherent elementary pulses. If both λ(u) and p(z) are proportional to delta functions, however, (4) describes a fully coherent pulse.

We now decompose (4) into fully and partially-coherent constituents. Suppose that both p(z) and λ(u) contain impulses q_n, σ_r > 0 at z_n, u_r together with continuous components p_m(z), λ_s(u) > 0 with connected support [10]. We write

p (z) = \sum_{n} q_{n} δ (z - z_{n}) + \sum_{m} p_{m} (z) λ (u) = \sum_{r} σ_{r} δ (u - u_{r}) + \sum_{s} λ_{s} (u)

and using (5) in (4) the MCF results in a weighted sum of MCFs of mutually incoherent pulses which are either fully coherent (n, r) or partially coherent (n, s), (m, r) and (m, s):

\begin{array}{l} Γ (t_{1}, t_{2}) = \sum_{n, r} q_{n} σ_{r} e_{0}^{*} (t_{1} - u_{r} | z_{n}) e_{0} (t_{2} - u_{r} | z_{n}) \\ + \sum_{n, s} q_{n} \int d u λ_{s} (u) e_{0}^{*} (t_{1} - u | z_{n}) e_{0} (t_{2} - u | z_{n}) + \sum_{m, r} σ_{r} \int d z p_{m} (z) e_{0}^{*} (t_{1} - u_{r} | z) e_{0} (t_{2} - u_{r} | z) \\ + \sum_{m, s} \int d u λ_{s} (u) \int d z p_{m} (z) e_{0}^{*} (t_{1} - u | z) e_{0} (t_{2} - u | z) \end{array}

This decomposition generalizes similar separations of the elementary-field model (1), see [11] and references therein. Since a sum with positive coefficients of nonnegative kernels is a genuine MCF we can focus on the construction of pulse models for each constituent in (6).

3. Stochastic pulse models reproducing the mutual coherence function

For each partially-coherent constituent in (6) we reproduce the corresponding MCF (4) with three different stochastic pulse models. The first is a fluctuating pulse of the form,

e_{1} (t) = \sqrt{N_{0}} e_{0} (t - U | Z) e^{j Θ}

with independent rv U and Z distributed according to pdfs λ(u)/N₀ and p(z), respectively. Phase Θ is uniformly distributed in [0, 2π) and assures circularity and mutual incoherence among constituents. The amplitude normalization

\sqrt{N_{0}}

in (7) accounts for the normalization of the pdf λ(u)/N₀, and each member of the ensemble (7) contains a single pulse. Here λ(u) is not an emission rate. The relevant physical quantity is pdf λ(u)/N₀ and its interpretation is simply the timing jitter distribution of the emission instants of wave e₁(t).

The second model is the emission driven by noise which, in contrast to (3), requires a random assignment of parameters z to e₀(t|z) for each realization of noise. To this end we use a complete orthonormal basis φ_k(u) of an interval containing the pulse duration to construct a Karhunen-Loéve expansion of white noise n(u) in terms of zero-mean, independent circular complex rv a_k with unit variance [3, 12]: n(u) = Σ_k a_k φ_k(u), with 〈a_k〉 = 0 and 〈a_k*a_p〉 = δ_kp. To each k we attach parameters Z_k to the emitted elementary pulse e₀(t|z). Z_k are also assumed independent and identically distributed (iid), and also independent of a_k. The optical field is:

e_{2} (t) = \int d u \sqrt{λ (u)} \sum_{k} e_{0} (t - u | Z_{k}) a_{k} φ_{k} (u)

Then, using 〈a_k*a_p〉 = δ _kp the correlation function is:

{〈 e_{2}^{*} (t_{1}) e_{2} (t_{2}) 〉}_{Z_{k} a_{k}} = \iint d u_{1} d u_{2} \sqrt{λ (u_{1}) λ (u_{2})} \sum_{k} φ_{k}^{*} (u_{1}) φ_{k} (u_{2}) {〈 e_{0}^{*} (t_{1} - u_{1} | Z_{k}) e_{0} (t_{2} - u_{2} | Z_{k}) 〉}_{Z_{k}}

The average over Z_k gives the same value for any k since Z_k are iid, so it can be taken out from the sum, and using the completeness of the basis functions φ_k(u), Σ_k φ_k*(u₁)φ_k(u₂) = δ (u₂−u₁), we recover (4). If λ(u) is proportional to δ (u) the square root in (8) is to be applied to an approximation to the delta function, such as a rectangular or a Gaussian function, and then the limit to the delta function is taken in (9) after using the completeness of φ_k(u). Here, as in the original noise-driven representation in [5], λ(u) has the interpretation of an emission rate.

The third is an emission model triggered by a Poisson impulse process. In these models the elementary optical fields are emitted probabilistically according to the emission rate λ(u), so that dN₀(u) = λ(u)du represents the average number of emissions in the interval (u, u + du). To be more precise, the number of emissions in an interval (u_a, u_b) is a rv that fluctuates according to a Poisson distribution with mean and variance ∫ $_{u_{a}}^{u_{b}}$ duλ(u) and, in particular, the total number of emissions is not fixed, but a Poisson rv with mean and variance N₀ = ∫duλ(u) [12]. The representation considered here can be viewed as a generalization to nonstationary pulsed light of the model of stationary emission of random fields in [13], where λ(u) is constant, or the model of nonstationary emission of nonrandom fields in [14], where λ(u) is time-varying and non-integrable. We work first the simpler case of the standard elementary-field MCF (1), which is reproduced by the emission of nonrandom fields e₀(t) triggered by a nonstationary Poisson process with λ(u) time-varying and integrable [15–17]. The field is:

e_{3} (t) = \sum_{k} e_{0} (t - τ_{k}) = e_{0} (t) \otimes \sum_{k} δ (t - τ_{k}) = e_{0} (t) \otimes Δ (t)

where the emission instants τ_k are distributed according to λ (u). The analysis of this model is particularly simple as (10) can be understood as the transit of a nonstationary Poisson impulse process Δ(t) through a linear filter with complex impulse response e₀(t). The computation of the mean and autocorrelation of Δ(t) is standard [15, 16] and yields 〈Δ(t)〉_P = λ(t) and

{〈 Δ (t_{1}) Δ (t_{2}) 〉}_{P} = {〈 Δ (t_{1}) 〉}_{P} {〈 Δ (t_{2}) 〉}_{P} + λ (t_{1}) δ (t_{2} - t_{1})

with P standing for average over the Poisson process. We get a mean optical field 〈e₃(t)〉_P = λ(t) ⊗ e₀(t), which simply describes the emission of elementary fields according to rate λ(t). However, this average optical field takes into account the random emission times of the triggering events, so that 〈e₃(t)〉_P becomes zero under quite general conditions. On the one hand, one can assume that e₀(t) is random and circular symmetric [13], a property that can be formalized by attaching a phase exp(jΘ) to e₀(t). On the other, we can use the following spectral argument: in the spectral domain the convolution is the product 〈E₃(ν)〉_P = Λ(ν)E₀(ν), which is nonzero only if these two functions overlap. E₀(ν) is centered at frequency ν₀ and we denote its spectral half-width by Δν₀. Λ(ν) is centered at zero frequency and its bandwidth (from dc to the highest frequency) is Δν_λ. Spectral overlap requires Δν_λ >ν₀− Δν₀ which cannot be the case unless Δν_λ is an optical frequency, a condition that would require emission rates with temporal features of extremely short duration. In other words, if we assume that the emission rate is smooth at the scales of the oscillation of the emitted elementary field, the average field is zero. Since e₃(t) is zero-mean only the second term in (11) contributes to the MCF 〈e₃*(t₁)e₃(t₂)〉_P = e₀*(t₁)⊗ 〈Δ(t₁)Δ(t₂)〉_P ⊗e₀(t₂). The computation then becomes equivalent to that of (3) and leads directly to MCF (1).

To construct the third stochastic pulse representation of the generalized model (4) we must assume now that the elementary pulses in (10) are random and parameterized by a set of rv Z_k, one for each triggering event, which are assumed iid with pdf p(z_k). The wave is:

e_{3} (t) = \sum_{k} e_{0} (t - τ_{k} | Z_{k})

We also assume that emission instants τ_k are independent of Z_k. Mean and MCF are calculated by performing two independent averages over the Poisson impulses P and over rv Z_k. The computation of these two averages is best performed by use of method of the characteristic function. The first (Φ) and second (Ψ) two-time characteristic functions are defined as

Φ (s_{1}, s_{2}) = \exp Ψ (s_{1}, s_{2}) = {〈 \exp (s_{1} e_{3}^{*} (t_{1}) + s_{2} e_{3} (t_{2})) 〉}_{P Z_{k}}

and the MCF is obtained by derivation of Φ over s₁, s₂ and subsequent restriction to s₁ = s₂ = 0 [15–17]. The characteristic functions of model (12) are well-known in the theory of Poisson point processes [16, 17]:

Ψ (s_{1}, s_{2}) = \log Φ (s_{1}, s_{2}) = \int d u λ (u) [{〈 e^{s_{1} e_{0}^{*} (t_{1} - u | Z) + s_{2} e_{0} (t_{2} - u | Z)} 〉}_{Z} - 1]

where again the subscript Z denotes probabilistic average over a single rv with pdf p(z). Now we can show the desired result: performing the derivative ∂/∂s₂ ≡ ∂₂ we get

{〈 e_{3} (t_{2}) 〉}_{P Z_{k}} = \partial_{2} Φ (s_{1}, s_{2}) |_{s_{1} = s_{2} = 0} = (\partial_{2} Ψ) e^{Ψ (s_{1}, s_{2})} |_{s_{j} = 0} = \partial_{2} Ψ |_{s_{j} = 0} = λ (t_{2}) \otimes {〈 e_{0} (t_{2} | Z) 〉}_{Z} = 0

which is zero by extending the spectral argument above to the ensemble e₀(t|Z) or by assuming that e₀(t|Z) is circular symmetric, as was done in [13]. Finally, derivation of (13) over s₁, s₂ and restriction to s₁ = s₂ = 0 gives

{〈 e_{3}^{*} (t_{1}) e_{3} (t_{2}) 〉}_{P Z_{k}} = \partial_{1} \partial_{2} Φ (s_{1}, s_{2}) |_{s_{j} = 0} = (\partial_{1} Ψ \partial_{2} Ψ + \partial_{1} \partial_{2} Ψ) e^{Ψ (s_{1}, s_{2})} |_{s_{j} = 0} = \partial_{1} \partial_{2} Ψ |_{s_{j} = 0}

and using the explicit form (14) of Ψ(s₁, s₂) we recover MCF (4). Note also that the limit of λ(u) proportional to a delta function is regular in the characteristic function (14).

4. Higher-order coherence theory

We now explore the differences among models (7), (8) and (12), which lead to the same second-order MCF (4), by analyzing their fourth-order correlation functions. Starting from (12) the computation can be done by use of the four-time characteristic function,

Φ (s_{1}, s_{2}, s_{3}, s_{4}) = \exp Ψ (s_{1}, s_{2}, s_{3}, s_{4}) = 〈 \exp [\sum_{q = 1, 3} s_{q} e_{3}^{*} (t_{q}) + \sum_{p = 2, 4} s_{p} e_{3} (t_{p}) {] 〉}_{P Z_{k}}

The form of the fourth-order characteristic function is similar to (14) but with a four-term sum in the exponential function [15–17]:

Ψ (s_{1}, s_{2}, s_{3}, s_{4}) = \log Φ (s_{1}, s_{2}, s_{3}, s_{4}) = \int d u λ (u) [{〈 e}^{\sum_{q = 1, 3} s_{q} e_{0}^{*} (t_{q} - u | Z) + \sum_{p = 2, 4} s_{p} e_{0} (t_{p} - u | Z)} 〉_{Z} - 1]

Derivation over s₁,… s₄ leads to a number of terms, some of which vanish due to the assumed circularity of e₀(t|Z). In particular, terms involving one or three derivatives, ∂_jΨ and ∂_i∂_j∂_kΨ, vanish, and also those non-circular terms with two derivatives, ∂₁∂₃Ψ and ∂₂∂₄Ψ. We get:

\begin{array}{l} Γ_{3} (t_{1}, t_{2}, t_{3}, t_{4}) = {〈 e_{3}^{*} (t_{1}) e_{3} (t_{2}) e_{3}^{*} (t_{3}) e_{3} (t_{4}) 〉}_{P Z_{k}} = Γ (t_{1}, t_{2}) Γ (t_{3}, t_{4}) + Γ (t_{1}, t_{4}) Γ (t_{3}, t_{2}) \\ + \int d u λ (u) {〈 e_{0}^{*} (t_{1} - u | Z) e_{0} (t_{2} - u | Z) e_{0}^{*} (t_{3} - u | Z) e_{0} (t_{4} - u | Z) 〉}_{Z} \end{array}

where

Γ (t_{i}, t_{j})

is given by (4) and we have introduced a subscript in

Γ_{3} (t_{1}, ... t_{4})

to highlight that this correlation function is for model e₃(t) (12). For λ(u) = const., t₁ = t₂, t₃ = t₄ and 〈.〉_Z Gaussian elementary-field correlations, (19) is the intensity correlation function computed as Eq. (30) in [13]. Here, the first two terms in (19) describe nonstationary Gaussian correlations and, being proportional to the squared emission rate, arise from the emission of two elementary fields. In turn, the last term in (19) describes an aggregate contribution to the correlation originated from single emissions of elementary fields at instant u and rate λ(u).

In order to compare (19) with the fourth-order correlation function of model (8), we first assume that the driving noise n(u) in (8) is circular Gaussian, 〈a_k*a_p a_r*a_s〉 = 〈a_k*a_p〉〈a_r*a_s〉 + 〈a_k*a_s〉〈a_r*a_p〉 [12]. This assumption is natural since n(u) is a noise process. Using this and performing the same algebra as in (9) we get Gaussian correlations,

Γ_{2} (t_{1}, t_{2}, t_{3}, t_{4}) = {〈 e_{2}^{*} (t_{1}) e_{2} (t_{2}) e_{2}^{*} (t_{3}) e_{2} (t_{4}) 〉}_{Z_{k} a_{k}} = Γ (t_{1}, t_{2}) Γ (t_{3}, t_{4}) + Γ (t_{1}, t_{4}) Γ (t_{3}, t_{2})

so that the first two terms in (19) coincide with the fourth-order correlation of (8). As for the fluctuating pulse model (7) the fourth-order correlation function is given by:

Γ_{1} (t_{1}, t_{2}, t_{3}, t_{4}) = N_{0} \int d u λ (u) {〈 e_{0}^{*} (t_{1} - u | Z) e_{0} (t_{2} - u | Z) e_{0}^{*} (t_{3} - u | Z) e_{0} (t_{4} - u | Z) 〉}_{Z}

and equals N₀ times the last term in (19). Using (20) and (21) in (19) we finally get:

Γ_{3} (t_{1}, t_{2}, t_{3}, t_{4}) = Γ_{2} (t_{1}, t_{2}, t_{3}, t_{4}) + \frac{1}{N_{0}} Γ_{1} (t_{1}, t_{2}, t_{3}, t_{4})

The Poisson model thus accounts for the higher-order correlations of both pulse emissions and fluctuations. First, if the number of emissions N₀ is large the theory becomes Gaussian; this is a well-known limit in the theory of Poisson processes [15–17]. In our case, Eq. (22) shows that in this limit the fourth-order correlations of the Poisson model (12) coincides with that of emission driven by noise (8). In words, a large number of random emissions works as a noise driving function, where the structure of fourth-order correlations is dominated by independent pairs of random emissions. In the opposite limit of sparse emission N₀ < 1 the dominant term in (22) is the last, which describes the contribution of the single emissions. In this case the fourth-order correlation function becomes proportional to that of a single fluctuating pulse (7). Equation (22) also shows three ways of extending the elementary-field representation to higher-order. When there is a noise-like elementary pulse generation mechanism with a large number of independent emissions the theory should be modeled as Gaussian, Γ₂. If the system emits single fluctuating pulses, the extension should be by Γ₁. And if the number of emissions is independent, random but not necessary large, the system should be modeled as Poisson, Γ₃.

5. Conclusion

We have introduced a representation (4) of the pulse MCF as an incoherent superposition of partially-coherent elementary fields and shown that its random constituents can be modeled as a fluctuating pulse (7), noise driven emission (8) or Poisson triggered emission (12). At fourth order this last model shows the wider class of correlations. The generalized elementary field representation and the pulse models provide a connection with descriptions based on pulse emissions or fluctuations, and also represent a means of extending elementary-field models to higher-order coherence theory.

Acknowledgments

This paper has been supported by Ministerio de Economía y Competitividad, Spain, under project TEC2011-29120-C05-02.

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Stochastic pulse models of a partially-coherent elementary field representation of pulse coherence

Abstract