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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 9390–9396
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Stochastic pulse models of a partially-coherent elementary field representation of pulse coherence

Carlos R. Fernández-Pousa  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 9390-9396 (2013)
http://dx.doi.org/10.1364/OE.21.009390


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Abstract

A representation of the mutual coherence function (MCF) of a light pulse as an incoherent sum of partially-coherent elementary pulses is introduced. It is shown that this MCF can be decomposed into fully and partially-coherent constituents and three different pulse models of partially-coherent constituents are constructed: single elementary-pulse fluctuations, emission of elementary fields driven by white noise, and elementary pulses triggered by Poisson impulses. The fourth-order correlation function of this last model includes as limit cases those of the fluctuating-pulse and noise-driven-emission models. These results provide a means of extending elementary-field models to higher-order coherence theory.

© 2013 OSA

1. Introduction

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

1. M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon. 3(4), 272–362 (2011). [CrossRef]

], the list includes the matrix representation [2

2. J. Perina, Coherence of Light (Kluwer, 1985).

], the coherent-mode expansion [3

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

] and, more recently, the elementary-field representation [4

4. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional beams,” Opt. Commun. 27(2), 185–188 (1978). [CrossRef]

6

6. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007). [CrossRef] [PubMed]

], the reproducing kernel Hilbert space representation [7

7. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

, 8

8. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013). [CrossRef] [PubMed]

] and the complex Gaussian representation [9

9. S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express 19(18), 17086–17091 (2011). [CrossRef] [PubMed]

]. 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

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

]. Its construction entails the solution of a Fredholm integral equation with kernel Γ(t1,t2) but, as pointed out in [7

7. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

], 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

4. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional beams,” Opt. Commun. 27(2), 185–188 (1978). [CrossRef]

, 5

5. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14(12), 5007–5012 (2006). [CrossRef] [PubMed]

], where the MCF of an optical pulse is modeled as an incoherent sum of shifted MCFs of fully coherent elementary pulses:
Γ(t1,t2)=duλ(u)e0*(t1u)e0(t2u)
(1)
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

7. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

, 8

8. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013). [CrossRef] [PubMed]

]:
Γ(t1,t2)=dxp(x)e0*(t1|x)e0(t2|x)
(2)
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) = Σn cnδ (xn), with cn > 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(tx), and the dual model of (1) [6

6. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007). [CrossRef] [PubMed]

] for frequency shifts e0(t |x) = e0(t) exp(jxt).

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

5. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14(12), 5007–5012 (2006). [CrossRef] [PubMed]

] 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) = λ(t)n(t), so that its correlation is
a*(t1)a(t2)=λ(t1)λ(t2)n*(t1)n(t2)=λ(t1)δ(t2t1)
(3)
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):
Γ(t1,t2)=duλ(u)dzp(z)e0*(t1u|z)e0(t2u|z)=duλ(u)e0*(t1u|Z)e0(t2u|Z)Z=duλ(u)Γ0(t1u,t2u)
(4)
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 qn, σr > 0 at zn, ur together with continuous components pm(z), λs(u) > 0 with connected support [10

10. The support of p(z) is the set of values of z s.t. p(z) ≠ 0.The hypothesis of connectedness is necessary since the range of variation of a fluctuating pulse parameter is always assumed connected.

]. We write
p(z)=nqnδ(zzn)+mpm(z)λ(u)=rσrδ(uur)+sλs(u)
(5)
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):
Γ(t1,t2)=n,rqnσre0*(t1ur|zn)e0(t2ur|zn)+n,sqnduλs(u)e0*(t1u|zn)e0(t2u|zn)+m,rσrdzpm(z)e0*(t1ur|z)e0(t2ur|z)+m,sduλs(u)dzpm(z)e0*(t1u|z)e0(t2u|z)
(6)
This decomposition generalizes similar separations of the elementary-field model (1), see [11

11. M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary-field representation of supercontinuum,” J. Opt. Soc. Am. B 30(1), 21–26 (2013). [CrossRef]

] 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,
e1(t)=N0e0(tU|Z)ejΘ
(7)
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 N0in (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 φk(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 ak with unit variance [3

3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

, 12

12. J. W. Goodman, Statistical Optics (Wiley, 1985).

]: n(u) = Σk ak φk(u), with 〈ak〉 = 0 and 〈ak*ap〉 = δkp. To each k we attach parameters Zk to the emitted elementary pulse e0(t|z). Zk are also assumed independent and identically distributed (iid), and also independent of ak. The optical field is:
e2(t)=duλ(u)ke0(tu|Zk)akφk(u)
(8)
Then, using 〈ak*ap〉 = δ kp the correlation function is:
e2*(t1)e2(t2)Zkak=du1du2λ(u1)λ(u2)kφk*(u1)φk(u2)e0*(t1u1|Zk)e0(t2u2|Zk)Zk
(9)
The average over Zk gives the same value for any k since Zk are iid, so it can be taken out from the sum, and using the completeness of the basis functions φk(u), Σk φk*(u1)φk(u2) = δ (u2u1), 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 φk(u). Here, as in the original noise-driven representation in [5

5. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14(12), 5007–5012 (2006). [CrossRef] [PubMed]

], 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 (ua, ub) is a rv that fluctuates according to a Poisson distribution with mean and variance ∫uaubduλ(u) and, in particular, the total number of emissions is not fixed, but a Poisson rv with mean and variance N0 = ∫duλ(u) [12

12. J. W. Goodman, Statistical Optics (Wiley, 1985).

]. 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

13. B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27(1), 360–374 (1983). [CrossRef]

], where λ(u) is constant, or the model of nonstationary emission of nonrandom fields in [14

14. C. R. Fernández-Pousa, “Nonstationary elementary-field light randomly triggered by Poisson impulses,” J. Opt. Soc. Am. A (to be published).

], 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

15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.

17

17. E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

]. The field is:
e3(t)=ke0(tτk)=e0(t)kδ(tτk)=e0(t)Δ(t)
(10)
where the emission instants τk 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

15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.

, 16

16. B. Picinbono, Random Signals and Systems (Prentice-Hall, 1993), Chap. 8.

] and yields 〈Δ(t)〉P = λ(t) and
Δ(t1)Δ(t2)P=Δ(t1)PΔ(t2)P+λ(t1)δ(t2t1)
(11)
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

13. B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27(1), 360–374 (1983). [CrossRef]

], 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 Δνλ. Spectral overlap requires Δνλ >ν0− Δν0 which cannot be the case unless Δνλ 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)〉Pe0(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 Zk, one for each triggering event, which are assumed iid with pdf p(zk). The wave is:
e3(t)=ke0(tτk|Zk)
(12)
We also assume that emission instants τk are independent of Zk. Mean and MCF are calculated by performing two independent averages over the Poisson impulses P and over rv Zk. 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(s1,s2)=expΨ(s1,s2)=exp(s1e3*(t1)+s2e3(t2))PZk
(13)
and the MCF is obtained by derivation of Φ over s1, s2 and subsequent restriction to s1 = s2 = 0 [15

15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.

17

17. E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

]. The characteristic functions of model (12) are well-known in the theory of Poisson point processes [16

16. B. Picinbono, Random Signals and Systems (Prentice-Hall, 1993), Chap. 8.

, 17

17. E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

]:
Ψ(s1,s2)=logΦ(s1,s2)=duλ(u)[es1e0*(t1u|Z)+s2e0(t2u|Z)Z1]
(14)
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
e3(t2)PZk=2Φ(s1,s2)|s1=s2=0=(2Ψ)eΨ(s1,s2)|sj=0=2Ψ|sj=0=λ(t2)e0(t2|Z)Z=0
(15)
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

13. B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27(1), 360–374 (1983). [CrossRef]

]. Finally, derivation of (13) over s1, s2 and restriction to s1 = s2 = 0 gives
e3*(t1)e3(t2)PZk=12Φ(s1,s2)|sj=0=(1Ψ2Ψ+12Ψ)eΨ(s1,s2)|sj=0=12Ψ|sj=0
(16)
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(s1,s2,s3,s4)=expΨ(s1,s2,s3,s4)=exp[q=1,3sqe3*(tq)+p=2,4spe3(tp)]PZk
(17)
The form of the fourth-order characteristic function is similar to (14) but with a four-term sum in the exponential function [15

15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.

17

17. E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

]:
Ψ(s1,s2,s3,s4)=logΦ(s1,s2,s3,s4)=duλ(u)[eq=1,3sqe0*(tqu|Z)+p=2,4spe0(tpu|Z)Z1]
(18)
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 ∂ijkΨ, vanish, and also those non-circular terms with two derivatives, ∂13Ψ and ∂24Ψ. We get:
Γ3(t1,t2,t3,t4)=e3*(t1)e3(t2)e3*(t3)e3(t4)PZk=Γ(t1,t2)Γ(t3,t4)+Γ(t1,t4)Γ(t3,t2)+duλ(u)e0*(t1u|Z)e0(t2u|Z)e0*(t3u|Z)e0(t4u|Z)Z
(19)
where Γ(ti,tj)is given by (4) and we have introduced a subscript in Γ3(t1,...t4) 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

13. B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27(1), 360–374 (1983). [CrossRef]

]. 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, 〈ak*ap ar*as〉 = 〈ak*ap〉 〈ar*as〉 + 〈ak*as〉 〈ar*ap〉 [12

12. J. W. Goodman, Statistical Optics (Wiley, 1985).

]. 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,
Γ2(t1,t2,t3,t4)=e2*(t1)e2(t2)e2*(t3)e2(t4)Zkak=Γ(t1,t2)Γ(t3,t4)+Γ(t1,t4)Γ(t3,t2)
(20)
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:
Γ1(t1,t2,t3,t4)=N0duλ(u) e0*(t1u|Z)e0(t2u|Z)e0*(t3u|Z)e0(t4u|Z)Z
(21)
and equals N0 times the last term in (19). Using (20) and (21) in (19) we finally get:
Γ3(t1,t2,t3,t4)=Γ2(t1,t2,t3,t4)+1N0Γ1(t1,t2,t3,t4)
(22)
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

15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.

17

17. E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

]. 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.

References and links

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J. Perina, Coherence of Light (Kluwer, 1985).

3.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional beams,” Opt. Commun. 27(2), 185–188 (1978). [CrossRef]

5.

P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14(12), 5007–5012 (2006). [CrossRef] [PubMed]

6.

A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007). [CrossRef] [PubMed]

7.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

8.

H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013). [CrossRef] [PubMed]

9.

S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express 19(18), 17086–17091 (2011). [CrossRef] [PubMed]

10.

The support of p(z) is the set of values of z s.t. p(z) ≠ 0.The hypothesis of connectedness is necessary since the range of variation of a fluctuating pulse parameter is always assumed connected.

11.

M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary-field representation of supercontinuum,” J. Opt. Soc. Am. B 30(1), 21–26 (2013). [CrossRef]

12.

J. W. Goodman, Statistical Optics (Wiley, 1985).

13.

B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27(1), 360–374 (1983). [CrossRef]

14.

C. R. Fernández-Pousa, “Nonstationary elementary-field light randomly triggered by Poisson impulses,” J. Opt. Soc. Am. A (to be published).

15.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.

16.

B. Picinbono, Random Signals and Systems (Prentice-Hall, 1993), Chap. 8.

17.

E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(030.1640) Coherence and statistical optics : Coherence
(030.6600) Coherence and statistical optics : Statistical optics
(320.5550) Ultrafast optics : Pulses

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: February 22, 2013
Revised Manuscript: March 28, 2013
Manuscript Accepted: March 28, 2013
Published: April 9, 2013

Citation
Carlos R. Fernández-Pousa, "Stochastic pulse models of a partially-coherent elementary field representation of pulse coherence," Opt. Express 21, 9390-9396 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9390


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References

  1. M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon.3(4), 272–362 (2011). [CrossRef]
  2. J. Perina, Coherence of Light (Kluwer, 1985).
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional beams,” Opt. Commun.27(2), 185–188 (1978). [CrossRef]
  5. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express14(12), 5007–5012 (2006). [CrossRef] [PubMed]
  6. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express15(8), 5160–5165 (2007). [CrossRef] [PubMed]
  7. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett.32(24), 3531–3533 (2007). [CrossRef] [PubMed]
  8. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express21(1), 190–195 (2013). [CrossRef] [PubMed]
  9. S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express19(18), 17086–17091 (2011). [CrossRef] [PubMed]
  10. The support of p(z) is the set of values of z s.t. p(z) ≠ 0.The hypothesis of connectedness is necessary since the range of variation of a fluctuating pulse parameter is always assumed connected.
  11. M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary-field representation of supercontinuum,” J. Opt. Soc. Am. B30(1), 21–26 (2013). [CrossRef]
  12. J. W. Goodman, Statistical Optics (Wiley, 1985).
  13. B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A27(1), 360–374 (1983). [CrossRef]
  14. C. R. Fernández-Pousa, “Nonstationary elementary-field light randomly triggered by Poisson impulses,” J. Opt. Soc. Am. A (to be published).
  15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.
  16. B. Picinbono, Random Signals and Systems (Prentice-Hall, 1993), Chap. 8.
  17. E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

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