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

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

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]

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

*W*(

*x*

_{1},

*x*

_{2}) across an one-dimensional quasi-monochromatic source and the two-time mutual coherence function (MCF) Γ(

*t*

_{1},

*t*

_{2}) of a random light pulse.

*t*

_{1},

*t*

_{2}) 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]

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. 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]

*t*

_{1},

*t*

_{2}) in terms of a (possibly infinite) number of coherent modes [3]. Its construction entails the solution of a Fredholm integral equation with kernel Γ(

*t*

_{1},

*t*

_{2}) 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]

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. Express **14**(12), 5007–5012 (2006). [CrossRef] [PubMed]

*λ*(

*u*) a nonnegative and integrable function and

*e*

_{0}(

*t*) analytic signals describing elementary pulses with central optical frequency

*ν*

_{0}equal to that of the pulse represented by the MCF.

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]

*p*(

**) is a nonnegative function that may contain impulses and**

*x**e*

_{0}(

*t*|

**) 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**

*x**p*(

*x*) = Σ

*(*

_{n}c_{n}δ*x*−

*n*), with

*c*> 0 the eingenvalues and

_{n}*e*

_{0}(

*t*|

*n*) the corresponding eigenfunctions of kernel Γ(

*t*

_{1},

*t*

_{2}). Class (2) also includes (1) for

*p*(

*x*) =

*λ*(

*x*) and time shifts

*e*

_{0}(

*t*|

*x*) =

*e*

_{0}(

*t*−

*x*), 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]

*e*

_{0}(

*t*|

*x*) =

*e*

_{0}(

*t*) exp(

*j*2π

*xt*).

*e*(

*t*) whose second-order correlation function 〈

*e**(

*t*

_{1})

*e*(

*t*

_{2})〉 coincides with MCF (1) or (2). If

*λ*(

*u*) and

*p*(

**) 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 ~**

*x**λ*(

*u*),

*p*(

**) describing the distribution of emission instants**

*x**u*in the elementary field

*e*

_{0}(

*t*) or other internal pulse parameters

**in**

*x**e*

_{0}(

*t*|

**).**

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

*e*(

*t*) =

*e*

_{0}(

*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

*e*

_{0}(

*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*) =

*n*(

*t*), so that its correlation isand using

*e*(

*t*) =

*e*

_{0}(

*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.

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

**= (**

*x**u,z*),

*e*

_{0}(

*t*|

**) =**

*x**e*

_{0}(

*t −u| z*) and

*p*(

**) =**

*x**λ*(

*u*)

*p*(

*z*) in (2):

*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

*e*

_{0}(

*t*|

*z*) and so

*N*

_{0}= ∫

*du λ*(

*u*) is its total number. We stress that this interpretation of

*N*

_{0}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

*N*

_{0}. Pulses

*e*

_{0}(

*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(

*t*

_{1},

*t*

_{2}) in terms of the MCF Γ

_{0}(

*t*

_{1},

*t*

_{2}) of elementary fields

*e*

_{0}(

*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.

*p*(

*z*) and

*λ*(

*u*) contain impulses

*q*> 0 at

_{n}, σ_{r}*z*,

_{n}*u*together with continuous components

_{r}*p*(

_{m}*z*),

*λ*(

_{s}*u*) > 0 with connected support [10]. We writeand 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*):

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]

## 3. Stochastic pulse models reproducing the mutual coherence function

*U*and

*Z*distributed according to pdfs

*λ*(

*u*)/

*N*

_{0}and

*p*(

*z*), respectively. Phase Θ is uniformly distributed in [0, 2π) and assures circularity and mutual incoherence among constituents. The amplitude normalization

*λ*(

*u*)/

*N*

_{0}, 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*)/

*N*

_{0}and its interpretation is simply the timing jitter distribution of the emission instants of wave

*e*

_{1}(

*t*).

*z*to

*e*

_{0}(

*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*with unit variance [3, 12]:

_{k}*n*(

*u*) = Σ

*(*

_{k}a_{k}φ_{k}*u*), with 〈

*a*〉 = 0 and 〈

_{k}*a*〉 =

_{k}*a_{p}*δ*. To each

_{kp}*k*we attach parameters

*Z*to the emitted elementary pulse

_{k}*e*

_{0}(

*t*|

*z*).

*Z*are also assumed independent and identically distributed (iid), and also independent of

_{k}*a*. The optical field is:Then, using 〈

_{k}*a*〉 =

_{k}*a_{p}*δ*the correlation function is:The average over

_{kp}*Z*gives the same value for any

_{k}*k*since

*Z*are iid, so it can be taken out from the sum, and using the completeness of the basis functions

_{k}*φ*(

_{k}*u*), Σ

*(*

_{k}φ_{k}**u*

_{1})

*φ*(

_{k}*u*

_{2}) =

*δ*(

*u*

_{2}−

*u*

_{1}), 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

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.

*λ*(

*u*), so that

*dN*

_{0}(

*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*) is a rv that fluctuates according to a Poisson distribution with mean and variance ∫

_{b}*duλ*(

*u*) and, in particular, the total number of emissions is not fixed, but a Poisson rv with mean and variance

*N*

_{0}= ∫

*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

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]

*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

*e*

_{0}(

*t*) triggered by a nonstationary Poisson process with

*λ*(

*u*) time-varying and integrable [15–17]. The field is:where the emission instants

*τ*are distributed according to

_{k}*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

*e*

_{0}(

*t*). The computation of the mean and autocorrelation of Δ(

*t*) is standard [15, 16] and yields 〈Δ(

*t*)〉

*=*

_{P}*λ*(

*t*) andwith

*P*standing for average over the Poisson process. We get a mean optical field 〈

*e*

_{3}(

*t*)〉

*=*

_{P}*λ*(

*t*) ⊗

*e*

_{0}(

*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 〈

*e*

_{3}(

*t*)〉

*becomes zero under quite general conditions. On the one hand, one can assume that*

_{P}*e*

_{0}(

*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]

*j*Θ) to

*e*

_{0}(

*t*). On the other, we can use the following spectral argument: in the spectral domain the convolution is the product 〈

*E*

_{3}(

*ν*)〉

*= Λ(*

_{P}*ν*)

*E*

_{0}(

*ν*), which is nonzero only if these two functions overlap.

*E*

_{0}(

*ν*) is centered at frequency

*ν*

_{0}and we denote its spectral half-width by Δ

*ν*

_{0}. Λ(

*ν*) is centered at zero frequency and its bandwidth (from dc to the highest frequency) is Δ

*ν*. Spectral overlap requires Δ

_{λ}*ν*>

_{λ}*ν*

_{0}− Δ

*ν*

_{0}which cannot be the case unless Δ

*ν*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

_{λ}*e*

_{3}(

*t*) is zero-mean only the second term in (11) contributes to the MCF 〈

*e*

_{3}*(

*t*

_{1})

*e*

_{3}(

*t*

_{2})〉

*=*

_{P}*e*

_{0}*(

*t*

_{1})⊗ 〈Δ(

*t*

_{1})Δ(

*t*

_{2})〉

*⊗*

_{P}*e*

_{0}(

*t*

_{2}). The computation then becomes equivalent to that of (3) and leads directly to MCF (1).

*Z*, one for each triggering event, which are assumed iid with pdf

_{k}*p*(

*z*). The wave is:We also assume that emission instants

_{k}*τ*are independent of

_{k}*Z*. Mean and MCF are calculated by performing two independent averages over the Poisson impulses

_{k}*P*and over rv

*Z*. 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 asand the MCF is obtained by derivation of Φ over

_{k}*s*

_{1},

*s*

_{2}and subsequent restriction to

*s*

_{1}=

*s*

_{2}= 0 [15–17]. The characteristic functions of model (12) are well-known in the theory of Poisson point processes [16, 17]: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 ∂/∂

*s*

_{2}≡ ∂

_{2}we getwhich is zero by extending the spectral argument above to the ensemble

*e*

_{0}(

*t|Z*) or by assuming that

*e*

_{0}(

*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]

*s*

_{1},

*s*

_{2}and restriction to

*s*

_{1}=

*s*

_{2}= 0 givesand using the explicit form (14) of Ψ(

*s*

_{1},

*s*

_{2}) 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

*s*

_{1},…

*s*

_{4}leads to a number of terms, some of which vanish due to the assumed circularity of

*e*

_{0}(

*t|Z*). In particular, terms involving one or three derivatives, ∂

*Ψ and ∂*

_{j}*∂*

_{i}*∂*

_{j}*Ψ, vanish, and also those non-circular terms with two derivatives, ∂*

_{k}_{1}∂

_{3}Ψ and ∂

_{2}∂

_{4}Ψ. We get:

*e*

_{3}(

*t*) (12). For

*λ*(

*u*) = const.,

*t*

_{1}=

*t*

_{2},

*t*

_{3}=

*t*

_{4}and 〈.〉

*Gaussian elementary-field correlations, (19) is the intensity correlation function computed as Eq. (30) in [13*

_{Z}**27**(1), 360–374 (1983). [CrossRef]

*u*and rate

*λ*(

*u*).

*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*〉 [12]. This assumption is natural since

_{r}*a_{p}*n*(

*u*) is a noise process. Using this and performing the same algebra as in (9) we get Gaussian correlations,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:and equals

*N*

_{0}times the last term in (19). Using (20) and (21) in (19) we finally get:The Poisson model thus accounts for the higher-order correlations of both pulse emissions and fluctuations. First, if the number of emissions

*N*

_{0}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

*N*

_{0}< 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, Γ

_{2}. If the system emits single fluctuating pulses, the extension should be by Γ

_{1}. And if the number of emissions is independent, random but not necessary large, the system should be modeled as Poisson, Γ

_{3}.

## 5. Conclusion

## Acknowledgments

## References and links

1. | M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon. |

2. | J. Perina, |

3. | L. Mandel and E. Wolf, |

4. | F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional beams,” Opt. Commun. |

5. | P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express |

6. | A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express |

7. | F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. |

8. | H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express |

9. | S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express |

10. | The support of |

11. | M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary-field representation of supercontinuum,” J. Opt. Soc. Am. B |

12. | J. W. Goodman, |

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 |

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, |

16. | B. Picinbono, |

17. | E. Parzen, |

**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

- 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]
- J. Perina, Coherence of Light (Kluwer, 1985).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
- F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional beams,” Opt. Commun.27(2), 185–188 (1978). [CrossRef]
- P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express14(12), 5007–5012 (2006). [CrossRef] [PubMed]
- 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]
- F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett.32(24), 3531–3533 (2007). [CrossRef] [PubMed]
- H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express21(1), 190–195 (2013). [CrossRef] [PubMed]
- S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express19(18), 17086–17091 (2011). [CrossRef] [PubMed]
- 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.
- 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]
- J. W. Goodman, Statistical Optics (Wiley, 1985).
- 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]
- C. R. Fernández-Pousa, “Nonstationary elementary-field light randomly triggered by Poisson impulses,” J. Opt. Soc. Am. A (to be published).
- A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965), Chap. 12.
- B. Picinbono, Random Signals and Systems (Prentice-Hall, 1993), Chap. 8.
- E. Parzen, Stochastic Processes (Society for Industrial and Applied Mathematics, 1999), Chap. 4.

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