Complex Gaussian representation of statistical pulses

We develop a general representation for ensembles of non-stationary random pulses in terms of statistically uncorrelated, timedelayed, frequency-shifted Gaussian pulses which are classical counterparts of coherent states of a quantum harmonic oscillator. We show that the two-time correlation function describing second-order statistics of the pulses can be expanded in terms of the complex Gaussian pulses. We also demonstrate how the novel formalism can be applied to describe recently introduced Gaussian Schell-model pulses and pulse trains generated by typical mode-locked lasers. © 2011 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (320.0320) Ultrafast optics; (320.5550) Pulses. References and links 1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). 2. E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007). 3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). 4. M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995). 5. L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998). 6. S. A. Ponomarenko, G. P. Agrawal and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004). 7. B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007). 8. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002). 9. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporarily modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003). 10. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). 11. Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003). 12. M. Brunel and S. Coëtlemec, “Fractional-order Fourier formulation of the propagation of partially coherent light pulses,” Opt. Commun. 230, 1–5 (2004). 13. R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705–4711 (2009). 14. G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970). 15. Although the slowly-varying envelope approximation breaks down for few-cycle long femtosecond pulses, the decomposition into the envelope and carrier wave makes sense even in this case, see Ref. [19] for details. 16. S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002). 17. E. Wolf, “New theory of partial coherence in space-frequency domain. Part I: Spectra and cross-spectra of steadystate sources,” J. Opt. Soc. Am. 72, 343–351 (1982). #150755 $15.00 USD Received 8 Jul 2011; revised 29 Jul 2011; accepted 29 Jul 2011; published 16 Aug 2011 (C) 2011 OSA 29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17086 18. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988). 19. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006). 1. P-representation of statistical pulses: introduction and preliminaries Relentless recent progress in ultrafast optics [1] has motivated the quest for a better insight into statistical features of non-stationary sources generating ultrashort optical pulses. To this end, the extension of the usual optical coherence theory, which deals with statistically stationary light [2, 3], is required. The pioneering work in this direction [4, 5] was followed by the exploration into non-stationary spectra of generic statistical pulses [6] and the development of coherence theory of cyclostationary random pulses [7]. Concurrently, particular models of partially coherent non-stationary sources, notably Gaussian Schell-model one, were introduced and the corresponding optical fields were explored [8–10]. The evolution of partially coherent pulses in linear dispersive media was also examined [11, 12] and the generalization of correlationinduced spectral changes to non-stationary random pulses was presented in Ref. [13]. The purpose of this work is to formulate a statistical theory of random pulses in the language that is sufficiently flexible to describe a variety of partially coherent pulse models on the one hand, and on the other hand, establishes a clear link with experimentally realizable ultrashort pulses. To this end, we represent each statistical pulse as a linear superposition of uncorrelated, time-delayed, frequency-shifted Gaussian pulses–which can be routinely produced in the laboratory by standard lasers–with a statistical distribution of emission times and carrier frequency shifts. The complex Gaussian pulses are classical analogues of coherent states of a quantum harmonic oscillator. By analogy with the Glauber-Sudarshan P-representation in quantum optics [3], we can then express the second-order two-time correlation function of any statistical pulse as an integral over an over-complete non-orthogonal set of complex Gaussian pulses. We then discuss the application of the advanced representation to several particular cases of practical interest. We start by considering a time-delayed by ts Gaussian pulse with the carrier frequency shifted to ωs; the pulse has the temporal profile ψ(t; ts,ωs) = Aexp [ − (t − ts) 2 2t2 ∗ ] eiωst , (1) where A and t∗ are a real amplitude and width of the pulse. Transforming to dimensionless variables, T = t/t∗, Ts = ts/t∗, and Ωs = ωst∗–which we are going to use hereafter unless we indicate otherwise–we obtain, after elementary algebra, the following expression ψα(T ) = e−(Imα) π1/4 exp [ − (T − √ 2α)2 2 ]


P-representation of statistical pulses: introduction and preliminaries
Relentless recent progress in ultrafast optics [1] has motivated the quest for a better insight into statistical features of non-stationary sources generating ultrashort optical pulses.To this end, the extension of the usual optical coherence theory, which deals with statistically stationary light [2,3], is required.The pioneering work in this direction [4,5] was followed by the exploration into non-stationary spectra of generic statistical pulses [6] and the development of coherence theory of cyclostationary random pulses [7].Concurrently, particular models of partially coherent non-stationary sources, notably Gaussian Schell-model one, were introduced and the corresponding optical fields were explored [8][9][10].The evolution of partially coherent pulses in linear dispersive media was also examined [11,12] and the generalization of correlationinduced spectral changes to non-stationary random pulses was presented in Ref. [13].
The purpose of this work is to formulate a statistical theory of random pulses in the language that is sufficiently flexible to describe a variety of partially coherent pulse models on the one hand, and on the other hand, establishes a clear link with experimentally realizable ultrashort pulses.To this end, we represent each statistical pulse as a linear superposition of uncorrelated, time-delayed, frequency-shifted Gaussian pulses-which can be routinely produced in the laboratory by standard lasers-with a statistical distribution of emission times and carrier frequency shifts.The complex Gaussian pulses are classical analogues of coherent states of a quantum harmonic oscillator.By analogy with the Glauber-Sudarshan P-representation in quantum optics [3], we can then express the second-order two-time correlation function of any statistical pulse as an integral over an over-complete non-orthogonal set of complex Gaussian pulses.We then discuss the application of the advanced representation to several particular cases of practical interest.
We start by considering a time-delayed by t s Gaussian pulse with the carrier frequency shifted to ω s ; the pulse has the temporal profile where A and t * are a real amplitude and width of the pulse.Transforming to dimensionless variables, T = t/t * , T s = t s /t * , and Ω s = ω s t * -which we are going to use hereafter unless we indicate otherwise-we obtain, after elementary algebra, the following expression Here the complex displacement conveniently combines time delay and frequency shift viz., In Eq. ( 2) we chose the amplitude A such that the pulse profile function is normalized to unity: Let us now look at an unnormalized coherent state of the quantum harmonic oscillator [3] It follows from Eq. ( 5) that the coordinate-representation wave function of the coherent state is given by ψ where the number state |n can be expressed in the coordinate representation as Performing summation over n on the r.h.s of Eq. ( 6) with the aid of Eq. ( 7) and the generating function for Hermite polynomials H n (x) in the form [14] we arrive at the final expression for the normalized coherent state wave function in the coordinate representation as Equation ( 9) is identical with Eq. ( 2) apart from the notation.Thus, the (normalized) complex Gaussian pulses have the same profiles as the coherent states.
As a consequence of the outlined mathematical equivalence, the complex Gaussian pulses form a complete set such that Alternatively, by introducing time bra-and ket-vectors, T | and |T , we can re-write the completeness relation explicitly in the temporal representation as where we denoted ψ α (T ) = T |α .

P-representation of statistical pulses: general formalism
Consider an ensemble of random pulses {E(T )}.Hereafter, we find it convenient to decompose the electric field E(T ) into a (usually) slowly-varying envelope U(T ) and a carrier wave [15] such that where Ω c is a deterministic carrier frequency of the pulse.The second-order statistical properties of the ensemble {U(T )} are specified by the cross-correlation function where the angle brackets denote ensemble averaging.Similarly to the spatial case [16], we can introduce a statistical operator Γ such that its matrix elements correspond to the two-time correlation function, Following the quantum optical development, we can represent the statistical operator in a diagonal form as Γ = d 2 αP(α)|α α|.
Equation ( 15) is a classical analogue of the well-known Glauber-Sudarshan Prepresentation of the quantum density operator [3]; it can be written in the matrix form as implying in more "classical" notations that Notice that one can formally invert Eq. ( 17) to determine the classical P-distribution in terms of the two-time correlation function.With this purpose, we can again borrow the known expression for P(α) in the matrix form [3] and expand the rhs in terms of complete sets of time ket-vectors to yield In principle, Eqs. ( 17) and ( 19) solve the problem of finding the appropriate complex Gaussian representation for any statistical pulse.In practice, of course, the integrals in Eq. ( 19) can fail to converge in the space of ordinary functions, making the P-representation cumbersome in the case.
An alternative-yet equivalent-statistical representation of random pulses is arrived at by examining an expansion of a statistical ensemble member U(T ) in terms of the complex Gaussian pulses with random amplitudes c(α) as We conclude with the help of Eq. ( 13) that for the expansion (Eq.( 20)) to be compatible with the P-representation (Eq.( 17)), {c(α)} must be uncorrelated, obeying The just derived stochastic expansion can serve as a good starting point for synthesizing new partially coherent pulses from complex Gaussian ones.
It is instructive to compare the developed representation with a coherent-mode decomposition of optical coherence theory, originally formulated for spatial fields [17].According to the latter, the cross-correlation function of the pulse can be expanded into a Mercer-type series as where the modes φ n (T ) form a complete orthonormal set such that Each mode and the corresponding eigenvalue λ n are determined by solving the linear integral equation in the form Equivalently, each ensemble member U(T ) can be represented using the Karhunen-Loève expansion where the stochastic amplitudes are uncorrelated and normalized such that Notice that the modes of the coherent-mode theory can have arbitrary temporal profiles, depending on particulars of the pulse statistics.The modes are determined by solving the integral equation (Eq.( 24)), which can be a formidable mathematical task.In contrast, the advanced Prepresentation is always formulated in terms of complex Gaussian pulses.Not only are the latter mathematically well-behaved and physically realizable, but they also remain shape-invariant on propagation through linear temporal elements-including time-lenses-and linear dispersive media.Hence, the advanced P-representation is expected to be superior to the coherent-mode approach whenever P-distributions of pulses turn out to be well-behaved ordinary functions.
It is also instructive to compare the complex Gaussian representation with the elementarypulse-representation introduced in Ref. [10].While the former can be used either to concoct novel sources or to represent the sources with given cross-correlation functions using Eqs.( 17) and ( 19), the elementary pulse envelopes do not, in general, form a complete set and hence do not allow for a general cross-correlation function representation.

Examples and discussion
As the first example, we examine the P-representation of a recently introduced [8,9] nonstationary Gaussian Schell-model (GSM) source with the cross-correlation function where we introduced the dimensionless pulse width and coherence time: σ p = t p /t * and σ c = t c /t * .Substituting from Eq. ( 27) into Eq.( 19), we obtain, after lengthy but straightforward algebra, the P-distribution of the GSM pulsed source in the general form Here we expressed the answer in physical variables T s and Ω s related to α by Eq. ( 5).As is seen from Eq. (28), the scaling factor t * -the width of a complex Gaussian pulse-serves as an additional degree of freedom in choosing the most adequate P-distribution for a given source model.In this case, the choice σ p → 1, implying that t * → t p and σ c = t c /t p , leads to the simplest and most physically transparent representation In physics terms, Eq. ( 29) implies that a nonstationary GSM source of width t p can be represented by a superposition of statistically uncorrelated Gaussian pulses of the same width, with no time delay and a Gaussian distribution of frequency shifts; the coherence time of the source determines the distribution width.We note in passing that a somewhat similar representation to Eq. ( 29) can also be synthesized using the elementary-pulse-representation technique of Ref. [10].
Another instructive application of the new formalism lies in the area of partially coherent source modeling.In particular, novel partially coherent sources can be straightforwardly synthesized by mixing a countable number of uncorrelated complex Gaussian pulses such that where w n ≥ 0 specifies the energy carried by the nth Gaussian pulse.It then follows from Eqs. ( 17) and (30) that in the dimensional variables, the two-time cross-correlation function takes the form Let us specialize to the case of α n = n(t 0 /t * + iω 0 t * )/ √ 2, where t 0 and t * characterize the individual Gaussian pulse peak time and width, respectively, and −N ≤ n ≤ N. The averaged intensity profile of the resulting random pulse train takes the form Provided N 1 and t * t 0 , Eq. (32) describes rather well the intensity of a random train of realistic ultrashort mode-locked pulses with the individual pulses centered at the integer multiples of t 0 , having a width of t * ; further, if w n = w 0 = const and t * = t 0 /N, we have a periodic train of identical mode-locked pulses generated in a cavity with a round-trip transit time of t 0 [18,19].
We note in passing that in the fully coherent case, the pulse field can be expressed by Eq. (20) with deterministic amplitudes {c(α)}.In particular, considering with α n = n(t 0 /t * + iω 0 t * )/ √ 2, we obtain a train of coherent Gaussian pulses with the field profile Equation ( 34) represents an ideal train of identical coherent Gaussian pulses provided that c n = c 0 = const, ω 0 = 2π/t 0 and t * = t 0 /N.
To summarize, we presented a novel formalism for describing statistical properties of ultrashort random pulses.The proposed approach is based on the expansion-diagonal representation akin to the Glauber-Sudarshan P-representation of quantum optics-of the cross-correlation function of any statistical pulse in terms of complex Gaussian pulses with the appropriately distributed emission times and carrier frequencies.We showed how the complex Gaussian representation can describe statistical features of Gaussian Schell-model pulses and the output of realistic mode-locked lasers.The new representation is anticipated to find applications in ultrafast optics and temporal imaging with ultrashort pulses.