A broad class of partially coherent non-stationary fields can be expressed in terms of the recently proposed independent-elementary-pulse model. In this work we first introduce a corresponding dual representation in the frequency domain and then extend this concept by considering shifted and weighted elementary spectral coherence functions. We prove that this method, which closely describes practical optical systems, leads to properly defined correlation functions. As an example, we demonstrate that our new model characterizes, in a natural way, trains of ultra-short pulses, affected by noise and timing jitter, emitted by usual modulators employed in telecom applications.
© 2007 Optical Society of America
In practice, varying stochastic phenomena alter the ideal behavior of all light sources . In optical communication, for example, the effects of the finite line width of the laser constitute a main limitation to the achievable bit rate at the output port of the fiber link [2–4]. This can be understood because the finite spectrum of the laser source degrades the spectral correlations of the pulses launched by the external modulator . Also, owing to the effects of spontaneous emission, many fiber amplifiers in optical communication reduce the temporal coherence of the pulses in the process of amplification . In ultrafast mode-locked lasers, the random changes from pulse to pulse within a train lead to a broadening of the ideal comb structure [6–8], which can be pernicious for various spectral domain applications . These, and many other potential novel uses motivate the research on the mathematical description of pulsed radiation in terms of the concepts of optical coherence theory.
Taking guidance from its spatial counterpart, the notion of Gaussian Schell-model pulse was some time ago introduced to account for possible spectral correlations in pulses whose light intensity profile is a Gaussian . Very recently, a new representation was proposed for partially coherent, non-stationary fields in terms of an incoherent superposition of temporally delayed and properly weighted coherent replicas of an elementary source pulse , in analogy with the corresponding spatial model . The goal of that mathematical model was two-fold: First, it gives a simple and rigorous way to construct a wide variety of arbitrary correlation functions of pulses, and second, it appears to be practically useful, since many light-wave devices operate in this regime .
In this work we put forward a different model that can be considered as a generalization of the independent-elementary-pulse representation mentioned above. We first introduce the dual spectral representation, i.e., we consider an incoherent superposition, in the frequency domain, of spectrally shifted and weighted replicas of an elementary spectrally coherent pulse. The advantage of the spectral domain model is that it is closely associated with practical optical situations. In this way, for instance, we recover the well-known effects of finite source line width on stationary light sources that are externally modulated. The spectral representation also provides a basis for the understanding of how temporal partial coherence distorts the ideal comb structure expected from a fully coherent pulse train . We then extend the model by considering, instead of an elementary coherent pulse, an elementary set of spectrally random fields. We demonstrate that this model is consistent with the non-negative definiteness condition. Finally, we show that our new model appears in a natural way when assessing the stochastic variations, such as timing jitter or pulse period fluctuations, of external modulators.
2. Correlation functions in terms of elementary coherent pulses
Partially coherent plane-wave pulses can be studied through coherence functions that measure the correlations of the field at two instants of time or at two frequencies. Simple mathematical models, such as the concept of Gaussian Schell-model pulses , are useful in assessing the implications of partial coherence, but in reality the correlation functions of non-stationary fields may also take on various complicated forms. In this section we first briefly recall the recently suggested method for constructing correlation functions by a superposition of uncorrelated elementary pulses in the time domain . After that, we introduce an analogous representation in the frequency domain and discuss its physical interpretation.
2.1. Time-domain approach
Let us introduce an elementary pulse as a temporally fully coherent complex field, Ue(t) = a(t)exp(-iωot), where ωo is the carrier frequency and a(t) represents the complex field envelope. If the realizations U(t) of a partially coherent, random field are assumed to consist of a continuum of such identical, but uncorrelated, elementary pulses, the mutual coherence function can be expressed in the form 
where the asterisk denotes complex conjugation, Γe(t 1,t 2) = a *(t 1)a(t 2)exp[iω0(t 1 −t 2)] is the mutual coherence function of the individual deterministic elementary pulses, and g(t) is a (normalized) real and positive weighting function. The angle brackets in the general definition of the mutual coherence function, shown in Eq. (1), denote an ensemble average over the field realizations. Considering the same pulses in the frequency domain, it can be shown that the corresponding cross-spectral density function takes on the form 
where A(ω) and G(ω) are the Fourier transforms of the functions a(t) and g(t), respectively.
It follows from Eq. (2) that the above representation is valid in all cases in which the cross-spectral density function is of the Schell-model type, i.e., the modulus of the spectral degree of coherence depends on Δω= ω2 −ω1 only. As discussed in Ref. , this model provides an intuitive description for radiation phenomena in terms of classical coherence theory.
2.2. Frequency-domain approach
In the following, we introduce an alternative, dual representation, starting from the frequency domain, for correlation functions in terms of elementary pulses. Let us begin by considering the cross-spectral density function W e(ω1,ω2) = Ũ * e(ω1)Ũ e(ω2), where Ũ e(ω) = A(ω−ω0) denotes the Fourier transform of U e(t). Note that since the complex field is taken coherent, the cross-spectral density is a separable function. Now, a broad class of non-stationary coherence functions can be created by spectrally shifting and weighting the above expression with an arbitrary real and positive function WN(ω),
This model corresponds to a spectrally incoherent superposition of coherent fields Ũ e(ω). In this way, the energy spectrum of the non-stationary field , S(ω), can readily be obtained as S(ω) = W(ω, ω), viz.,
in other words, it is a convolution between the weight function WN(ω) and the energy spectrum of the coherent pulse. Of course, the spectral degree of coherence can be obtained at once with the definition
which satisfies 0 ≤ ∣μ(ω1,ω2)∣ = 1.
The time-domain coherence function in our model is obtained by recalling the generalized Wiener-Khintchine theorem, i.e.,
Hence, on substituting from Eq. (3) and after some calculations we find that
The average intensity, by definition, then is I(t) = Γ(t,t) = ΓN(0)∣a(t)∣2; thus the temporal profile of the waveform is determined solely by the complex envelope of the elementary pulses. We can also directly see that the degree of coherence, defined as the absolute value of
depends only on the time-domain form of the weight function, ΓN. This shows that the model introduced in Eq. (3) is valid for all fields that are of the Schell-model form in the time domain, rather than in the frequency domain as in the previous case. We emphasize that these classes of fields are not mutually exclusive and thus in many situations the coherence functions can be represented in either way.
As a practical example of the cross-spectral density functions defined by Eq. (3), we may consider a stationary light source of a mutual coherence function ΓN(t 2 −t 1) that is externally modulated according to a deterministic temporal format a(t). Such a situation appears frequently, for example, in telecom devices . In this case, the function WN(ω) corresponds to the spectral density (or spectrum) of the light source. In particular, when both the modulation format and the spectral density are Gaussian, Gaussian Schell-model pulses are obtained .
3. Spectral elementary-coherence-function representation
In this section we extend the above model by considering elementary sources that, instead of fully coherent, are partially coherent in the spectral domain. To this end we define an elementary spectral coherence function as
where Ũ pc(ω) is a field realization of a non-stationary ensemble that is described by the cross-spectral density function W pc(ω1,ω2) . In the time domain the ensemble can be thought of as being constituted of partially coherent fields given by U pc(t) = a pc(t)exp(−iω0 t), where a pc(t) is a random complex field envelope. In the case of pulses, each field realization can be taken to contain a finite energy and thus be square-integrable. Now, using Eq. (10), let us rewrite Eq. (3) as
which gives us the total cross-spectral density in terms of a weighted superposition of shifted elementary coherence functions. The energy spectrum is still found from the convolution S(ω) = WN(ω)⊗S pc(ω), where now S pc(ω) = W pc(ω,ω). It can be shown that the spectral coherence function obtained by Eq. (11) satisfies the non-negative definiteness condition (see Appendix).
In the time domain, recalling Eq. (6), we find that the mutual coherence function in this case is expressed as
The intensity is now I(t) = ΓN(0)/I pc(t), where I pc(t) = Γpc (t,t). Thus, the average temporal profile of the ensuing pulses depends only on the chosen elementary coherence function. Moreover, γ(t 1,t 2) equals the product of the complex degrees of coherence related to functions ΓN and Γpc. Additionally, it can readily be shown, on the basis of Eq. (A2) and the properties of Fourier transforms, that Eq. (12) also satisfies the corresponding non-negative definiteness condition required of time-domain correlation functions.
4. Practical example
As an example of fields described by the elementary-coherence-functions representation, let us now assume that we have a source in which CW laser light is externally modulated. In any realistic situation, the laser is affected by random phase and amplitude changes. Even more, the modulator launches optical pulses with pulse-period fluctuations. In this way, the complex field of the output pulse train can be written as
where ω0 is the carrier frequency of the laser source and N(t) is a random function that describes its amplitude and phase fluctuations. Further, M(t) = ∑n ψ(t - nT), where ψ(t) denotes the deterministic complex envelope of the pulses and T is the mean period, and j(t) is a real, dimensionless random function of zero mean that describes the timing fluctuations. Now, if the pulse-period fluctuations are small compared with the mean period, we may make a first-order Taylor approximation for the complex field, M[t 1 − Tj(t)] ≈ M(t) − Tj(t)M(t), where the dot means the time derivative.
Going one step further, we may calculate the mutual coherence function of the global source (laser plus modulator), Γ(t 1,t 2) = ⟨U *(t 1)U(t 2)⟩, by assuming that the noise term N(t) is stationary and taking into account that, since the jitter and the noise originate from different devices, the random processes N(t) and j(t) can be taken as statistically independent. In this way, we find that 
with ΓN(t 2 −t 1) = 〈N*(t 1)N(t 2)〉 and Γj(t 1,t 2) = 〈j(t 1)j(t 2)). The point to emphasize now is that this equation perfectly matches the structure of Eq. (12), by identifying
Consequently, the (elementary) cross-spectral density function is determined by Eq. (11), with
where Ωi = ωi − ω0 (i = 1,2), ⊗2 denotes two-dimensional convolution, M͂(ω) is the Fourier transform of M(t), and Wj is obtained from Γj by the inverse of Eq. (6).
So, in this practical example, the resulting cross-spectral density function is given by an incoherent superposition of shifted and weighted elementary spectral coherence functions. The weight function is real and positive and corresponds to the power spectrum of the CW laser light, WN(ω) = ∫ΓN(τ)exp[−i(ω −ω0)τ]dτ. On the other hand, if we consider the case with no jitter, i.e., j(t) = 0, the elementary coherence functions become fully coherent, corresponding the situations of Section 2.2.
5. Summary and conclusions
We have introduced novel mathematical models for the coherence functions of non-stationary plane-wave fields and pointed out how they appear in practical physical situations dealing with random pulses and pulse trains, First, the space–frequency domain analogue of the recent independent-elementary-pulse representation was discussed. The concept was then extended by considering a new representation based on a weighted superposition of shifted elementary spectral coherence functions. It was further shown that the resulting representation satisfies the sufficiency condition required of correlation functions. Several examples of actual physical situations that correspond to these new frequency domain models were briefly discussed.
Finally, a corresponding elementary-coherence-function representation could naturally also be introduced starting from the space–time domain. As with the models dealing with coherent elementary pulses, the two dual approaches together could cover a wider class a partially coherent wave fields. However, as was demonstrated by the examples above, a clear physical association can be found for the frequency domain representation in many cases. Hence, in our opinion, it corresponds to the representation that is more closely anchored to practical experimental arrangements.
Appendix A: Proof of the non-negative definiteness condition
In order to be a proper correlation function, the cross-spectral density defined by Eq. (11) must satisfy the non-negative definiteness condition
where, in the second step, we have changed the order of integration and ensemble averaging, and the last inequality follows from the fact that WN(ω) is a real and positive function. Thus any function obtained by the representation of Eq. (11) is indeed a proper correlation function. Furthermore, this proof is valid also for the independent-elementary-pulse representation discussed in section 2, since that constitutes a special case (of coherent elementary fields) of the more general approach.
V. Torres is grateful for funding from the Dirección General de Investigación Cientifica y Técnica, Spain, and FEDER, under project FIS2004-02404. He also acknowledges financial support from a FPU grant of the Ministerio de Educación y Ciencia, Spain. H. Lajunen thanks the Academy of Finland (project 111701), the Network of Excellence on Micro-Optics (NEMO, http://www.micro-optics.org), the Helsingin Sanomat Centennial Foundation, and the Vilho, Yrjö, and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. A. T. Friberg acknowledges funding from the Swedish Foundation for Strategic Research (SSF).
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