1. Introduction

In comparison with the well-understood theory of optical coherence of stationary electromagnetic fields [1], non-stationary fields have received surprisingly little attention given that all real fields are strictly speaking non-stationary (ultrashort pulses being the most striking example). The problem of measuring spectra of non-stationary fields has been approached from time to time (see [2, 3, 4] and references cited therein), and rather recently the general theory of non-stationary fields has been formulated [5, 6, 7]. This theory reveals an important aspect not present in coherence theory of stationary light, namely partial correlation between the spectral components of light fields. This aspect was clearly illustrated by the introduction of a temporally and spectrally partially coherent ‘model source’ with Gaussian distributions on intensity and spectrum [8]. The field emitted by such a source was named a Gaussian Schell-model pulse (or pulse train) in analogy with spatially partially coherent Gaussian Schell-model fields [1]. The model is capable of bridging the gap between stationary fields and coherent pulses and it often admits analytic expressions as does its spatial counterpart. For instance, it has been shown that Gaussian Schell-model pulses can be generated by temporal modulation of stationary fields [9], their propagation in linear dispersive media has been studied [10, 11, 12], and the space-analogy has been investigated in some detail [9, 13].

In this paper we present a new model for temporally and spectrally partially coherent sources, which is based on the representation of the non-stationary field as a superposition of individually fully coherent but mutually uncorrelated elementary pulses. The model is intuitively attractive in the sense that many real light sources can be imagined to radiate in this manner, at least approximately. The new model contains the Gaussian Schell-model as a special case, and it is the temporal analogue of a recently-introduced elementary-source model for spatially partially coherent radiation [14].

2. Identical-elementary-pulse source representation

Let us assume that the electromagnetic field, with each scalar component represented by a random analytic signal U(t), consists of a continuum of identical, temporally fully coherent elementary pulses with center frequency ω ₀. Then the elementary pulse may be written as [15]

Full temporal coherence implies that the cross-correlation function of the (deterministic) elementary pulse is separable [4], i.e.,

If the elementary pulses originate at instants of time t ^′, are mutually uncorrelated and weighted by some real and positive function g(t ^′) [16], normalized such that

the cross-correlation function associated with the entire wave field takes the form

Thus the (generally time-dependent) intensity of the entire field is given by

and the normalized temporal cross-correlation function, or the complex degree of coherence

satisfies |γ((t ₁,t ₂)|≤1.

The physical meaning of the field representation (2), when constructed as explained above, is intuitively clear. However, it can be obtained more formally using the basic definition

where the angular brackets denote ensemble averaging. Let us write a single field realization in the form

where p(t ^′) is a random function. The latter form of Eq. (2) then follows immediately by inserting (8) into (7) if p(t) has the property

(here δ is the Dirac delta function) and we use the fact that U _e is a deterministic function. Expression (8) shows that elementary pulses of the form U _e(t) are generated continuously with a random temporal weight, and (9) demands that the elementary pulses generated at different instants of time t ^′ are independent.

The assumptions just made may appear rather restrictive, and apparently only a narrow class of temporally partially coherent fields can be represented in the form of Eq. (4). Relaxation of these restrictions is of course possible in various ways; as a simple example, one could make the center frequency ω ₀ time-dependent, which would obviously lead to a time-dependent source spectrum. In this paper we do not consider such extensions because the present assumptions in fact allow the modelling of a rich class of phenomena, for instance a smooth transition from fully coherent pulses to stationary fields, as we shall shortly see. The model (4) can often approximate real fields that do not satisfy conditions such as (9) rigorously — the situation is analogous to the use of ‘model sources’ in studies of spatial coherence properties of stationary fields [1]. Moreover, Eq. (4) can be viewed as a type of coherent-mode decomposition of a more general class of fields.

Obviously, the field described by Eq. (4) is fully temporally coherent if and only if g(t ^′)=δ (t ^′), in which case it consists of just a single elementary pulse and Eq. (6) gives |γ(t ₁,t ₂)|=1. A strictly stationary field is obtained in the limit g(t ^′)= constant [17], and quasi-stationary fields are characterized by slowly varying distributions of g(t ^′) in comparison with the effective temporal width of a(t).

3. Spectral source representation

The spectral coherence properties, i.e., the correlations between different frequency components of a field with a temporal cross-correlation function Γ, are determined by the two-frequency cross-correlation function

from which one obtains the spectral density

and the normalized form of the spectral correlation function, or the spectral degree of coherence

which satisfies |µ(ω ₁,ω ₂)|≤1.

Inserting Eq. (4) into Eq. (10) we obtain, after some simple algebraic manipulation, the important result

where A and G are the Fourier transforms of a and g, respectively. It is immediately seen from Eqs. (11) and (12) that

and

where Φ(ω ₁,ω ₂)=arg[A(ω ₂-ω ₀)]-arg[A(ω ₁-ω ₀)].

In view of these results we may conclude that the spectral density of the total field is determined by the functional form of the elementary pulse alone, while the modulus of the spectral degree of coherence is determined solely by the weight function of the elementary pulses. Moreover, the modulus of µ is a function of the frequency difference Δω=ω ₂-ω ₁ only.

4. Gaussian Schell-model pulses

Let us assume that the functional forms of both the elementary pulse and the weight function are Gaussian, i.e.,

where, in view of Eq. (3), g ₀=(2/π)^1/2 $T_{\mathrm{g}}^{-1}$. Then straightforward calculations using Eqs. (4)–(6) show that the time-dependent intensity of the field is given by

with I ₀=|a ₀|² T/T _e and

while the complex degree of coherence takes the form

where

Thus T and T _c represent the pulse duration and the coherence time, respectively, of the entire field. They are related to the parameters of the elementary-pulse representation by simple algebraic formulas (19) and (21).

Turning our attention to the spectral properties of the field, we immediately see from Eq. (14) that the spectrum is given by

where S ₀=${\pi T}_{\mathrm{e}}^2$|a ₀|² and

represents the spectral bandwidth of the field. Furthermore, use of Eq. (15) leads to

where

is the spectral coherence width of the field. Thus the elementary-pulse model with a Gaussian elementary field and a Gaussian weight function leads to a Gaussian Schell-model pulse introduced in Ref. [8].

5. Lorentz-Gaussian Schell-model pulses

Let us next consider elementary pulses with an exponential temporal decay,

still assuming the weight function to be given by Eq. (17). Now, according to Eq. (15), the modulus of the spectral degree of coherence is obviously still of the same Gaussian form as in (24) but a non-vanishing phase term

appears. The spectrum takes the Lorentzian form

where the bandwidth is

and S ₀=T _e|a ₀|².

In the temporal domain the intensity and the complex degree of coherence are given by expressions

and (if t ₂>t ₁)

respectively, where erfc denotes the complementary error function and $I_0=\frac{1}{2}\exp (T_g^2⁄2T_e^2)$.

Referring to the spectral representation, we name the model pulses introduced in this section Lorentz-Gaussian Schell-model pulses. We note, however, that in this model only the modulus of the spectral degree of coherence obeys the Schell model, i.e., depends on the frequency difference only.

6. Discussion

The elementary-pulse model discussed here provides an intuitive link between classical coherence theory and atomic-level emission phenomena. The Lorentzian spectral line shape arises from the finite lifetime of an excited state in spontaneous emission, or from collision broadening, and it is accompanied with an exponential elementary pulse as discussed in Sect. 5. The Gaussian spectral line, which implies a Gaussian elementary pulse considered in Sect. 4, is familiar from the Doppler broadening mechanism. An incoherent superposition of elementary pulses with a weight function g may be understood to simulate, e.g., electric pumping of a light-emitting diode if the pump pulse is longer than the spontaneous emission lifetime. The results could be extended further by considering elementary pulses that give rise to Voight line shapes. The presence of a resonator can be modelled using an appropriate spectral filter to determine the temporally extended shape of the elementary pulse. Another way to simulate lasers (and stimulated emission) is to extend the model by introducing correlations between the elementary pulses instead of using a delta function in Eq. (9). We plan to study these aspects in more detail elsewhere.

In this paper only plane-wave fields at a single spatial point were considered. Due to the nature of the superposition in Eq. (4), the propagation of any partially coherent field in a linear (e.g., dispersive) medium can be governed easily, provided that the propagation characteristics of the elementary coherent pulse are known. Furthermore, generalization to spatially partially coherent fields is possible by combining the present formalism with its spatial analog [14]. This would pave the path to an illustrative study of statio-temporal properties of partially coherent non-stationary fields. Work is underway to relate the source spectra of these model sources with spectra measurable by physical experiments [2], and to obtain precise understanding of interferometric measurement of the coherence properties of non-stationary fields.

7. Conclusions

A new model for temporally and spectrally partially coherent non-stationary plane-wave fields based on a superposition of mutually uncorrelated but individually fully coherent elementary pulses was introduced. The modulus of the spectral degree of coherence was shown to depend on the frequency difference only. Fully coherent pulses as well as stationary fields were obtained naturally from the general representation at the appropriate limits. Gaussian Schell-model sources were considered as an example and a new type of partially coherent source, named a Lorentz-Gaussian Schell-model source, was introduced. The temporal and spectral coherence properties of fields emitted by these model sources were evaluated explicitly, and an illustrative connection to atomic-level emission phenomena was pointed out.