1. Introduction

The propagation of radiation from a planar partially coherent source is well known to involve, in general, the evaluation of four-dimensional (4D) integrals (Sect. 5.3.2 in Ref. [1]), which is a tremendous computational task. To reduce the computational complexity of propagating such radiation, we recently employed a decomposition of a certain class of partially coherent fields into a superposition of mutually independent coherent elementary fields, which are identical but originate from different positions in the source plane [2]. The source model is based on Gori’s Gaussian-beam decomposition of Gaussian Schell-model beams and other more general classes of spatially partially coherent fields [3, 4, 5]. This model has been successfully applied to describe light propagation from pulsed surface-emitting lasers, which produce spatially partially coherent light [6].

In essence, the model of Ref. [2] reduces the 4D propagation integrals for the partially coherent field into 2D integrals for the coherent elementary mode, followed by the construction of the full partially coherent field by a simple procedure of shifting and adding elementary-mode contributions. In a follow-up paper [7] we formulated an analogous method for dealing with non-stationary plane-wave fields as trains of identical but independent coherent pulses; see also Refs. [8, 9]. Though not yet explicitly demonstrated in the literature, these models can be combined to propagate efficiently fields with partial spatial, temporal, and spectral coherence properties, with certain restrictions concerning the functional forms of the correlation functions associated with the field.

Numerical modeling of radiation from 3D partially coherent sources is an even greater computational challenge than modeling of 2D sources: in general the evaluation of 6D integrals is required (Sect. 5.3.1 in Ref. [1]). In this paper we show that also in the 3D case the problem can be reduced to the calculation of 2D integrals provided that the correlation functions satisfy certain constraints. Again the partially coherent field is decomposed into a set of independent elementary modes of identical form, but now the modes originate from a certain three-dimensional volume. This volume is not necessarily directly linked to the physical source volume, but nevertheless related to it in a manner that can in principle be determined from far-field coherence measurements. In practical modeling of field propagation into another volume, it is necessary to compute only the propagation of the single coherent elementary field and perform a simple 3D “copy and paste” procedure.

2. Theoretical formulation

Let us begin with a coherent field f(x,y) at frequency ω (which we leave implicit for brevity) and at plane z=0, with angular spectrum

where k_x and k_y are the transverse components of the wave vector k=(k_x,k_y,k_z),

k=|k|=ω/c is the wave number and c denotes the speed of light in vacuum. If the field f(x,y) is shifted laterally from the origin into a center position (x′,y′) and longitudinally into the plane z=z′, and associated with a random overall (complex) amplitude c(x′,y′,z′), the angular spectrum of the resulting field at the plane z=0 is

where the first phase term arises from the shift theorem of the Fourier transform and the origin of the second term is the propagation law of the angular spectrum.

Let us now consider a three-dimensional ensemble of elementary fields, all with the same f(x,y) but centered at different positions (x′,y′,z′). The angular correlation function at the plane z=0 of the entire partially coherent field generated by this distribution of elementary fields, each with an angular spectrum defined by Eq. (3), is given by the 6D superposition integral

in which the asterisk denotes complex conjugation and the angular brackets indicate ensemble averaging. We further assume that the elementary fields are independent (uncorrelated) in the sense that

where p(x′,y′,z′) is a real and positive weight function, assumed to vanish in the half-space z>0, and δ denotes the Dirac delta function. Noting that c(x′,y′,z′) is the only random function in Eq. (3), inserting from Eq. (3) into Eq. (4), and using Eq. (5) we see straightforwardly that the angular correlation function takes the form

where F(k_x,k_y) is given by Eq. (1) and

is the three-dimensional Fourier transform of the weight function p(x′,y′,z′).

In view of Eq. (6), we have in fact reduced a three-dimensional source distribution into an equivalent planar (secondary) source at z=0, and can therefore use the formalism of Sect. 5.3.1 in Ref. [1] to determine the field properties in the space-frequency domain. Denoting the unit position vector by ŝ=r/r=(s_z,s_y,s_z) with r=|r|, the cross-spectral density function in the far zone is

Thus the radiant intensity [1]

is the same as that produced by a single coherent elementary field f(x,y), as in the case of a planar source [2]. The complex degree of spectral coherence [1] in the far zone is

Thus its absolute value depends only on (the Fourier transfrom of) the weight function p(x′,y′,z′). This offers some means to determine the three-dimensional weight function from far-field coherence measurements as we will illustrate below. Finally, the angular spectrum representation of the cross-spectral density function readily gives a three-dimensional space-frequency domain field representation in the form

from which the spectral density S(x,y,z)=W(x,y,z,x,y,z) as well as the transverse (z ₁=z ₂) and longitudinal (x ₁=x ₂, y ₁=y ₂) coherence properties at any frequency ω may be determined. The space-time properties of either stationary or non-stationary fields can be obtained by assuming appropriate spectral correlations and taking the (one- or two-dimensional) Fourier transformations into the temporal domain.

3. Source modeling and numerical propagation procedure

Obviously the model considered here is restricted to cross-spectral density functions of the form of Eq. (11), which is a general expression for a three-dimensional superposition of identical but shifted independent elementary planar sources. Thus our model is not general, but nevertheless applicable to a wide class of fields. In particular, if we observe the far-field distribution to obey an expression of the form of Eq. (8), the method is expected to be valid at least for engineering purposes. The most important criterion in this respect is the functional form of the complex degree of coherence in the far field, given by Eq. (10). In view of the upper branch of Eq. (2) and the definition k_z=ks_z, the absolute value of µ ^(∞) at a given (large) distance r is a measure of transverse spatial coherence on the surface of a half-sphere of radius r. This quantity can be measured straightforwardly using a goniometric Young’s interference arrangement. The result does not depend on r because the far-field condition is assumed to be satisfied, and it contains information about the three-dimensional weight distribution p(x′,y′,z′). In addition, the radiant intensity distribution can be measured straightforwardly, yielding information about the elementary-mode field distribution f(x,y). The information about the source obtained with these (in principle simple) measurements does not allow a unique construction of the source, but especially with some a priori knowledge about the source properties (such as its physical dimensions and the physics of light production) an adequate source model can often be built. Here ‘adequate’ means that the predictions of the model on propagation of the field over any arbitrary distance are in agreement with experimental results is within a desired tolerance. Thus, fundamentally, we aim at a model for the source capable of reproducing the propagation features of the radiated field at all distances, rather than a model to describe the true physical nature of the source. However, we stress that the model is capable of dealing with sources of any state of (transverse and longitudinal) spatial coherence, as long as the cross-spectral density function in the far field is of the form of Eq. (8).

Once the source model exists, propagation of the partially coherent light it produces is easily governed numerically. Instead of employing Eq. (11), one uses directly the elementary-mode decomposition

where f(x,y,z) is the propagated version (into the plane z= constant) of the elementary field f(x,y) at z=z′. In practice, one assumes a discrete distribution of planar sources with field distribution f(x,y) throughout the volume defined and weighted by p(x′,y′,z′). The coherent field f(x,y) to all distances needed using 2D integration across the xy plane, and then shifted and weighted elementary-field contributions are added incoherently. As illustrated in Fig. 1, the elementary field f(x,y) needs to be propagated over the range [d-L _T,d+L _S] in steps Δz small enough to represent the source and the target regions with sufficient resolution. Of course, one also needs to ensure sufficiently dense transverse sampling (Δx) of the weight function to obtain convergent results.

 

Fig. 1. Procedure of propagating the elementary-mode superposition from the source volume V_S into the target volume V_T. Points P′=(x′,y′,z′) within V_S are the centers of origin of the elementary fields f(x,y), which are propagated over the set of distances indicated by the arrows along the z-axis to map V_S onto V_T.

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4. Determination of model-source parameters

The coherent elementary wave form f (x,y) associated with the 3D source can be determined from measurement of the radiant intensity just as in the case of planar partially coherent sources. Thus only the problem of determining the weight function p(x′,y′,z′) remains. Before considering some ways to approach the problem (a complete solution to this problem is not attempted here and may not exist), we stress that p(x′,y′,z′) does not necessarily coincide with the intensity distribution of the source at all. This is clear in the transverse direction: considering a planar source, the intensity distribution at any plane z= constant is a convolution of the weight function and the intensity distribution of the (propagated) elementary mode. In the longitudinal direction the situation is less easy to visualize, but a convincing illustration can be provided as follows. In a single-mode gas or solid-state laser the primary source distribution (of radiating atoms) fills the gas column or laser rod and thus has a considerable length. However, the correct weight distribution in this case is a delta function p(x′,y′,z′)=δ(x′,y′,z′) because the field is fully coherent. The opposite extreme is a fully incoherent source: in this case the function p(x′,y′,z′) coincides with the true source volume. These interpretations may appear contradictory at first. However, the (longitudinally) single-mode field is fully deterministic and thus information about the (coherent) field distribution at any single plane is sufficient for full specification of the field everywhere in space by use of coherent propagation formulas. In the incoherent case, the field at any single plane is contributed by elementary sources at every transverse plane inside the source volume. An excimer laser is an example of the intermediate case with a transversely and longitudinally extended effective source distribution p(x′,y′,z′), which is not equal to the primary source volume. Typically an excimer laser produces an anisotropic Gaussian far-field distribution and hence the elementary field is of an anisotropic Gaussian form. Because of the low reflectivity of the cavity output mirror, the excimer beam effectively traverses the cavity only a small number of times before being coupled out, thus gaining a limited degree of spatial (and temporal) coherence. As a result, the longitudinal extension of p(x′,y′,z′) is smaller than the length of the laser rod.

Let us consider the possibility to determine the weight function by far-field coherence measurements. We assume goniometric Young’s interference arrangement with one of the pinholes on axis and the other rotated with an arm of length r. Denoting s _x1=s _y1=0, s _x2=s_x, s _y2=s_y, a fringe visibility measurement provides (after normalization) the function

where we have used Eq. (7). Measurement of V gives information on the weight function p, but does not determine it unambiguously. Thus one needs additional a priori information about the source, or at least some educated guesses. Consider, as an example, the (often reasonable) assumption that the weight function is of a box-like form

with unknown dimensional constants D and L, which we wish to determine from far-field measurements to build at least the first approximation of the source model. Now goniometric coherence measurements in xz and xy planes would give the same information. Rotating the interferometer arm in the xz plane one would measure

where sincx=sin(πx)/(πx) and θ is the angle defined by s_x=sinθ and s_z=cosθ. Hence both the transverse and the longitudinal extension of the weight distribution have an effect in the spatial coherence properties of the field on the surface of a sphere. If the measurement yields a result of the form of Eq. (15), our box model is justified and the effective source parameters D and L can be determined by a simple fitting procedure.

5. Illustration: longitudinal superposition of Gaussian elementary sources

To illustrate the 3D elementary-source model, we consider the propagation and coherence properties of a partially coherent volume source consisting of Gaussian elementary fields. The effects of partial transverse coherence of a planar source are well known from a thorough study of the transverse and longitudinal coherence of Gaussian Schell-model beams in Refs. [10] and an elementary-mode analysis in Ref. [3]. Therefore we concentrate on purely longitudinal distributions of elementary sources, of the form p(x′,y′,z′)=p _L(z′)δ(x′,y′). Assuming that the radiant intensity is a narrow Gaussian pattern, so that the elementary mode is a Gaussian with waist size w ₀ and then, using the well-known propagation law of Gaussian beams (Ref. [1], Sect. 5.6.2) and Eq. (12), the cross-spectral density function of the entire independent-field superposition can be written as

where z _R=kw ² ₀/2 is the Rayleigh range of the elementary Gaussian source, we normalize

and indicate explicitly that the sources are confined to the half-space z<0.

The expression (16) can be used to evaluate numerically, e.g, the complex degree of transverse coherence coherence at any plane z= constant,

where I(x,y,z)=W(x,y,z,x,y,z) is the transverse intensity distribution, as well as the complex degree of longitudinal spatial coherence along the optical axis,

where I(0,0,z)=W(0,0,z,0,0,z) is the axial intensity distribution.

Let us assume a uniform longitudinal distribution of Gaussian elementary sources, i.e., p _L(z′)=1/L when -L<z′<0 and p _L(z′)=0 otherwise. Now we obtain (if z ₁>0, z ₂>0, and Δz=z ₂-z ₁) an analytical expression for the axial longitudinal coherence in the form

from which the axial intensity distribution is

If z ₁=0 and the asymptotic limit z ₂→∞ is considered, we have

i.e., a non-zero value is obtained for the complex degree of axial coherence between points in the source plane and in the far field. No analytical expressions can apparently be obtained for transverse coherence or intensity.

For numerical illustrations, we assume that w ₀=100λ, which ensures that the paraxial approximation is valid. Now the value L=0 indicates a planar, coherent Gaussian source, and increasing values of L reduce the spatial coherence of the field in both transverse and longitudinal directions, modifying simultaneously the transverse and axial beam profiles. These features are illustrated in Figs. 2–5, in which the appropriate quantities are plotted for four representative values of L, namely L=0.1z _R, L=z _R, L=10z _R, and L=100z _R, where z _R=10⁴ πλ.

 

Fig. 2. Longitudinally extended volume source consisting of a superposition of axial Gaussian elementary sources in the interval -L<z<0. The distributions of normalized (to the value at z=0) axial intensity are shown for different values L, assuming w ₀=100λ.

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Fig. 3. Source models as in Fig. 2, but distributions of longitudinal axial spatial coherence between points z ₁=0 and and arbitrary plane z ₂=z are shown.

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Fig. 4. Source models as in Fig. 2, but normalized (to the value at the origin) transverse intensity distributions at the exit plane z=0 of the source volume are shown.

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Fig. 5. Source models as in Fig. 2, but distributions of the absolute value of the complex degree of transverse spatial coherence at the end plane of the source volume z=0 are shown between points (x ₁,y ₁)=(0,0) and (x ₂,y ₂)=(x,0).

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Several illustrative observations can be made by inspection of Figs. 2–5. Consider first the influence of the longitudinal source dimension in the axial beam profile in Fig. 2. The gradual accumulation of axial intensity in the range -L<z<0 is evident (not shown in totality for L=10z _R, and L=100z _R), and an increasing L leads to a slower decay of the axial intensity in the half-space z>0. The distribution of longitudinal spatial coherence, illustrated in Fig. 3, shows somewhat more interesting (yet logical) trends. Within the source region -L<z<0, the longitudinal coherence increases monotonously towards unity at z=0 and then decreases towards a constant asymptotic value given by Eq. (22) when z→∞. Such a behavior is highly reminiscent of the effect of partial transverse coherence of a planar source in the longitudinal spatial coherence of the beam it generates, studied in Ref. [10] in the case of Gaussian Schell-model beams.

The increase with L of the width of the spatial transverse intensity distribution of the field at the end plane of the active region z=0, illustrated in Fig. 4, is clearly understandable from the geometry of the 3D source: each elementary source radiates a diverging beam, all of which overlap (incoherently) at z=0, with the inevitable result of an increase in beam width. Because of the incoherent superposition of mutually uncorrelated beams of the Gaussian form but different widths at z=0, the distribution of the complex degree of spatial coherence at this plane decreases when the distance between the two points increases, as illustrated in Fig. 5. This effects grows stronger when L increases. Evidently, when L≈z _R or larger, the effect of the longitudinal extent of the source distribution has an appreciable effect in transverse coherence at point separations that are of the order of beam width. The same trend is seen also at other planes z>0 and in the far zone.

6. Conclusions and outlook

In this paper we have generalized the independent-elementary-field representation of spatially partially coherent planar sources to 3D volume sources. We investigated some effects caused by the longitudinal distribution of elementary sources, as well as the possibility to determine the effective source weight distribution from far-field coherence measurements. The discussion was restricted to scalar, stationary fields in the space-frequency-domain. An extension of the elementary-field representation to electromagnetic fields is under development; such a model must be able to model also the (partial) polarization of the field and hence appears to require two coherent, vectorial elementary fields (polarization modes) instead of a single scalar function f(x,y). In addition, we are developing a combination of the spatial [2] and temporal-domain [7] elementary-field models to obtain a convenient model for non-stationary partially coherent fields.

In the numerical examples presented in the previous section we considered only free-space propagation. However, the model can be easily applied to propagation in paraxial optical systems. The propagation principle illustrated in Fig. 1 can be extended to such situations using the Collins integral [11] for the coherent elementary beam. Non-paraxial space-invariant systems are also simple to treat since the propagation of the coherent elementary field needs to be calculated only once. In space-variant systems the response to the elementary field depends on the lateral source point position (x′,y′), but also then it is often possible to divide the effective source volume into domains where the system is space-invariant, and thus reduce the computational complexity significantly.

As already speculated in Sect. 4, the model introduced here may prove applicable to, e.g., excimer lasers, which have previously been modeled as planar Gaussian Schell-model sources [12]. Such a model ignores the longitudinal extension of the effective source distribution, which may be non-negligible since light effectively traverses the cavity just a few times and thus an effective longitudinal source with a length of some fraction of the cavity length is expected. Work remains to be done on methods to determine the effective source. Here we only considered an approach based on far-field coherence measurements, but it is also possible to obtain useful information by measuring transverse intensity profiles in focal regions of a paraxial system (focal caustics). One objective of future work is therefore to model excimer lasers (and other practical sources in which the longitudinal effects may play a significant role) using the techniques presented here, and test the results against experimental measurements.