An efficient method is presented for propagating light emitted by a wide class of spatially partially coherent sources. This class includes all quasihomogeneous sources with slowly varying intensity distributions in comparison to their spatial coherence areas. The method is based on the construction of a set of individually coherent but mutually uncorrelated, identical, laterally shifted elementary sources of finite extent.
© 2006 Optical Society of America
Propagation of spatially partially coherent light in free space or in optical systems is far more difficult to treat than the propagation of either coherent or incoherent light: in general the evaluation of four-dimensional rather than two-dimensional integrals is required . Analytical calculations are possible only rarely even if the paraxial approximation is made, and numerical computations are burdened by computation time and computer memory limitations. Thus any models that would ease the computational complexity are welcome in wave-optical engineering with partially coherent light.
The calculation of four-dimensional propagation integrals can be reduced to the evaluation of a set of two-dimensional (2D) integrals using the Karhunen-Loewe -type coherent-mode representation of partially coherent fields . The determination of these coherent modes involves the solution of a Fredholm integral equation of the second kind. Moreover, since the coherent modes have different functional forms, they must be propagated individually before the final incoherent superposition can be made. For these reasons the method is efficient only if the number of modes with substantial weights is fairly small, i.e., if the source is relatively coherent. Unfortunately many sources of practical importance in photonics (such as light-emitting diodes) are not of this type, but are instead rather incoherent.
In this paper we develop an efficient propagation model for partially coherent light, which is based on a new type of coherent-mode expansion involving an incoherent superposition of a set of identical, but spatially displaced coherent elementary fields. The method may be viewed as an extension of the decomposition presented by Gori and Palma  for a special class of Gaussian fields. Our extension is not generally applicable but it is valid for, e.g., quasihomogeneous sources (such as light-emitting diodes), for which the radius of the effective coherence area is much smaller than the effective source area. Since our coherent modes are of identical functional form, their free-space propagation requires just a single two-dimensional integration. Moreover, the modal wave form can be determined straightforwardly from the source properties - no integral equation needs to be solved.
The paper is organized as follows. The general formalism and the new coherent-mode field representation are introduced in Sect. 2. The particular case of quasimonochromatic sources is treated in Sect. 3. In Sect. 4 we discretize the formalism and provide numerical example of partially coherent field propagation in free space to demonstrate its practical applicability to engineering problems. Finally, conclusions are drawn and extensions of the method are discussed in Sect. 5.
2. Field representation
Using scalar coherence theory in the space-frequency domain  we represent the field at the source plane z = z 0 by the cross-spectral density function
where ρj = (xj,yj) with j = 1,2, S(ρ, z 0) is the spectral intensity of the source field, and μ(ρ1,ρ2,z 0) is its spectral degree of coherence. The propagated cross-spectral density function in the positive half-space z > z 0 is then given by 
where r j = (xj,yj,zj - z 0), k j⊥ = (kjx,kjy), k j = (kjx,kjy,kjz), and
is the angular correlation function of the field at the source plane. In the far zone we have
where k = 2π/λ is the wave number, λ is the wavelength, rj = |r j|, s j = r j/rj are unit position vectors, s j⊥ are their projections into the plane z = z 0, and θj are the angles between s j and the z-axis.
where ∆k ⊥ = k 1⊥ - k 1⊥ and F is assumed to be a real and positive function related to the intensity in the far field in analogy with the function S at the source plane; the radiant intensity  of the field is given by (s 1 = s 2 = s, r 1 = r 2 = r, θ 1 = θ 2 = θ)
The expression (5) is by no means generally valid, but it nevertheless contains a wide class of partially coherent fields.
The key observation of this paper that the cross-spectral density function of any field with a Schell-model angular correlation function can be represented in the form
To see this, we define the Fourier transforms f͂(k ⊥) of f(ρ) and s͂(k ⊥) of s(ρ) and apply the definition (3) as well as the convolution theorem to Eq. (7) to obtain
This is of the form of Eq. (5) if we associate the function s͂ with γ and if the function f(ρ) satisfies the relation
Obviously the function
satisfies Eq. (9) and has the property f(-ρ) = f *(ρ) because F is real and positive.
Inspection of Eq. (7) immediately shows that it is a continuous, incoherent linear superposition of fully coherent elementary fields, each with the same wave form f(ρ, z 0) and cross-spectral density function
These elementary fields originate from different lateral locations, being centered at positions ρ = ρ′ and weighted by the factor
In view of Eqs. (10) and (12) both the elementary function f and the weight distribution s can be evaluated by (inverse) Fourier transforms if the coherence properties of the field in the far zone are known or measured (sometimes there exists some a priori information about the source, which helps to avoid full measurements). Obviously, Eq. (7) reduces to a representation of a coherent field when s(ρ′) → δ(ρ′) and to a representation of an incoherent field when f(ρ) → δ(ρ), where δ is the Dirac delta function.
An important practical result follows immediately from the considerations presented above and the linearity of the superposition in Eq. (7): The free-space propagation of every partially coherent field with a Schell-model angular correlation function can be governed by propagating a single coherent field and then linearly combining the contributions from the entire source area. This result reduces the task of evaluating the four-dimensional integrals in Eqs. (2) and (3) into the evaluation of two-dimensional integrals. Moreover, since all elementary sources are identical, these two-dimensional integrations need to be performed only once for a given partially coherent source.
3. Quasihomogeneous fields
Let us now consider so-called quasihomogeneous fields, for which the source intensity distribution is a slowly varying function in comparison with the modulus of the complex degree of coherence. Mathematically, Eq. (1) then reduces to
where and ∆ρ = ρ 1 - ρ 2 are the average and difference coordinates at the source plane. Partially coherent sources of this type are common in practise: the class contains, e.g., various lamps, light-emitting diodes, multimode fibers and lasers. Moreover, new interesting optoelectronic devices emitting such radiation are emerging, such as the recently-introduced pulsed broad-area vertical-cavity surface-emitting laser .
where is the Fourier transform (FT) of S(ρ̅), and μ͂(k̅⊥) is the FT of μ(∆ρ). Therefore, in view of Eq. (6), the radiant intensity is determined by the FT of the complex degree of coherence μ at z = z 0. In addition, the complex degree of coherence associated with the angular correlation function is given by the FT of the source intensity distribution S. Since, in view of Eqs. (5), (9), and (14)
the supports of the functions f and μ are approximately the same and much smaller than that of S. Therefore we may replace Eq. (7) by
Finding the field representation is now particularly easy because there is no need to determine the weight function s from the far-field coherence properties.
4. Numerical illustration
where the sampling points ρm form a regular (e.g., cartesian or hexagonal) grid that covers the area in which S(ρ, z 0) differs substantially from zero. In case of a quasihomogeneous source
From a numerical point of view, the question of sampling density is of interest even though the method scales well when this density is increased. Intuitively it is clear that the area of the sampling cell should be some small fraction of the effective coherence area at z = z 0. Numerical simulations indicate that for well-behaved functions μ it is typically sufficient to choose the distance between the sampling points smaller than one half of the rms width of μ. Increasing the sampling density and observing the convergence is of course a safe way to make sure of the correctness of the numerical results (at least within the assumptions made).
We illustrate the method with a simple numerical example. Let us imagine that the intensity distribution has been measured to be a supergaussian function of order N and half-width w, and that the radiant intensity is observed to be a Gaussian function with a paraxial 1/e2 half-width θ 0. If the space-bandwidth product wθ 0 is observed to be much larger than the ‘diffraction limit’ (wθ 0)min ≈ λ/π in the coherent case, we know that the source must be quasihomogeneous. In view of Eqs. (4)–(6) and (13)–(15) we then know that μ͂(k̅⊥) is a Gaussian function with 1/e2 half-width θ 0 and therefore μ(∆ρ,z 0) is a Gaussian function with rms width σ = λ/πλ 0  so that (assuming a y-invariant geometry for simplicity)
Therefore the well-known analytical propagation laws of Gaussian beams can be used to determine the partially coherent field at any distance z > z 0.
Figure 1 illustrates the propagation of the field generated by a supergaussian source with N = 3 evaluated by the elementary-source model for two different choices of w and σ. The evolution from a supergaussian near-field distribution to a Gaussian far-field distribution is clearly seen in both cases; when σ is reduced, the far-zone behavior is reached at increasingly short propagation distances. This is consistent with the far-field condition z ≫ zR for quasi-homogeneous Gaussian Schell-model fields (N = 1), for which the Rayleigh range is given by zR = πσW/λ.
5. Discussion and conclusions
We have introduced an efficient model for propagating a large class of partially coherent fields in free space using a new coherent-mode expansion involving a set of laterally displaced modes with the same functional form. We have recently applied  the method presented here (in a more restricted form) to analyze radiation from a novel pulsed vertical-cavity surface-emitting laser, which emits a Gaussian-correlated quasihomogeneous field . The good agreement between the experimental and theoretical results indicates the usefulness of the proposed method in modelling real sources.
The extension of the method to propagation through paraxial optical systems is straightforward with the aid of the Collins propagation formula  in place of the Fresnel formula; in this way fields near the image plane of the source or the best focus of the system can be analyzed. Non-paraxial optical systems can be treated as well if the coherent response to the elementary field is known. This is particularly simple for space-invariant systems. The method loses some of its inherent efficiency in case of space-variant systems because then the coherent response must in principle be determined separately for each elementary source. However, using symmetry considerations and assuming that the system is piecewise space-invariant it is often sufficient to determine just a small number of coherent responses to approximate the partially coherent response adequately.
In this paper we have emphasized quasihomogeneous fields, but the general formalism of Sec. 2 shows that the method is applicable to a much wider class of fields, namely those with Schell-model angular correlation functions. It is therefore worth stressing that a low degree of coherence is not required if the field is of this form. In fact, a Gaussian-beam elementary-source decomposition has been known for a long time for Gaussian Schell-model sources of any degree of coherence . In Ref  it was also shown that a decomposition based on identical elementary Gaussian beams, originating from a single center point but propagating in different directions, is possible for Gaussian Schell-model fields. In view of the Fourier-transform relationship between the cross-spectral density function of the source and its angular correlation function it is clear that such an elementary-field decomposition is in fact applicable to any Schell-model source: an elementary-field decomposition is now applied to the angular correlation function instead of the cross-spectral density function at z = z 0. Explicitly, if μ(ρ 1,ρ 2,z 0) = μ(∆ρ,z 0) in Eq. (1), we may write
where G(k ⊥) is the FT of [S(ρ,z 0)]1/2 and F(k ⊥) is the FT of μ(∆ρ,z 0). Now the cross-spectral density function at the source plane is
Thus the field at z = 0 is indeed a weighted superposition of coherent elementary fields with the same intensity distribution but propagating in different directions.
Finally it should be noted that the concept presented here is not restricted to spatially partially coherent fields only, but it can be formulated also for temporally partially coherent light. Although this is a subject of a separate paper, we note here that in the temporal/spectral domain the elementary-field approach provides an illustrative link between ideal coherent pulses and stationary optical fields.
This work was supported by the Academy of Finland (contracts 205683 and 207523) and the Network of Excellence in Micro-optics (NEMO). Discussions with M. Peeters are appreciated.
References and links
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