## Independent-elementary-field model for three-dimensional spatially partially coherent sources

Optics Express, Vol. 16, Issue 9, pp. 6433-6442 (2008)

http://dx.doi.org/10.1364/OE.16.006433

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### Abstract

The independent-elementary-source model recently developed for planar sources [Opt. Express **14**, 1376 (2006)] is extended to volume source distributions. It is shown that the far-field radiation pattern is independent of the three-dimensional (3D) distribution of the coherent elementary sources, but the absolute value of the complex degree of spectral coherence is determined by the 3D Fourier transform of the weight function of the elementary fields. Some methods to determine an ‘effective’ three-dimensional source distribution with partial transverse and longitudinal spatial coherence properties are outlined. Especially in its longitudinal extension, this effective source volume can be very different from the primary emitting volume of the source. The application of the model to efficient numerical propagation of partially coherent fields is discussed.

© 2008 Optical Society of America

## 1. Introduction

2. P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express **14**, 1376 (2006). [CrossRef] [PubMed]

3. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional fields,” Opt. Commun. **27**, 185–187 (1978). [CrossRef]

4. F. Gori, “Directionality and spatial coherence,” Opt. Acta **27**, 1025–1034 (1980). [CrossRef]

5. F. Gori and M. Santarsiero, “Devising genuine correlation functions,” Opt. Lett. **32**, 3531–3133 (2007). [CrossRef] [PubMed]

6. M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, P. Vahimaa, and J. Turunen, “Propagation of spatially partially coherent emission from a vertical-cavity surface-emitting laser,” Opt. Express **13**, 9337–9345 (2005). [CrossRef] [PubMed]

2. P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express **14**, 1376 (2006). [CrossRef] [PubMed]

7. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express **14**, 5007–5012 (2006). [CrossRef] [PubMed]

8. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express **15**, 5160–5165 (2007).
http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-5160 [CrossRef] [PubMed]

9. V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. **32**, 1608–1010 (2007). [CrossRef] [PubMed]

## 2. Theoretical formulation

*f*(

*x*,

*y*) at frequency

*ω*(which we leave implicit for brevity) and at plane

*z*=0, with angular spectrum

*k*and

_{x}*k*are the transverse components of the wave vector

_{y}**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

*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

*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

*p*(

*x*′,

*y*′,

*z*′).

*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*) with

_{z}*r*=|

**r**|, the cross-spectral density function in the far zone is

*f*(

*x*,

*y*), as in the case of a planar source [2

2. P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express **14**, 1376 (2006). [CrossRef] [PubMed]

*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

*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

*k*=

_{z}*ks*, the absolute value of

_{z}*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).

*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 Δ

*z*small enough to represent the source and the target regions with sufficient resolution. Of course, one also needs to ensure sufficiently dense transverse sampling (Δ

*x*) of the weight function to obtain convergent results.

## 4. Determination of model-source parameters

*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.

*r*. Denoting

*s*

_{x1}=

*s*

_{y1}=0,

*s*

_{x2}=

*s*,

_{x}*s*

_{y2}=

*s*, a fringe visibility measurement provides (after normalization) the function

_{y}*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

*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

*x*=sin(

*πx*)/(

*πx*) and

*θ*is the angle defined by

*s*=sin

_{x}*θ*and

*s*=cos

_{z}*θ*. 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

10. A. T. Friberg and R. J. Sudol, “The spatial coherence properties of gaussian Schell-model beams,” Opt. Acta **30**, 1075–1097 (1983). [CrossRef]

3. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional fields,” Opt. Commun. **27**, 185–187 (1978). [CrossRef]

*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

*z*

_{R}=

*kw*

^{2}

_{0}/2 is the Rayleigh range of the elementary Gaussian source, we normalize

*z*<0.

*z*= constant,

*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,

*I*(0,0,

*z*)=

*W*(0,0,

*z*,0,0,

*z*) is the axial intensity distribution.

*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 Δ

*z*=

*z*

_{2}-

*z*

_{1}) an analytical expression for the axial longitudinal coherence in the form

*z*

_{1}=0 and the asymptotic limit

*z*

_{2}→∞ is considered, we have

*w*

_{0}=100

*λ*, 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*=0.1

*z*

_{R},

*L*=

*z*

_{R},

*L*=10

*z*

_{R}, and

*L*=100

*z*

_{R}, where

*z*

_{R}=10

^{4}

*πλ*.

*L*<

*z*<0 is evident (not shown in totality for

*L*=10

*z*

_{R}, and

*L*=100

*z*

_{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

10. A. T. Friberg and R. J. Sudol, “The spatial coherence properties of gaussian Schell-model beams,” Opt. Acta **30**, 1075–1097 (1983). [CrossRef]

## 6. Conclusions and outlook

*f*(

*x*,

*y*). In addition, we are developing a combination of the spatial [2

**14**, 1376 (2006). [CrossRef] [PubMed]

7. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express **14**, 5007–5012 (2006). [CrossRef] [PubMed]

11. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. **60**, 1168–1177 (1970). [CrossRef]

*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.

## Acknowledgments

## References and links

1. | L. Mandel and E. Wolf, |

2. | P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express |

3. | F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional fields,” Opt. Commun. |

4. | F. Gori, “Directionality and spatial coherence,” Opt. Acta |

5. | F. Gori and M. Santarsiero, “Devising genuine correlation functions,” Opt. Lett. |

6. | M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, P. Vahimaa, and J. Turunen, “Propagation of spatially partially coherent emission from a vertical-cavity surface-emitting laser,” Opt. Express |

7. | P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express |

8. | A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express |

9. | V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. |

10. | A. T. Friberg and R. J. Sudol, “The spatial coherence properties of gaussian Schell-model beams,” Opt. Acta |

11. | S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. |

12. | J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(050.1940) Diffraction and gratings : Diffraction

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: February 20, 2008

Revised Manuscript: March 31, 2008

Manuscript Accepted: March 31, 2008

Published: April 22, 2008

**Citation**

Jari Turunen and Pas Vahimaa, "Independent-elementary-field model for
three-dimensional spatially partially
coherent sources," Opt. Express **16**, 6433-6442 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6433

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
- P. Vahimaa and J. Turunen, "Finite-elementary source model for partially coherent radiation," Opt. Express 14,1376 (2006). [CrossRef] [PubMed]
- F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional fields," Opt. Commun. 27, 185-187 (1978). [CrossRef]
- F. Gori, "Directionality and spatial coherence," Opt. Acta 27, 1025-1034 (1980). [CrossRef]
- F. Gori and M. Santarsiero, "Devising genuine correlation functions," Opt. Lett. 32, 3531-3133 (2007). [CrossRef] [PubMed]
- M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, P. Vahimaa, and J. Turunen, "Propagation of spatially partially coherent emission from a vertical-cavity surface-emitting laser," Opt. Express 13, 9337-9345 (2005). [CrossRef] [PubMed]
- P. Vahimaa and J. Turunen, "Independent-elementary-pulse representation for non-stationary fields," Opt. Express 14, 5007-5012 (2006). [CrossRef] [PubMed]
- A. T. Friberg, H. Lajunen, and V. Torres-Company, "Spectral elementary-coherence-function representation for partially coherent light pulses," Opt. Express 15, 5160-5165 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-5160 [CrossRef] [PubMed]
- V. Torres-Company, G. Minguez-Vega, J. Lancis, and A. T. Friberg, "Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper," Opt. Lett. 32, 1608-1010 (2007). [CrossRef] [PubMed]
- A. T. Friberg and R. J. Sudol, "The spatial coherence properties of gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983). [CrossRef]
- S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168- 1177 (1970). [CrossRef]
- J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

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