## Propagation of partially coherent light through a light pipe |

Optics Express, Vol. 21, Issue 14, pp. 17007-17019 (2013)

http://dx.doi.org/10.1364/OE.21.017007

Acrobat PDF (1814 KB)

### Abstract

In laser projection applications, laser light modules are often combined with rotating diffusers in order to reduce the appearance of speckle on the projection screen. The rotation of a diffuser in a laser beam generates a beam of partially coherent light. Propagation of this light through the different optical components constituting the laser projector is thus essential when investigating the appearance of speckle. In this paper, a computationally efficient simulation model is presented to propagate partially coherent light through a homogenizing rectangular light pipe. The light pipe alters the coherence properties of the light and different consequences are discussed. The outcomes of the simulation model are experimentally verified using a reversing wavefront Michelson interferometer.

© 2013 OSA

## 1. Introduction

1. K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. **49**, 79–98 (2010) [CrossRef] .

3. U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID **18**, 813–820 (2010) [CrossRef] .

5. S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. **61**, 847–851 (1971) [CrossRef] .

11. F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees of coherence through complex optical systems,” Appl. Opt. **48**, 1527–1534 (2009) [CrossRef] [PubMed] .

## 2. Optical simulation model

^{2}and a length of 13cm.

### 2.1. Source model

*elementary fields*[8

8. P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express **14**, 1376–1381 (2006) [CrossRef] [PubMed] .

12. P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. **29**, 256–260 (1979) [CrossRef] .

14. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. **67**, 245–250 (1988) [CrossRef] .

15. M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express **13**, 9337–9345 (2005) [CrossRef] [PubMed] .

*U*. This complex scalar function must satisfy the Helmholtz wave equation, i.e. (∇

^{2}+

*k*

^{2})

*U*= 0, with

*k*= 2

*π*/

*λ*as the wave number. Making use of Green’s theorem, the Rayleigh-Sommerfeld formulation is derived from the wave equation for the propagation of light in free-space [16] where

*z*= 0) to an observation plane and the integration is performed over the area of the aperture. This formulation is valid as long as both the propagation distance and the aperture size are greater than the wavelength of the light. As can be deducted from Eq. (2), the propagation of the partially coherent light with the Rayleigh-Sommerfeld formulation is not quite much different from a ray-tracing principle. Every point in the aperture plane influences every point in the observation plane and all these individual influences are added. In order to further reduce the computation time of our simulations, we approximated the field as a spatially separable field in

*x*and

*y*dimensions with

*U*(

*x*,

*y*,

*z*) =

*U*(

_{x}*x*,

*z*)

*U*(

_{y}*y*,

*z*). Such an approximation should not reduce the accuracy of the results significantly, as the light pipe is of a rectangular shape. Combination of propagated fields

*U*and

_{x}*U*provides a 2-D wavefront at the observation plane.

_{y}17. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. **45**, 1102–1110 (2006) [CrossRef] [PubMed] .

### 2.2. Simulation model of the light pipe

### 2.3. Calculation of the complex degree of coherence

*U*, the mutual coherence function at points

**r**and

_{1}**r**, and at times

_{2}*t*and

*t*+

*τ*is given by [6

6. L. Mandel and E. Wolf, *Optical Coherence and Quantum Optics* (Cambridge university press, Cambridge, 1995) [CrossRef] .

^{*}denotes the complex conjugate,

*τ*denotes the time difference and 〈〉 denotes an ensemble average [18]. The

*complex degree of coherence γ*(

**r**

_{1},

**r**

_{2},

*τ*) is now defined as where

*I*(

**r**) = Γ(

**r**,

**r**, 0) denotes the intensity at a point

**r**[18]. In case |

*γ*(

**r**

_{1},

**r**

_{2},

*τ*)| = 1, the field is said to be fully

*coherent*at the examined pair of points and at a time difference

*τ*; in the case |

*γ*(

**r**

_{1},

**r**

_{2},

*τ*)| = 0, the field is called

*incoherent*. In intermediate cases, i.e. 0 < |

*γ*(

**r**

_{1},

**r**

_{2},

*τ*)| < 1, the field is said to be partially coherent.

*l*and

*m*in the detector plane takes on the form and, consequently, the complex degree of coherence is Here,

*U*is the field of one elementary mode at the plane of the detector at pixel

^{l}*l*. Essentially, the field at pixel

*l*is multiplied with the conjugate of the same field at pixel

*m*. This is performed for every elementary beam and then the results are summed. In the remainder of this paper, we visualize the coherence properties of a field by taking the magnitude of the complex degree of coherence between the center point of the field and other points along its transverse extent, i.e.

*τ*= 0. The resulting graph thus describes the spatial coherence properties of the light as a function of the transverse extent.

## 3. Experimental evaluation of the simulation model

19. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. **31**, 861–863 (2006) [CrossRef] [PubMed] .

21. G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. **45**, 249–255 (2009) [CrossRef] .

**r**= (

_{1}*x*

_{1},

*y*

_{1}) and

**r**= (−

_{2}*x*

_{1}, −

*y*

_{1}), where the position (0, 0) corresponds to the center of the interferogram. Therefore, radially symmetric points with respect to the interferogram’s center are compared. If the field in

**r**is coherent with the field in

_{1}**r**, interference fringes occur at those positions in the CCD plane that correspond to

_{2}**r**and

_{1}**r**.

_{2}*γ*is possible by acquiring four images. We measure the interference pattern

20. M. Imai, Y. Ohtsuka, and S. Satoh, “Spatial coherence analysis of light propagation in optical fibers by interferometric methods,” J. Opt. Soc. Am. A **3**, 1059–1064 (1986) [CrossRef] .

*γ*only provide a 2-D slice of the 4-D degree of coherence. A full mapping of the 4-D degree of coherence can be obtained by measuring several inter-ferograms for various shifts of the interferogram’s center. However, in the case of a rotating diffuser, the source is known to be a Schell-model source. In such a situation, the degree of coherence only depends on the difference in position vector, i.e.

*γ*(

**r**,

_{1}**r**) ≡

_{2}*γ*(

**r**−

_{1}**r**). We have experimentally verified the latter condition for the diffuser used in our setup using the reversing wavefront interferometer.

_{2}## 4. Comparison of the simulation results with the experiments

### 4.1. Modelling of the source

*W*at the diffuser plane is also an important parameter in the simulation model. This was first estimated by taking an image of the laser beam at the plane of the diffuser with a CCD camera and measuring its size. A diameter of 780μm was obtained. This measurement is verified making use of the interferometer setup. By measuring the diameter of the coherence area at a position in the far field, the coherence angle can be found as follows where

*D*

_{coh}is the diameter of the degree of coherence in the far-field and

*z*is the optical path length between the diffuser and the far-field plane where

*D*

_{coh}is measured. The coherence angle can then be used in in order to estimate the beam diameter. At a distance of 13cm, the coherence area diameter is found to be

*D*

_{coh}= 114 μm. Solving Eqs. (8)–(9) for this situation results in an extent of the source beam at the diffuser plane of 780μm. This leads us to conclude that this measured value for the extent of the source beam is accurate.

### 4.2. Free space propagation

*L*

_{LP}/

*n*= 13cm/1.5 = 8.67cm. Remark that this distance does not correspond with the optical path length but that we simply want to correlate this situation with the propagation through the glass light pipe.

*γ*[−

**r**

_{1},

**r**

_{1}, 0] between points that are radially symmetric with respect to the correlation measurement’s center (which is the center of Fig. 6). The diameter of the interference pattern in Fig. 6 thus corresponds to the diameter of the coherence area in Fig. 5. There are still some small variations in the pattern of Fig. 6 because the image

### 4.3. Influence of the light pipe on the complex degree of coherence

_{coh}is defined by the Fourier transform ℱ of the near-field intensity. The near field intensity distribution at the rotating diffuser NF

_{int}can be described as the convolution between a Gaussian profile (originating from the Gaussian single-mode beam impinging at the diffuser) and a spatial comb (as a result of the internal reflection inside the light pipe), i.e. in 1-D this can be written as where exp(−

*ax*

^{2}) is a Gaussian function with a width equal to the width of the single-mode beam impinging on the diffuser,

*D*is the transverse size of the light pipe and

*x*is the transverse coordinate in the near field. As a result, the profile of the complex degree of coherence in Fig 7 is the product between a Gaussian envelope and a comb, where

*ξ*is the spatial frequency in the far-field. Consequently, the far-field grid spacing

*x*

_{grid}is given by where

*λ*is the wavelength,

*z*is the longitudinal propagation distance from near field to the exit facet of the light pipe. As a result, if the width of the light pipe is reduced, the distance between the peaks in Fig. 7 will increase. Note that the factor 1/2 is the result of the fact that the reversing wavefront interferometer measures the coherence of radially symmetric points (cfr. Section 3).

^{2}, so the interference pattern in both directions has a different period. A cross-section profile of the degree of coherence is taken in the horizontal direction and is depicted in Fig. 8(b).

*λ*/4. The measurements are not taken instantly, so some fluctuations arise resulting in a broadening of the grid pattern. Additionally, the grid-like pattern will be somewhat blurred due to the finite size of the CCD pixels and due to optical imperfections in the interferometer. The sum of these external factors results in a fringe visibility that does not reach zero for a low degree of coherence.

## 5. Summary & Conclusion

## Acknowledgments

## References and links

1. | K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. |

2. | J. Hecht, “A short history of laser development,” Appl. Opt. |

3. | U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID |

4. | J. W. Goodman, |

5. | S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. |

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

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

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

9. | H. Gross, |

10. | F. Wyrowski, “Field Tracing for Unified Optical Modeling,” in |

11. | F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees of coherence through complex optical systems,” Appl. Opt. |

12. | P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. |

13. | J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. |

14. | Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. |

15. | M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express |

16. | J. Goodman, |

17. | F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. |

18. | M. Born and E. Wolf, |

19. | M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. |

20. | M. Imai, Y. Ohtsuka, and S. Satoh, “Spatial coherence analysis of light propagation in optical fibers by interferometric methods,” J. Opt. Soc. Am. A |

21. | G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(110.4980) Imaging systems : Partial coherence in imaging

(110.3175) Imaging systems : Interferometric imaging

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: March 6, 2013

Revised Manuscript: May 15, 2013

Manuscript Accepted: May 25, 2013

Published: July 10, 2013

**Citation**

Stijn Roelandt, Jani Tervo, Youri Meuret, Guy Verschaffelt, and Hugo Thienpont, "Propagation of partially coherent light through a light pipe," Opt. Express **21**, 17007-17019 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17007

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### References

- K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt.49, 79–98 (2010). [CrossRef]
- J. Hecht, “A short history of laser development,” Appl. Opt.49, F99–F122 (2010). [CrossRef] [PubMed]
- U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID18, 813–820 (2010). [CrossRef]
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, Englewood, 2007).
- S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am.61, 847–851 (1971). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge university press, Cambridge, 1995). [CrossRef]
- F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun.27, 185–187 (1978). [CrossRef]
- P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express14, 1376–1381 (2006). [CrossRef] [PubMed]
- H. Gross, Handbook of Optical Systems: Aberration Theory and Correction of Optical Systems (Wiley, 2007).
- F. Wyrowski, “Field Tracing for Unified Optical Modeling,” in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW4A.1.
- F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees of coherence through complex optical systems,” Appl. Opt.48, 1527–1534 (2009). [CrossRef] [PubMed]
- P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun.29, 256–260 (1979). [CrossRef]
- J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun.32, 203–207 (1980). [CrossRef]
- Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun.67, 245–250 (1988). [CrossRef]
- M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express13, 9337–9345 (2005). [CrossRef] [PubMed]
- J. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).
- F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt.45, 1102–1110 (2006). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge university press, Cambridge, 1999).
- M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett.31, 861–863 (2006). [CrossRef] [PubMed]
- M. Imai, Y. Ohtsuka, and S. Satoh, “Spatial coherence analysis of light propagation in optical fibers by interferometric methods,” J. Opt. Soc. Am. A3, 1059–1064 (1986). [CrossRef]
- G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron.45, 249–255 (2009). [CrossRef]

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