## Non-approximated numerical modeling of propagation of light in any state of spatial coherence |

Optics Express, Vol. 19, Issue 25, pp. 25022-25034 (2011)

http://dx.doi.org/10.1364/OE.19.025022

Acrobat PDF (2486 KB)

### Abstract

Due to analytical and numerical difficulties, the propagation of optical fields in any state of spatial coherence is traditionally computed under severe approximations. The paraxial approach in the Fresnel–Fraunhofer domain is one of the most widely used. These approximations provide a rough knowledge of the actual light behavior as it propagates, which is not enough for supporting applications, such as light propagation under a high numerical aperture (NA). In this paper, a non-approximated model for the propagation of optical fields in any state of spatial coherence is presented. The method is applicable in very practical cases, as high-NA propagations, because of its simplicity of implementation. This approach allows for studying unaware behaviors of light as it propagates. The light behavior close to the diffracting transmittances can also be analyzed with the aid of the proposed tool.

© 2011 OSA

## 1. Introduction

1. F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica **5**(8), 785–795 (1938). [CrossRef]

5. D. C. Alvarez-Palacio and J. Garcia-Sucerquia, “Lensless microscopy technique for static and dynamic colloidal systems,” J. Colloid Interface Sci. **349**(2), 637–640 (2010). [CrossRef] [PubMed]

6. K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun. **283**(24), 4859–4865 (2010). [CrossRef]

*π*rays that respectively propagate the radiant and the modulating energies of the field, emitted by point sources distributed on two different layers associated to the emitting surface, named the radiant and the virtual layers [8

8. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A **27**(6), 1322–1330 (2010). [CrossRef] [PubMed]

9. R. Castañeda, H. Muñoz-Ossa, and J. Garcia-Sucerquia, “Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation,” Appl. Opt. **49**(31), 6063–6071 (2010). [CrossRef]

## 2. Exact calculation of the marginal power spectrum

*z*to each other, is [3]where

## 3. Diffraction features shown by the exact calculation software

### 3.1 The free-space diffraction envelope

*z*> 0. Both distributions coincide only in the paraxial region, i.e., within a relatively slight region around the optical axis. It is illustrated by the comparison shown in Fig. 2 for

*free-space diffraction envelope*, because it depends only on the geometrical parameters of the propagation. Figure 3(b) shows the comparison between the free-space diffraction envelope and conventional Lorentzian and Gaussian distributions. The free-space diffraction envelope deviates about 1.5% from the Lorentzian function and about 5% from the Gaussian distribution.

*R*, which is directly provided by the encircled energy curve for the considered

*z*, the aperture angle of the cone is

### 3.2 Spatial frequency modulation by interference

**a**is the midpoint between the two pinholes and

**b**is their separation vector. Consequently, Eqs. (3) and (2) yield

*π*, Fig. 5(a) shows the exact calculation of the power spectrum propagation along the first

*z*axis, and Fig. 5(b) sketches the exactly calculated profile of the power spectrum at the OP in the Fresnel–Fraunhofer domain

*z*axis are determined by the complex degree of spatial coherence. The decay of the fringe brightness and the increase in the fringe width from the cone center to the cone edge is due to the free-space diffraction envelope and to the spatial frequency chirping, respectively.

### 3.3 Diffraction with discrete sets of point sources

8. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A **27**(6), 1322–1330 (2010). [CrossRef] [PubMed]

9. R. Castañeda, H. Muñoz-Ossa, and J. Garcia-Sucerquia, “Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation,” Appl. Opt. **49**(31), 6063–6071 (2010). [CrossRef]

8. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A **27**(6), 1322–1330 (2010). [CrossRef] [PubMed]

*N*point sources with pitch

**b**, allocated in the radiant layer. The power spectrum that this array provides at the OP takes the form

*classes of radiator pairs*[12

12. R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. **226**(1-6), 45–55 (2003). [CrossRef]

*m*th class of radiator pairs, i.e., the class with separation vector

**27**(6), 1322–1330 (2010). [CrossRef] [PubMed]

*L*and pitch

### 3.4 Modulating energy at the AP

*z*between the AP and the OP, including

## 4. Conclusion

*z*instead of the application of the conventional Fourier transform methods. The modulating energy in this case points out interesting details of the source correlations. The simplicity and the efficiency of the numerical algorithm are remarkable. It makes this strategy especially useful for modeling practical applications beyond the conventional paraxial approach in the Fresnel–Fraunhofer domain, such as partially coherent imaging, digital holography microscopy, and beam shaping.

## Acknowledgments

## References and links

1. | F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica |

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

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

4. | Z. Jaroszewicz, |

5. | D. C. Alvarez-Palacio and J. Garcia-Sucerquia, “Lensless microscopy technique for static and dynamic colloidal systems,” J. Colloid Interface Sci. |

6. | K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun. |

7. | R. Castañeda, “The optics of spatial coherence wavelets, in |

8. | R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A |

9. | R. Castañeda, H. Muñoz-Ossa, and J. Garcia-Sucerquia, “Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation,” Appl. Opt. |

10. | R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. , |

11. | K. Iizuka, |

12. | R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.5630) Coherence and statistical optics : Radiometry

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 9, 2011

Revised Manuscript: June 21, 2011

Manuscript Accepted: July 4, 2011

Published: November 22, 2011

**Citation**

Román Castañeda and Jorge Garcia-Sucerquia, "Non-approximated numerical modeling of propagation of light in any state of spatial coherence," Opt. Express **19**, 25022-25034 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25022

Sort: Year | Journal | Reset

### References

- F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica5(8), 785–795 (1938). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon Press, 1993), Chap. 10.
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), Chap. 4.
- Z. Jaroszewicz, Axicons: Design and Propagation Properties (Research and development treatises, SPIE Polish chapter) (SPIE, 1997), Vol. 5.
- D. C. Alvarez-Palacio and J. Garcia-Sucerquia, “Lensless microscopy technique for static and dynamic colloidal systems,” J. Colloid Interface Sci.349(2), 637–640 (2010). [CrossRef] [PubMed]
- K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun.283(24), 4859–4865 (2010). [CrossRef]
- R. Castañeda, “The optics of spatial coherence wavelets, in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic Press, 2010), Vol. 164.
- R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A27(6), 1322–1330 (2010). [CrossRef] [PubMed]
- R. Castañeda, H. Muñoz-Ossa, and J. Garcia-Sucerquia, “Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation,” Appl. Opt.49(31), 6063–6071 (2010). [CrossRef]
- R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. , 58(11) (2011), doi: . [CrossRef]
- K. Iizuka, Engineering Optics (Springer Verlag, 1985).
- R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun.226(1-6), 45–55 (2003). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.