## Spectral coherence properties of temporally modulated stationary light sources

Optics Express, Vol. 11, Issue 16, pp. 1894-1899 (2003)

http://dx.doi.org/10.1364/OE.11.001894

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### Abstract

It is shown that partially spectrally coherent pulses of light with controlled spectral coherence properties can be generated by temporal modulation of beams emitted by stationary light sources. A method for generation of spectrally Gaussian Schell-model-type pulses is presented.

© 2003 Optical Society of America

## 1. Introduction

1. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. **204**, 53–58 (2002). [CrossRef]

1. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. **204**, 53–58 (2002). [CrossRef]

2. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. **AP-15**, 187–188 (1967). [CrossRef]

3. J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. **26**, 297–300 (1976). [CrossRef]

8. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A **5**, 713–720 (1988). [CrossRef]

1. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. **204**, 53–58 (2002). [CrossRef]

## 2. Fundamentals

*S*

_{s}(

*ω*) the spectrum of the stationary light source at

*z*=0. According to the Wiener-Khintchine theorem [9], the mutual coherence function of the source is given by

*τ*=

*t*

_{2}-

*t*

_{1},

*t*

_{1}and

*t*

_{2}represent two arbitrary instants of time, and we have used the subscript s to indicate that the field is stationary. The complex degree of coherence of the light field is then given by

_{s}(0) is the (constant) intensity of the light beam.

*M*(

*t*). Since

*M*(

*t*) is deterministic, the modulator does not affect the complex degree of coherence. The transmission of each temporal realization of the field through the modulator is obtained by multiplication with

*M*(

*t*) and, according to the definition of the mutual coherence function, the transmitted field at

*z*

_{1}=

*z*

_{2}=0 is described by

*c*the mutual coherence function takes the form

*z*

_{1},

*z*

_{2};

*t*

_{1},

*t*

_{2})=arg

*M*(

*t*

_{1}-

*z*

_{1}/

*c*)-arg

*M*(

*t*

_{2}-

*z*

_{2}/

*c*). Obviously we can control the pulse intensity be means of the absolute value of

*M*(

*t*) and the complex degree of temporal coherence by the functional form of

*S*

_{s}(

*ω*) and the argument of

*M*(

*t*). For real-valued

*M*, we always obtain a Schell-model temporal pulse, i.e., the degree of temporal coherence is a function of the time difference only, not depending on the choice of the origin of time.

## 3. Gaussian temporal modulation of sources with Gaussian spectra

_{s}is the characteristic spectral width of the spectrum of the original source and

*T*

_{m}is a measure of the pulse duration (the subscript m refers to the modulator). The model for the spectrum is rather realistic if we consider typical light-emitting diodes, for which the spectral width Ω is of the order of tens of nanometers. While the temporal modulation function

*M*(

*t*) implemented by real electro-optic or acousto-optics modulators is typically more like a flat-top function than a Gaussian function, the Gaussian model used here still provides useful order-of-magnitude estimates for modelling various phenomena.

_{0}=√

*π*Ω

_{s}

*S*

_{0}. We next insert Eqs. (12) and (11) into Eq. (4). In view of Eqs. (5) and (6), we see that a temporal Gaussian Schell-model pulse with an intensity distribution of the form

*γ*(

*z*

_{1},

*z*

_{2};

*t*

_{1},

*t*

_{2}) with

*µ*(

*z*

_{1},

*z*

_{2};

*ω*

_{1},

*ω*

_{2}) of the form

## 4. Interpretation of the results

_{c}, to the parameters

*T*

_{m}and Ω

_{s}that we may control, at least over certain intervals, by experimental means. These expressions may be called equivalence relations in the sense that there exists an infinite number of combinations of

*T*

_{m}and Ω

_{s}that yield either the same spectrum or the same spectral degree of coherence (but not both at the same time).

## 5. Discussion

**204**, 53–58 (2002). [CrossRef]

10. 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]

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

*f*system with a reflective spatial filter

*F*(

**r**) (see Fig. 1(b)). Here one starts with an incoherent Gaussian source in the spatial frequency domain. The intensity distribution of this source (at certain frequency) is a Gaussian function

*S*

_{s}(

**f**). The field in the spatial domain has a constant intensity distribution but a Gaussian Schell-model-type distribution of the complex degree of spatial coherence

*µ*(

**r**

_{1}–

**r**

_{2}). When truncated spatially by the (reflective) filter function

*F*(

**r**), we obtain a field with a Gaussian spectrum

*S*(

**f**) and a Gaussian distribution of the degree of spatial coherence

*µ*(

**f**

_{1},

**f**

_{2}) in the spatial frequency domain.

## 6. Conclusions

**204**, 53–58 (2002). [CrossRef]

## Acknowledgments

## References and links

1. | P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. |

2. | A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. |

3. | J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. |

4. | E. Wolf and E. Collett, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. |

5. | F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. |

6. | A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. |

7. | J. Deschamps, D. Courjon, and J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. |

8. | A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A |

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

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

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

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 23, 2003

Revised Manuscript: July 30, 2003

Published: August 11, 2003

**Citation**

Hanna Lajunen, Jani Tervo, Jari Turunen, Pasi Vahimaa, and Frank Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express **11**, 1894-1899 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1894

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### References

- P. P äkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, �??Partially coherent Gaussian pulses,�?? Opt. Commun. 204, 53�??58 (2002). [CrossRef]
- A. C. Schell, �??A technique for the determination of the radiation pattern of a partially coherent aperture,�?? IEEE Trans. Antennas Propag. AP-15, 187�??188 (1967). [CrossRef]
- J. T. Foley and M. S. Zubairy, �??The directionality of Gaussian Schell-model beams,�?? Opt. Commun. 26, 297�??300 (1976). [CrossRef]
- E. Wolf and E. Collett, �??Beams generated by Gaussian quasi-homogeneous sources,�?? Opt. Commun. 32, 27�??31 (1980). [CrossRef]
- F. Gori, �??Collett�??Wolf sources and multimode lasers,�?? Opt. Commun. 34, 301�??305 (1980). [CrossRef]
- A. T. Friberg and R. J. Sudol, �??Propagation parameters of Gaussian Schell-model beams,�?? Opt. Commun. 41, 383�??387 (1982). [CrossRef]
- J. Deschamps, D. Courjon, and J. Bulabois, �??Gaussian Schell-model sources: an example and some perspectives,�?? J. Opt. Soc. Am. 73, 256�??261 (1983). [CrossRef]
- A. T. Friberg and J. Turunen, �??Imaging of Gaussian Schell-model sources,�?? J. Opt. Soc. Am. A 5, 713�??720 (1988). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), p. 59.
- 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]

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