### Abstract

Stochastic fields do not generally possess a Fourier transform. This makes the second-order statistics calculation very difficult, as it requires solving a fourth-order stochastic wave equation. This problem was alleviated by Wolf who introduced the coherent mode decomposition and, as a result, space-frequency statistics propagation of wide-sense stationary fields. In this paper we show that if, in addition to wide-sense stationarity, the fields are also wide-sense statistically homogeneous, then monochromatic plane waves can be used as an eigenfunction basis for the cross spectral density. Furthermore, the eigenvalue associated with a plane wave, exp[i(k·r-ωt)], is given by the spatiotemporal power spectrum evaluated at the frequency (k, ω). We show that the second-order statistics of these fields is fully described by the spatiotemporal power spectrum, a real, positive function. Thus, the second-order statistics can be efficiently propagated in the wavevector-frequency representation using a new framework of deterministic signals associated with random fields. Analogous to the complex analytic signal representation of a field, the deterministic signal is a mathematical construct meant to simplify calculations. Specifically, the deterministic signal associated with a random field is defined such that it has the identical autocorrelation as the actual random field. Calculations for propagating spatial and temporal correlations are simplified greatly because one only needs to solve a deterministic wave equation of second order. We illustrate the power of the wavevectorfrequency representation with calculations of spatial coherence in the far zone of an incoherent source, as well as coherence effects induced by biological tissues.

Original language | English (US) |
---|---|

Pages (from-to) | 20806-20820 |

Number of pages | 15 |

Journal | Optics Express |

Volume | 21 |

Issue number | 18 |

DOIs | |

State | Published - Sep 9 2013 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Optics Express*,

*21*(18), 20806-20820. https://doi.org/10.1364/OE.21.020806

**Deterministic signal associated with a random field.** / Kim, Taewoo; Zhu, Ruoyu; Nguyen, Tan H.; Zhou, Renjie; Edwards, Chris; Goddard, Lynford L.; Popescu, Gabriel.

Research output: Contribution to journal › Article

*Optics Express*, vol. 21, no. 18, pp. 20806-20820. https://doi.org/10.1364/OE.21.020806

}

TY - JOUR

T1 - Deterministic signal associated with a random field

AU - Kim, Taewoo

AU - Zhu, Ruoyu

AU - Nguyen, Tan H.

AU - Zhou, Renjie

AU - Edwards, Chris

AU - Goddard, Lynford L.

AU - Popescu, Gabriel

PY - 2013/9/9

Y1 - 2013/9/9

N2 - Stochastic fields do not generally possess a Fourier transform. This makes the second-order statistics calculation very difficult, as it requires solving a fourth-order stochastic wave equation. This problem was alleviated by Wolf who introduced the coherent mode decomposition and, as a result, space-frequency statistics propagation of wide-sense stationary fields. In this paper we show that if, in addition to wide-sense stationarity, the fields are also wide-sense statistically homogeneous, then monochromatic plane waves can be used as an eigenfunction basis for the cross spectral density. Furthermore, the eigenvalue associated with a plane wave, exp[i(k·r-ωt)], is given by the spatiotemporal power spectrum evaluated at the frequency (k, ω). We show that the second-order statistics of these fields is fully described by the spatiotemporal power spectrum, a real, positive function. Thus, the second-order statistics can be efficiently propagated in the wavevector-frequency representation using a new framework of deterministic signals associated with random fields. Analogous to the complex analytic signal representation of a field, the deterministic signal is a mathematical construct meant to simplify calculations. Specifically, the deterministic signal associated with a random field is defined such that it has the identical autocorrelation as the actual random field. Calculations for propagating spatial and temporal correlations are simplified greatly because one only needs to solve a deterministic wave equation of second order. We illustrate the power of the wavevectorfrequency representation with calculations of spatial coherence in the far zone of an incoherent source, as well as coherence effects induced by biological tissues.

AB - Stochastic fields do not generally possess a Fourier transform. This makes the second-order statistics calculation very difficult, as it requires solving a fourth-order stochastic wave equation. This problem was alleviated by Wolf who introduced the coherent mode decomposition and, as a result, space-frequency statistics propagation of wide-sense stationary fields. In this paper we show that if, in addition to wide-sense stationarity, the fields are also wide-sense statistically homogeneous, then monochromatic plane waves can be used as an eigenfunction basis for the cross spectral density. Furthermore, the eigenvalue associated with a plane wave, exp[i(k·r-ωt)], is given by the spatiotemporal power spectrum evaluated at the frequency (k, ω). We show that the second-order statistics of these fields is fully described by the spatiotemporal power spectrum, a real, positive function. Thus, the second-order statistics can be efficiently propagated in the wavevector-frequency representation using a new framework of deterministic signals associated with random fields. Analogous to the complex analytic signal representation of a field, the deterministic signal is a mathematical construct meant to simplify calculations. Specifically, the deterministic signal associated with a random field is defined such that it has the identical autocorrelation as the actual random field. Calculations for propagating spatial and temporal correlations are simplified greatly because one only needs to solve a deterministic wave equation of second order. We illustrate the power of the wavevectorfrequency representation with calculations of spatial coherence in the far zone of an incoherent source, as well as coherence effects induced by biological tissues.

UR - http://www.scopus.com/inward/record.url?scp=84884557459&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884557459&partnerID=8YFLogxK

U2 - 10.1364/OE.21.020806

DO - 10.1364/OE.21.020806

M3 - Article

C2 - 24103953

AN - SCOPUS:84884557459

VL - 21

SP - 20806

EP - 20820

JO - Optics Express

JF - Optics Express

SN - 1094-4087

IS - 18

ER -