## Simulating fields of arbitrary spatial and temporal coherence

Optics Express, Vol. 14, Issue 17, pp. 7567-7578 (2006)

http://dx.doi.org/10.1364/OE.14.007567

Acrobat PDF (608 KB)

### Abstract

Optical coherence theory typically deals with the average properties of randomly fluctuating fields. However, in some circumstances the averaging process can mask important physical aspects of the field propagation. We derive a new method of simulating partially coherent fields of nearly arbitrary spatial and temporal coherence. These simulations produce the expected coherence properties when averaged over sufficently long time intervals. Examples of numerous fields are given, and an analytic formula for the intensity fluctuations of the field is given. The method is applied to the propagation of partially coherent fields through random phase screens.

© 2006 Optical Society of America

## 1. Introduction

04. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. **37**, 671–684 (1990). [CrossRef]

05. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A **19**, 1592–1598 (2002). [CrossRef]

06. J. Wu and A. D. Boardman, “Coherence length of a Gaussian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. **38**, 1355–1363 (1991). [CrossRef]

07. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian Beam: implications for free-space laser communication,” J. Opt. Soc. Am. A **19**, 1794–1802 (2002). [CrossRef]

08. O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom,” Opt. Eng. **43**, 330–341 (2004). [CrossRef]

13. W. Wang and E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E **52**, 5532–5539 (1995). [CrossRef]

## 2. Generating realizations of fields of arbitrary spatial and temporal coherence

*z*= 0, which emits optical pulses of fixed spatial and temporal shape at random times. We initially restrict ourselves to a time interval

*t*∈ [-

*T*/2,

*T*/2]. Assuming that the pulses are emitted independently of one another, the probability

*p*(

*N*) that

*N*are emitted in this interval is dictated by Poissonian statistics, i.e.

*N*pulses are emitted in this interval. The field of these

*N*pulses is then given by

**r**,

*t*) is the field amplitude of a single pulse in the plane

*z*= 0 at transverse position

**r**= (

*x*,

*y*), aside from a linear phase term,

*t*

_{j}is the time of emission of the

*j*th pulse and

**K**

_{j}is the angle of inclination of the

*j*th pulse. The time of arrival is assumed to be a random variable uniformly distributed throughout the interval, and the angle of inclination is a random variable whose probability distribution

*P*(

**K**) is for now unspecified. This representation of the field is very similar to that used in Ref. [13

13. W. Wang and E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E **52**, 5532–5539 (1995). [CrossRef]

**K**

_{j}and its probability distribution allows us to control the spatial coherence properties of the field as well as the temporal properties.

*V*(

**r**,

*t*) is defined as

*τ*=

*t*

_{2}-

*t*

_{1}and the angle brackets denote ensemble averaging. It is to be noted that this ensemble average is equivalent to three independent averages: the average over the arrival times

*t*

_{j}of the pulses, the average over the inclination factors

**K**

_{j}, and the average over the number of pulses per interval

*N*. The instantaneous form of this function for our field of

*N*pulses is

*t*

_{i}. To do so, we express Λ(

**r**,

*t*) in terms of its temporal Fourier transform, i.e.

*t*

_{i}and

*t*

_{j}over the interval can be written as

*x*] = sin[

*x*]/

*x*. We may separate our expression (6) into two distinct sums: one for which

*i*=

*j*and one for which

*i*≠

*j*. We may then write

**K**

_{j}. We now use our probability density function to evaluate this average. We have

*P̃*(

**r**) represents the two-dimensional Fourier transform of the probability density function, defined by

*t*

_{j}and

**K**

_{j}for each pulse, and the only random variable remaining is the number of pulses in the interval. We may average over this quantity as well, to get the mutual coherence function as

*p*(

*N*) is the Poisson distribution. We need the well-established results

*T*→ ∞, the functions

*f*(

*ω*)

*T*and

*g*(

*ω*)

*T*reduce to

*δ*(

*ω*) is the Dirac delta function and

*δ*

^{(e)}is the even half-delta function, defined such that

*W*(

**r**

_{1},

**r**

_{2},

*ω*) of the field, we have

*ω*)|

^{2}, an average field profile Θ

^{*}(

**r**

_{1}) and a spectral degree of coherence

*P̃*(

**r**

_{2}-

**r**

_{1}). These three functions can be chosen independently of one another, and we may therefore construct a realization of a partially coherent field which has quite general spatial and temporal coherence.

## 3. Intensity fluctuations of partially coherent field realizations

15. L.C. Andrew and R.L. Phillips, *Laser Beam Scintillation with Applications* (SPIE Press, Bellingham, Washington, 2001). [CrossRef]

*I*(

**r**,

*t*) = 〈|

*V*(

**r**,

*t*)|

^{2}〉 is the intensity of the field which is on average independent of time. The quantity 〈

*I*(

**r**,

*t*)〉

^{2}can be derived from the results of the previous section, so we focus on the quantity 〈

*I*(

**r**,

*t*)

^{2}〉. Using our collection of pulses over a finite interval

*T*again, we have

*V*

_{N}(

**r**,

*t*) is defined by Eq. (2). Looking at the temporal Fourier decomposition of this equation, we find that

*N*

^{4}terms to the summation, but most of them result in either a zero-frequency contribution which will be neglected or a negative-frequency term which is identically zero. The only non-zero terms are those for which

*I*(

**r**,

*t*)〉

^{2}, we may write

*σ*

_{I}~ 1. This result is consistent with a light field said to be

*chaotic*or

*Gaussian*[16, chapter 3]. Our simulation method therefore cannot produce a pure coherent laser field. However, partially coherent fields derived from laser light, for instance by passing coherent light through a rotating ground glass plate, are well-known to be chaotic [17

17. F.T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. **20**, 27–29 (1966). [CrossRef]

*η*(small rate of pulse emission) the value of

*η*must be to reach the ideal limit by considering Gaussian pulses,

*ησ*

_{t}to be sufficiently large.

## 4. Examples

### 4.1. Gaussian Schell-model fields

*I*(

**r**) is the average field intensity,

*σ*

_{I}being the beam width,

*μ*(

**r**

_{2}-

**r**

_{1}) is the spatial correlation function (equivalent to the spectral degree of coherence in the frequency domain),

*σ*

_{g}being the correlation length, and

*γ*(

*τ*) is the temporal coherence function, to be taken as Lorentzian or Gaussian. On comparison with Eq. (27), it can be seen that our field generator should generate a Gaussian Schell-model field if we take and take |

*ω*)|

^{2}to be the temporal Fourier transform of

*γ*(

*τ*). Figure 1 illustrates the intensity of the field generated by our simulation method for several realizations, with

*σ*

_{I}= 2 cm,

*σ*

_{g}= 1 cm, and Gaussian spectrum of center frequency 1 × 10

^{15}Hz and 1% bandwidth. The average pulse rate is taken to be 5 pulses/cycle. The pictures show the gradual evolution of the field in time; the frames are each separated by 5 periods at the center frequency.

*y*= 0. The ideal Gaussian is shown as a dashed line, and it can be seen that there is excellent agreement. Convergence could be further improved by extending the duration of the time average.

*x*, -

*x*along the line

*y*= 0. The circles represent the results generated from our realization, while the dashed line represents the ideal Gaussian Schell-model case. Again it can be seen that there is excellent agreement.

*γ*(

*τ*) calculated at the center of a coherent Gaussian beam, for a Gaussian and Lorentian lineshape. Again there is excellent agreement between the results of the simulation and the expected average behavior.

### 4.2. Propagation through random phase screens

18. D.L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE **71**, 722–737 (1983). [CrossRef]

*V*(

**r**′,

*t*) from the plane

*z*= 0 in the form

*V*

_{0}(

**r**′,

*t*) is the field in the source plane

*z*= 0,

*R*= |

**r**-

**r**′| is the source point-field point distance, and the integration is carried out over the source plane. For quasi-monochromatic fields such as we are considering, we may separate the source field into the components

*ω*

_{0}is the central frequency of oscillation of the source and

*F*

_{0}(

**r**,

*t*) is the slowly-varying piece of the source field. We may then write

*F*

_{0}is assumed to be slowly-varying with respect to time and paraxial, the spatial argument which depends on the tranverse coordinates will be negligible compared to the other terms. We may simplify our field calculation to the form

*t̂*=

*t*-

*z*/

*c*. The field in any plane of constant

*z*is therefore simply the time-shifted Fresnel transform of the field in the source plane, and can be calculated straightforwardly with a fast Fourier transform.

*L*

_{0}is the outer scale of the phase screen. Such a screen is a poor model for quantitative studies of atmospheric turbulence (in such cases more sophisticated spectral models exist [9]), but will suffice for a brief illustration of partially coherent field propagation through random media. The screens are generated by the method described in Ref. [18

18. D.L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE **71**, 722–737 (1983). [CrossRef]

*L*

_{0}= 15cm and the screen is located at

*z*= 1.0km. A spatially coherent field with

*σ*

_{I}= 2cm is propagated through this screen and to a detector at

*z*= 2km; the intensity at the detector is shown in Fig. 5(a). It can be seen that the spot has been aberrated by the phase screen and, perhaps more important, is no longer centered on the center of the detector plane. This is an example of beam wander; it is to be noted that if the beam wanders sufficiently from the axis, it may not illuminate the detector at all, resulting in information loss (a ‘miss’). Temporal fluctuations of the coherent field do not fix the problem; as illustrated in Fig. 5(b), over a 200 cycle time average of the field, the spot shape and position remain essentially unchanged.

*σ*

_{g}= 1cm through the same phase screen. As illustrated in Fig. 6(a), this field is also significantly distorted by the phase screen. However, some field intensity is still present in the center of the detector plane. When a long time average is taken, as shown in Fig. 6(b), we see that there is still a tendency for the field to ‘wander’ from the center of the screen, but an appreciable amount of field intensity remains near the center of the detector plane.

**K**

_{j}) into the phase screen.

## 5. Conclusions

## Acknowledgements

## References and links

01. | È Verdet, |

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

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

04. | J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. |

05. | G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A |

06. | J. Wu and A. D. Boardman, “Coherence length of a Gaussian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. |

07. | J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian Beam: implications for free-space laser communication,” J. Opt. Soc. Am. A |

08. | O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom,” Opt. Eng. |

09. | L.C. Andrews and R.L. Phillips, |

10. | M.S. Soskin and M.V. Vasnetsov, |

11. | G. Gbur, T.D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields”, J. Opt. A |

12. | G. Gbur and T.D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. |

13. | W. Wang and E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E |

14. | S.O. Rice, |

15. | L.C. Andrew and R.L. Phillips, |

16. | R. Loudon, |

17. | F.T. Arecchi, E. Gatti, and A. Sona, “Time distribution of photons from coherent and Gaussian sources,” Phys. Lett. |

18. | D.L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(030.1640) Coherence and statistical optics : Coherence

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 20, 2006

Revised Manuscript: July 21, 2006

Manuscript Accepted: July 25, 2006

Published: August 21, 2006

**Citation**

Gregory J. Gbur, "Simulating fields of arbitrary spatial and temporal coherence," Opt. Express **14**, 7567-7578 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7567

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### References

- E Verdet, Lecons d’Optique Physique, (L’Imprimierie Imperiale, Paris, 1869), vol. 1.
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999, 7th (expanded) edition).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
- J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990). [CrossRef]
- G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002). [CrossRef]
- J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991). [CrossRef]
- J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian Beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002). [CrossRef]
- O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004). [CrossRef]
- L.C. Andrews and R.L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).
- M.S. Soskin and M.V. Vasnetsov, Singular Optics, in Progress in Optics, ed. E.Wolf (Elsevier, Amsterdam, 2001), vol. 42, p. 219-276.
- G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004). [CrossRef]
- G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006). [CrossRef]
- W. Wang and E. Wolf, "Invariance properties of random pulses and of other random fields in dispersive media," Phys. Rev. E 52, 5532-5539 (1995). [CrossRef]
- S.O. Rice, Mathematical analysis of random noise, in Selected Papers on Noise and Stochastic Processes, ed. N. Wax (Dover, NY, 1954).
- L.C. Andrew and R.L. Phillips, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Washington, 2001). [CrossRef]
- R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 1983, 2nd edition).
- F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966). [CrossRef]
- D.L. Knepp, "Multiple phase-screen calculation of the temporal behavior of stochastic waves," Proc. IEEE 71, 722-737 (1983). [CrossRef]

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