## Spatially partially coherent beam parameter optimization for free space optical communications |

Optics Express, Vol. 18, Issue 20, pp. 20746-20758 (2010)

http://dx.doi.org/10.1364/OE.18.020746

Acrobat PDF (819 KB)

### Abstract

The problem of coherence length optimization in a spatially partially coherent beam for free space optical communication is investigated. The weak turbulence regime is considered. An expression for the scintillation index in a series form is derived and conditions for obtaining improvement in outage probability through optimization in the coherence length of the beam are described. A numerical test for confirming performance improvement due to coherence length optimization is proposed. The effects of different parameters, including the phase front radius of curvature, transmission distance, wavelength and beamwidth are studied. The results show that, for smaller distances and larger beamwidths, improvements in outage probability of several orders of magnitude can be achieved by using partially coherent beams.

© 2010 Optical Society of America

## 1. Introduction

1. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. **43**(2), 330–341 (2004). [CrossRef]

2. 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**(9), 1794–1802 (2002). [CrossRef]

4. T. J. Schulz, “Optimal beam for propagation through random media,” Opt. Lett. **30**(10), 1093–1095 (2005). [CrossRef] [PubMed]

5. D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. **48**(3), (2009). [CrossRef]

*l*) as a function of beam size, wavelength, turbulence strength, and propagation distance under the heuristic metric criterion. For a more detailed discussion on PCBs, please see the references listed in [1

_{c}1. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. **43**(2), 330–341 (2004). [CrossRef]

5. D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. **48**(3), (2009). [CrossRef]

6. H.T. Eyyuboglu, Y. Baykal, E. Sermtlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A **26**(2), 387–394 (2009). [CrossRef]

7. H. T. Eyyuboglu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Applied Physics B - Laser and Optics **98**(4), 857–863 (2010). [CrossRef]

8. Y. Baykal, H. T. Eyyuboglu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Applied Optics **48**(10), 1943–1954 (2009). [CrossRef] [PubMed]

9. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Optics Express **18**(12), 12587–12598 (2010). [CrossRef] [PubMed]

4. T. J. Schulz, “Optimal beam for propagation through random media,” Opt. Lett. **30**(10), 1093–1095 (2005). [CrossRef] [PubMed]

5. D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. **48**(3), (2009). [CrossRef]

## 2. Beam model

10. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A **9**(5), 796–803 (1992). [CrossRef]

11. S. R. Seshadri, “Partially coherent Gaussian schell-model electromagnetic beams,” J. Opt. Soc. Am. A **16**(6), 1373–1380 (1999). [CrossRef]

*z*, the mean intensity profile,

*Ī*(

*ρ*), on the receiver plane is modeled as [12

12. L. C. Andrews and R. L. Phillips, *Laser beam propagation through random media*, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

*ρ*is the radial distance from the mean beam center on the transverse plane,

*W*is the reference beam radius or beamwidth,

_{r}*W*

_{0}is the effective beam radius at the transmitter,

*W*is the receiving beam size given by

*r*

_{0}= 1 −

*z*/

*F*

_{0}is a focusing parameter,

*F*

_{0}is the phase-front radius of curvature at the transmitter,

*k*= 2

*π*/

*λ*is the wave number,

*λ*is the wavelength,

*l*is the spatial coherence length,

_{c}^{−2/3}. Noting that the Rytov variance is

12. L. C. Andrews and R. L. Phillips, *Laser beam propagation through random media*, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

*ζ*

_{1}= 0.54,

*ζ*

_{2}= 8/9, and

*ζ*

_{3}= 0.5 for a focused beam, while

*ζ*

_{1}= 0.48,

*ζ*

_{2}= 1 and

*ζ*

_{3}= 1 for a collimated beam, and

*ρ*=

*σ*, and

_{pe}*ρ*.

## 3. Performance metric

*P*, which is the probability that the received signal-to-noise ratio (SNR) falls below a threshold SNR resulting in unacceptable error rates. For a given noise level, this occurs when the intensity

_{out}*I*becomes less than a threshold intensity

*I*. We will use this simplified approach in this work. Thus, we write

_{th}*p*(

*I*) is the probability density function of

*I*. Using the log-normal (LN) model for

*I*[12

12. L. C. Andrews and R. L. Phillips, *Laser beam propagation through random media*, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

*P*,

_{out}*y*is a large negative number. Under these conditions, we need to minimize

_{th}*I*<

_{th}*Ī*, the value of the above cost function is a negative number, and hence zero is not the minimum value of the above cost function in general.

## 4. PCB Parameter selection

*l*= ∞ will minimize (4), and PCB is not even necessary. However, this can be ascertained only after performing a non-linear optimization on (4) with respect to

_{c}*l*, or evaluating the cost function (4) or outage probability (3) for various values of

_{c}*l*. This becomes more difficult when one studies the effects of combinations of other parameters, such as

_{c}*F*

_{0}and

*W*

_{0}, over

*l*. We show in this section that this can be done without conducting extensive optimization. Toward that end, we first provide an alternative series expansion for the SI and develop interpretations.

_{c}### 4.1. Expressions for the Scintillation Index

*l*through trigonometric expressions. As

_{c}*l*increases or decreases, it is difficult to interpret from (2) whether

_{c}*x*| < 1, while

*x*| > 1, with the positive or negative sign selection before the sine function corresponding to the sign of

*x*. Note that, in (5), the non-wander part is

*x*| is close to unity, the series converges slowly, and more terms in the series expansion are required for accurate representation.

### 4.2. Coherence length optimization

*l*, we differentiate the cost function

_{c}*φ*(

*l*) with respect to

_{c}*l*and set it to zero to obtain

_{c}*l*is optimal, the derivative of the SI equals a target value,

_{c}*x*| < 1, and

*x*| > 1. In (9), we have to use

#### Proposition 1

*I*so that

_{th}*I*<

_{th}*Ī*, if the derivative

*T*(

*l*) at a high

_{c}*l*value, then an optimum finite

_{c}*l*exists.

_{c}#### Proof

*T*(

*l*) increases rapidly as

_{c}*l*→ 0, while

_{c}*T*(

*l*) → 0 as

_{c}*l*→ ∞. The SI derivative

_{c}*l*→ 0. This can be seen by observing that (12) and (13) both approach zero,

_{c}*x*→ 0.5

*z*

_{0}, and

*z*→ 1/

_{e}*z*

_{0}as

*l*→ 0. If the SI derivative becomes larger than the target value for a large

_{c}*l*, then it implies that the SI derivative has crossed the target curve at least at one

_{c}*l*value, where the condition (8) is satisfied. This gives an optimum

_{c}*l*value.

_{c}*l*. All we need to do is to calculate the SI derivative and the target value at a large

_{c}*l*value, say at

_{c}*l*= 100

_{c}*W*

_{0}. If the scintillation derivative is larger than the target value, then finite optimized

*l*exists.

_{c}*Ī*decreases with a decrease in

*l*, it may not be possible to guarantee

_{c}*I*<

_{th}*Ī*for smaller values of

*l*when the outage probability is larger. In that case,

_{c}*T*(

*l*) becomes negative when

_{c}*I*>

_{th}*Ī.*This occurs when

*l*becomes smaller than a certain minimum value

_{c}*l*

_{c}_{0}, which can be obtained from (1) in the form

*T*(

*l*) at an

_{c}*l*value close to but greater than

_{c}*l*

_{c}_{0}, making sure that

*T*(

*l*) is higher than the SI derivative at that point. Then Proposition 1 is used to verify if the SI derivative is larger than or equal to

_{c}*T*(

*l*) at a large

_{c}*l*value, in which case an optimum finite

_{c}*l*is guaranteed.

_{c}*W*

_{0}= 0.05 m,

*I*= 0.01,

_{th}*F*

_{0}= ∞, and

*λ*= 1.55

*μ*m at distances of 1000, 1500 and 2000 meters. The Rytov variance for these distances corresponds to 0.1991, 0.4187 and 0.7095 respectively. The reference beamwidth

*W*is 0.025 m, which will be used in all our numerical examples. Observe that the optimal

_{r}*l*increases with

_{c}*z*, and the benefit from optimizing

*l*decreases with increasing

_{c}*z*. We show the corresponding SI derivative and target values given by (8) in Fig. 2. Note that the meeting points of the SI derivative and target values correspond to the optimal

*l*values. As

_{c}*z*increases, both the SI derivative and target values increase but target values increase more than the SI derivative for any given increase in

*z*. Observe that for low outage probability or sufficiently small

*I*, the RHS of (8) is a positive quantity. Therefore, for optimal

_{th}*l*to exist, the derivative

_{c}*dx*/

*dl*helps to create a situation where an optimal finite

_{c}*l*value exists. This implies a small

_{c}*z*, a small

*λ*, and a large

*W*

_{0}.

### 4.3. Effects of phase front radius of curvature

*F*

_{0}, and setting it to zero, we obtain the following condition to find optimal

*F*

_{0}

*F*

_{0}less than but close to

*z*, the SI increases with an increase in

*F*

_{0}as the beam wander effects begin to dominate significantly when

*F*

_{0}approaches

*z*. So, the left-hand-side (LHS) of (14) is a positive quantity in this region. The RHS is also a positive quantity in this region for low threshold values (

*I*<

_{th}*Ī*), and the optimality condition is satisfied in this region.

*F*

_{0}for

*λ*= 1

*μ*m,

*W*

_{0}= 0.05 m,

*l*= ∞, and

_{c}*I*= 0.1. A larger

_{th}*I*than the previous figures is selected so that

_{th}*P*does not become too low. The optimal

_{out}*F*

_{0}values for the

*z*values of 1000 m, 1250 m, and 1500 m are 860 m, 1030 m and 1180 m respectively. These values are less than

*z*. A natural question with respect to this set of results would be: will there be further performance improvement by optimizing over

*l*? To answer this question, we can run optimization over

_{c}*l*for each possible value of

_{c}*F*

_{0}, requiring significant computations. Fortunately, we can simply use Proposition 1, and produce the SI derivative and the target values given by (8) for different values of

*F*

_{0}. The results so obtained are shown in Fig. 4. We observe that the SI derivative values are larger than the target values for all values of

*F*

_{0}. This implies that further improvement in

*P*can be achieved using

_{out}*l*optimization for each of these

_{c}*F*

_{0}values.

### 4.4. Effects of beamwidth

*W*

_{0}and setting it to zero. This produces

*z*,

*λ*) pairs under

*F*

_{0}= ∞,

*l*= ∞, and

_{c}*I*= 0.025. These cases correspond to (

_{th}*z*,

*λ*) values of (1500 m, 1

*μ*m), (1500 m, 1.55

*μ*m), (2000 m, 1

*μ*m), and (2000 m, 1.55

*μ*m). The optimized

*W*

_{0}values for the cases are found to be 0.016 m, 0.02 m, 0.018 m, and 0.024 respectively. In Fig. 6, we plot the SI derivative with respect to

*W*

_{0}, and the target values given by the RHS of (15) for

*l*= 0.02 m,

_{c}*z*= 1500 m,

*λ*= 1

*μm*,

*F*

_{0}= ∞, and

*I*= 0.025. The optimal value for

_{th}*W*

_{0}, obtained from calculating outage probability (3), is found to be 0.019 m. This agrees with Fig. 6, where the SI derivative and the target curve are found to meet around the optimal value. To gain insight on the effects of

*l*for a given value of

_{c}*W*

_{0}, consider two cases:

*F*

_{0}= ∞), and ignore beam wander effects. The SI can then be expressed as

*x*. Now, from (18), we can write

*W*

_{0}and

*l*affect SI via

_{c}*x*in nearly similar ways. As

*l*increases,

_{c}*x*also increases. From (6) and (19), we observe that when |

*x*| < 1, the SI will also increase if |

*x*| is close to unity, otherwise the SI will decrease. If |

*x*| > 1, then we can see from (7) and (19) that the SI will increase with increasing |

*x*|.

*x*is not affected much by

*l*. Therefore, SI is relatively unaffected by

_{c}*l*. However, the mean intensity increases with

_{c}*l*. Therefore, not much benefit can be obtained by optimizing

_{c}*l*for

_{c}*W*

_{0}selected in this region, and coherent beam tends to perform better.

*l*for different values of

_{c}*W*

_{0}, we consider the SI derivative with respect to

*l*and the target values given by (8). Figure 7 shows the derivative and the target values for different values of

_{c}*W*

_{0}under

*λ*= 1

*μm*,

*F*

_{0}= ∞ evaluated at a large value of

*l*= 0.5 m. For the

_{c}*z*= 1500 m case, we see from Fig. 5 that the optimal

*W*

_{0}is 0.016. For the same parameters, the target values just meet the derivative curve in Fig. 7 at about

*W*

_{0}= 0.016. Since the target curve does not exceed the SI derivative curve for this

*W*

_{0}, performance improvement by optimizing

*l*is not guaranteed. In fact, for all the optimal

_{c}*W*

_{0}values observed in Fig. 5, we could not obtain further improvement in performance by optimizing

*l*, i.e.,

_{c}*l*= ∞ gives best performance. Observe from Fig. 7 that for very small values of

_{c}*W*

_{0}, benefits from optimizing

*l*is not guaranteed. For larger values of

_{c}*W*

_{0}, there is guaranteed benefit from optimizing

*l*. This agrees with our theory observations that for small

_{c}*W*

_{0}, the SI becomes nearly independent of

*l*with no potential benefits from small

_{c}*l*values. Also, for longer distance, the minimum

_{c}*W*

_{0}required for benefit from

*l*optimization increases.

_{c}## 5. Conclusion

## Appendix

*y*)

*= 1 +*

^{q}*qy*+ (1/2!)

*q*(

*q*− 1)

*y*+ ⋯ in (24), and recalling (2), we get the following expression after a few simplification steps,

^{2}*x*| > 1, we substitute

*y*= 1/

*x*and proceed as follows. Observe that

*x*. Next, using

## Acknowledgments

## References and links

1. | O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. |

2. | 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 |

3. | V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk |

4. | T. J. Schulz, “Optimal beam for propagation through random media,” Opt. Lett. |

5. | D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. |

6. | H.T. Eyyuboglu, Y. Baykal, E. Sermtlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A |

7. | H. T. Eyyuboglu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Applied Physics B - Laser and Optics |

8. | Y. Baykal, H. T. Eyyuboglu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Applied Optics |

9. | S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Optics Express |

10. | E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A |

11. | S. R. Seshadri, “Partially coherent Gaussian schell-model electromagnetic beams,” J. Opt. Soc. Am. A |

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

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.7060) Coherence and statistical optics : Turbulence

(060.2605) Fiber optics and optical communications : Free-space optical communication

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: July 28, 2010

Revised Manuscript: August 24, 2010

Manuscript Accepted: September 3, 2010

Published: September 15, 2010

**Citation**

Deva K. Borah and David G. Voelz, "Spatially partially coherent beam parameter optimization for free space optical communications," Opt. Express **18**, 20746-20758 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20746

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### References

- O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. 43(2), 330-341 (2004). [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(9), 1794-1802 (2002). [CrossRef]
- V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk 54, 1054-1059 (1983).
- T. J. Schulz, "Optimal beam for propagation through random media," Opt. Lett. 30(10), 1093-1095 (2005). [CrossRef] [PubMed]
- D. G. Voelz, and X. Xiao, "Metric for optimizing spatially partially coherent beams for propagation through turbulence," Opt. Eng. 48(3), (2009). [CrossRef]
- H. T. Eyyuboglu, Y. Baykal, E. Sermtlu, O. Korotkova, and Y. Cai, "Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media," J. Opt. Soc. Am. A 26(2), 387-394 (2009). [CrossRef]
- H. T. Eyyuboglu, Y. Baykal and X. Ji, "Scintillations of Laguerre Gaussian beams," Applied Physics B - Laser and Optics 98(4), 857-863 (2010). [CrossRef]
- Y. Baykal, H. T. Eyyuboglu, and Y. Cai, "Scintillations of partially coherent multiple Gaussian beams in turbulence," Appl. Opt. 48(10), 1943-1954 (2009). [CrossRef] [PubMed]
- S. Zhu, Y. Cai, and O. Korotkova, "Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam," Opt. Express 18(12), 12587-12598 (2010). [CrossRef] [PubMed]
- E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9(5), 796-803 (1992). [CrossRef]
- S. R. Seshadri, "Partially coherent Gaussian Schell-model electromagnetic beams," J. Opt. Soc. Am. A 16(6), 1373-1380 (1999). [CrossRef]
- L. C. Andrews, and R. L. Phillips, Laser beam propagation through random media, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

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