## Flat topped beams and their characteristics in turbulent media

Optics Express, Vol. 14, Issue 10, pp. 4196-4207 (2006)

http://dx.doi.org/10.1364/OE.14.004196

Acrobat PDF (4323 KB)

### Abstract

The source and receiver plane characteristics of flat topped (FT) beam propagating in turbulent atmosphere are investigated. To this end, source size, beam power and *M*^{2} factor of source plane FT beam are derived. For a turbulent propagation medium, via Huygens Fresnel diffraction integral, the receiver plane intensity is found. Power captured within an area on the receiver plane is calculated. Kurtosis parameter and beam size variation along the propagation axis are formulated. Graphical outputs are provided displaying the variations of the derived source and receiver plane parameters against the order of flatness and propagation length. Analogous to free space behavior, when propagating in turbulence, the FT beam first will form a circular ring in the center. As the propagation length increases, the circumference of this ring will become narrower, giving rise to a downward peak emerging from the center of the beam, eventually turning the intensity profile into a pure Gaussian shape.

© 2006 Optical Society of America

## 1. Introduction

1. F. M. Dickey and S. C. Holswade, *Laser beam shaping: theory and techniques* (Marcel Dekker, New York, 2000). [CrossRef]

3. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. **27**, 1007–1009 (2002). [CrossRef]

3. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. **27**, 1007–1009 (2002). [CrossRef]

4. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. **206**, 225–234 (2002). [CrossRef]

5. F. Gori, “Flattened Gaussian beams,” Opt. Commun. **107**, 335–341 (1994). [CrossRef]

6. Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped-Gaussian beams,” Appl. Opt.45 (2006). [CrossRef] [PubMed]

7. X. Ji and B. Lü, “Propagation of a flattened Gaussian beam through multi-apertured optical *ABCD* systems,” Optik **114**, 394–400 (2003). [CrossRef]

14. D. Ge, Y. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. **43**, 4732–4738 (2004). [CrossRef] [PubMed]

15. J. Zhang and Y. LiD. Lu and G. G. Matvienko, “Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam,” in *Optical Technologies for Atmospheric, and Environmental Studies*, eds., Proc. SPIE5832, 48–55 (2005). [CrossRef]

## 2. Formulation

### 2.1. Formulation of the source field and its related parameters

3. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. **27**, 1007–1009 (2002). [CrossRef]

*u*

_{s}(

*s*) becomes

*s*

_{x}

*,s*

_{y}) designates the decomposition of the vector

**s**into

*x*and

*y*components of the transverse source plane,

*N*is the order parameter for flatness, such that at

*N*=1,

*u*

_{s}(

*s*) will yield the fundamental mode (Gaussian) source field, correspondingly

*α*

_{sx}and

*α*

_{sy}becoming the Gaussian source sizes along

*s*

_{x}and

*s*

_{y}directions. When

*N*≠1, on the other hand, the beam sizes will depart from Gaussian source sizes, and according to the definition developed by Carter [16

16. W.H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,”Appl. Opt. **19**, 1027–1029 (1980). [CrossRef] [PubMed]

*s*

_{x}direction will read

*α*

_{syN}is attained by simply interchanging

*α*

_{sx}with

*α*

_{sy}. From Eq. (5), it is clear that at

*N*=1, αsxN will appropriately reduce to

*α*

_{sx}.

*N*, denoted as

*P*

_{sN}, can be calculated by integrating the intensity over the entire transverse plane, which means that it is equivalent to the denominator in Eqs. (2) or (3). This way

*P*

_{sN}will become

*M*

^{2}is reported to be a measure of beam quality [13

13. B. Lü and H. Ma, “Coherent and incoherent off-axial Hermite-Gaussian beam combinations,” Appl. Opt. **39**, 1279–1289 (2000). [CrossRef]

*s*

_{x}and

*s*

_{y}components, is given by

*x*direction, by the following equations

*f*

_{x}and

*f*

_{y}are the variables of frequency spatial domain.

*I*

_{s}(

*s*

_{x}) and

*I*

_{f}(

*f*

_{x}) are attained by respectively decoupling the

*x*and

*y*components in the manner

*I*

_{s}(

*s*

_{x}

*,s*

_{y})=

*I*

_{s}(

*s*

_{x})

*I*

_{s}(

*s*

_{y}),

*I*

_{f}(

*f*

_{x}

*,f*

_{y})=

*I*

_{f}(

*f*

_{x})

*I*

_{s}(

*f*

_{y}) where the function,

*I*

_{f}(

*f*

_{x}

*,f*

_{y}), is retrieved from the two dimensional spatial frequency Fourier transform of

*I*

_{s}(

*s*

_{x}

*,s*

_{y}). Identical expressions will hold for the

*y*component. By applying the same steps as before, that is resorting to the use of binomial expansion, the

*M*

^{2}is confined to the source plane, therefore for an FT beam considered in this paper, Eq. (11) will be valid regardless of propagation medium.

*N*to be non-integer as pointed out by Li [4

4. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. **206**, 225–234 (2002). [CrossRef]

*N*being (positive) integer, since, the summations would otherwise turn into infinite series, thus preventing us gaining a proper insight into the subject.

### 2.2. Formulation of the average receiver intensity and its related parameters

*L*away from the source plane, will be supplied via the Huygens Fresnel integral in the following way

**P**)=(

*p*

_{x}

*,p*

_{y}) are the x and y components of the transverse receiver plane,

*k*=2

*π/λ*with

*λ*being the wavelength of propagation. The ensemble average term appearing on the second line of Eq. (12) is [17

17. H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A **22**, 2709–2718 (2005). [CrossRef]

*D*

_{ψ}(s

_{1}-s

_{2}) represents the wave structure function, and

*ρ*

_{0}=(0.545

*k*

^{2}

*L*)

^{-3/5}is the coherence length of a spherical wave propagating in the turbulent medium,

*y*counterparts of

*x*is replaced by

*y*.

*I*

_{s}(

*s*) is replaced with

*I*

_{r}(

**p**) from Eq. (14). Thus for the

*p*

_{x}direction, the beam size during propagation will be established as

*I*

_{r}(

**p**), we cannot proceed analytically; therefore numerical evaluations have to be sought for finding the beam size during propagation.

18. H. Mao, D. Zhao, F. Jing, and H. Liu, “Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens,” J. Opt. A: Pure Appl. Opt. **6**, 640–650 (2004). [CrossRef]

*K*

_{y}is found by switching the receiver plane coordinate parameter to

*p*

_{y}. Again, the complexity of the

*I*

_{r}(

**p**) expression does not allow us to acquire an analytic result for the kurtosis parameter.

*α*

_{r}, which is also known as power in bucket, that is

*P*

_{sN}, i.e. the term in the denominator of Eq. (17), it is envisaged that each individual FT beam of order

*N*, is normalized with respect to its own order of flatness.

## 3. Results and discussions

### 3.1. Source plane

*α*

_{sx}=

*α*

_{sy}=3 cm, against the flatness order N. As seen in this figure, the source intensity of the FT beam will become flatter with escalating N values, eventually assuming the shape of a cylinder if equal Gaussian source sizes are used. Figure 1 also makes it evident that higher

*N*values will give rise to larger beam sizes and source powers.

### 3.2. Receiver plane

*λ*=1.55 µm,

^{-15}m

^{-2/3}. Therefore, these parameters are not quoted in our figures. By taking Eq. (14) and setting the source sizes and the flatness order as stated in the inset of the figure, Fig. 6 provides the views of the FT beam at selected propagation distances. Figure 6 shows that, the FT beam will initially develop a circular ring in the center. As the propagation advances, the circumference of this ring will become narrower, while a downward peak will gradually emerge from the center of the beam. Eventually this peak will smooth out the initial ring, with the profile turning into a pure Gaussian shape. This phenomenon may be considered to be akin to acquiring the spot of Arago as described in [19

19. J. E. Harvey and J. L. Forgham, “The spot of Arago: New relevance for an old phenomenon,” Am. J. Phys. **52**, 243–247 (1984). [CrossRef]

4. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. **206**, 225–234 (2002). [CrossRef]

7. X. Ji and B. Lü, “Propagation of a flattened Gaussian beam through multi-apertured optical *ABCD* systems,” Optik **114**, 394–400 (2003). [CrossRef]

8. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. **6**, 390–395 (2004). [CrossRef]

9. Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A: Pure Appl. Opt. **6**, 1061–1066 (2004). [CrossRef]

11. N. Zhou, G. Zeng, and L. Hu, “Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial *ABCD* optical systems,” Opt. Commun. **240**, 299–306 (2004). [CrossRef]

*F*=

*λL*) with

*α*

_{s}=

*α*

_{sx}=

*α*

_{sy}being the source size for a symmetrical source, in the plots of Figs. 6 and 7, the propagation distances of

*L*=0.7 km, 1.4 km and 5 km correspond to

*F*of 0.83 (i.e., close to Fresnel diffraction region meaning near field), 0.42 and 0.12 (i.e., in the Fraunhoffer diffraction region meaning far field), respectively.

8. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. **6**, 390–395 (2004). [CrossRef]

*α*

_{sx},

*α*

_{sy}and

*α*

_{sxy}as shown in Ref. [9

9. Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A: Pure Appl. Opt. **6**, 1061–1066 (2004). [CrossRef]

20. Y. Cai and S. He “Average intensity and spreading of an elliptical Gaussian beam in a turbulent atmosphere,” Opt. Lett. **31**, 568–570 (2006). [CrossRef] [PubMed]

21. Y. Cai and S. He “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express **14**, 1353–1367 (2006). [CrossRef] [PubMed]

*α*

_{sx}≠

*α*

_{sy}.

*N*, while this trend later continues at much slower pace. Turning to power in bucket variation and choosing a receiving aperture size of

*α*

_{r}=0.5(

*N*values is slightly retarded. Of course we should be careful here in the interpretation of receiving less power with increasing

*N*. Less power occurs because, as explained underneath Eq. (17),

*P*

_{αN}is determined by dividing the received power of each individual FT beam by its own source power. If this normalization were to be carried out with respect the Gaussian case, i.e.,

*N*=1, then we would come across rising power in bucket values with increasing

*N*, instead of the presently falling trend.

18. H. Mao, D. Zhao, F. Jing, and H. Liu, “Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens,” J. Opt. A: Pure Appl. Opt. **6**, 640–650 (2004). [CrossRef]

*N*values and fixed Gaussian source sizes. It is to be noted from Figs. (1) and (2) that on the source plane, the beam profile becomes sharper with larger

*N*. This fact is reflected in Fig. 11 such that at short propagation distances, kurtosis parameter becomes smaller as

*N*gets bigger. On the other hand, kurtosis parameter of all beams, except for

*N*=1, begins to rise at medium propagation ranges, later reaching a maximum. This is a predictable behavior, where the rise corresponds to the formation of the ring, the subsequent narrowing action and then the appearance of the peak in Figs 6 and 7. When these curly surfaces on the beam start to erode with the beam evolving towards a Gaussian profile, the kurtosis parameter declines, finally approximating to the

*N*=1 (Gaussian) case for all

*N*values, as seen to the right of Fig. 11.

## 4. Conclusion

*M*

^{2}factor are formulated and plotted against flatness order. From these plots, it is seen that beam intensity profile will become flatter, while source size, beam power and

*M*

^{2}factor will increase against growing value of flatness order. Via the use of Huygens Fresnel integral, the average intensity on the receiver plane for a turbulent propagation environment is derived. Based on this intensity expression, the kurtosis parameter, beam size variation along the propagation axis and power in bucket formulations are stated. Plotting the receiver average intensity at selected source and propagation parameters, it is observed that, initially a circular ring will be formed in the center. As we go along the propagation axis however, the circumference of this ring will become narrower, while a downward peak will gradually emerge from the center of the beam. Eventually this peak will smooth out the initial ring, with the profile converging towards a pure Gaussian shape. These beam profile changes are also reflected in the plots of kurtosis parameter against propagation length. Furthermore, when the source beam is made highly asymmetric, the conversion of receiver intensity profile into an Airy function becomes more visible. The relative beam size becomes smaller, as the order parameter is raised, indicating that flatter beams will be subjected to less spreading during propagation in turbulence. For a fixed receiver aperture radius, less power is captured against the rising flatness order.

## References and links

1. | F. M. Dickey and S. C. Holswade, |

2. | D. L. Shealy and J. A. HoffnagleF. M. Dickey and D. L. Shealy, “Beam shaping profiles and propagation,” in |

3. | Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. |

4. | Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. |

5. | F. Gori, “Flattened Gaussian beams,” Opt. Commun. |

6. | Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped-Gaussian beams,” Appl. Opt.45 (2006). [CrossRef] [PubMed] |

7. | X. Ji and B. Lü, “Propagation of a flattened Gaussian beam through multi-apertured optical |

8. | Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. |

9. | Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A: Pure Appl. Opt. |

10. | V. Bagini, R. Borghi, F. Gori, M. Santarsiero, D. Ambrosini, and G. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A |

11. | N. Zhou, G. Zeng, and L. Hu, “Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial |

12. | B. Lü and S. Lou, “General propagation equation of flattened Gaussian beams,” J. Opt. Soc. Am. A. |

13. | B. Lü and H. Ma, “Coherent and incoherent off-axial Hermite-Gaussian beam combinations,” Appl. Opt. |

14. | D. Ge, Y. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. |

15. | J. Zhang and Y. LiD. Lu and G. G. Matvienko, “Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam,” in |

16. | W.H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,”Appl. Opt. |

17. | H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A |

18. | H. Mao, D. Zhao, F. Jing, and H. Liu, “Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens,” J. Opt. A: Pure Appl. Opt. |

19. | J. E. Harvey and J. L. Forgham, “The spot of Arago: New relevance for an old phenomenon,” Am. J. Phys. |

20. | Y. Cai and S. He “Average intensity and spreading of an elliptical Gaussian beam in a turbulent atmosphere,” Opt. Lett. |

21. | Y. Cai and S. He “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: March 13, 2006

Revised Manuscript: April 7, 2006

Manuscript Accepted: May 1, 2006

Published: May 15, 2006

**Citation**

Halil T. Eyyuboglu, Çaglar Arpali, and Yahya Kemal Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express **14**, 4196-4207 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4196

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### References

- F. M. Dickey and S. C. Holswade, Laser beam shaping: theory and techniques (Marcel Dekker, New York, 2000). [CrossRef]
- D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F. M. Dickey and D. L. Shealy, eds., Proc. SPIE 5876, 1-11 (2005).
- Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002). [CrossRef]
- Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002). [CrossRef]
- F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994). [CrossRef]
- Y. Baykal and H. T. Eyyuboğlu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. 45 (2006). [CrossRef] [PubMed]
- X. Ji and B. Lü, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik 114, 394-400 (2003). [CrossRef]
- Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004). [CrossRef]
- Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A: Pure Appl. Opt. 6, 1061-1066 (2004). [CrossRef]
- V. Bagini, R. Borghi, F. Gori, M. Santarsiero, D. Ambrosini, and G. S. Spagnolo, "Propagation of axially symmetric flattened Gaussian beams," J. Opt. Soc. Am. A 13, 1385-1394 (1996). [CrossRef]
- N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004). [CrossRef]
- B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A. 17, 2001-2004 (2000). [CrossRef]
- B. Lü and H. Ma, "Coherent and incoherent off-axial Hermite-Gaussian beam combinations," Appl. Opt. 39, 1279-1289 (2000). [CrossRef]
- D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43,4732-4738 (2004). [CrossRef] [PubMed]
- J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005). [CrossRef]
- W.H. Carter, "Spot size and divergence for Hermite Gaussian beams of any order,"Appl. Opt. 19,1027-1029 (1980). [CrossRef] [PubMed]
- H. T. Eyyuboğlu and Y. Baykal, "Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere," J. Opt. Soc. Am. A 22,2709-2718 (2005). [CrossRef]
- H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (2004). [CrossRef]
- J. E. Harvey and J. L. Forgham, "The spot of Arago: New relevance for an old phenomenon," Am. J. Phys. 52, 243-247 (1984). [CrossRef]
- Y. Cai and S. He "Average intensity and spreading of an elliptical Gaussian beam in a turbulent atmosphere," Opt. Lett. 31, 568-570 (2006). [CrossRef] [PubMed]
- Y. Cai and S. He "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006). [CrossRef] [PubMed]

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