## Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures

Optics Express, Vol. 17, Issue 15, pp. 12753-12766 (2009)

http://dx.doi.org/10.1364/OE.17.012753

Acrobat PDF (1247 KB)

### Abstract

A general model for different apertures and flat-topped laser beams based on the multi-Gaussian function is developed. The general analytical expression for the propagation of a flat-topped beam through a general double-lens system with apertures is derived using the above model. Then, the propagation characteristics of the flat-topped beam through a spatial filter are investigated by using a simplified analytical expression. Based on the Fluence beam contrast and the Fill factor, the influences of a pinhole size on the propagation of the flat-topped multi-Gaussian beam (FMGB) through the spatial filter are illustrated. An analytical expression for the propagation of the FMGB through the spatial filter with a misaligned pinhole is presented, and the influences of the pinhole offset are evaluated.

© 2009 OSA

## 1. Introduction

1. H. T. Eyyuboglu, Ç. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express **14**(10), 4196–4207 (2006). [CrossRef] [PubMed]

13. Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. **6**(12), 1061–1066 (2004). [CrossRef]

14. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. **13**(3), 201–203 (1988). [CrossRef] [PubMed]

15. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**(5), 1752–1756 (1988). [CrossRef]

16. F. Gori, “Flattened Gaussian beams,” Opt. Commun. **107**(5-6), 335–341 (1994). [CrossRef]

17. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A **18**(8), 1897–1904 (2001). [CrossRef]

6. J. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A **24**(1), 84–92 (2007). [CrossRef]

8. P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. **14**(6), 1130–1135 (2005). [CrossRef]

17. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A **18**(8), 1897–1904 (2001). [CrossRef]

## 2. Flat-topped multi-Gaussian beam modules and aperture modules

17. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A **18**(8), 1897–1904 (2001). [CrossRef]

*N*is the order of the multi-Gaussian function,

*ω*is the width of the individual Gaussian component, and

*n*represents the offset of the corresponding Gaussian component. For the two-dimensional formula, the expression becomes [18

18. H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. **40**, 156–166 (2008). [CrossRef]

*N*is higher, and the difference between any two successive orders is much smaller. The multi-Gaussian function can also be used to simulate other beam shapes, such as the circle, the ellipse, and the rectangle, as shown in Fig. 2 . The forms of the Eqs. (2) and (3) are invariable for different FMGB shapes, and what we just need do is to establish the relationship between N and M. For instance, we can get the elliptical shape as shown in Fig. 2(b) by inserting

*ω*,

*N*, and

*M*, we can get different kinds of aperture functions. In next section, a general analytical solution for the propagation of the FMGB through a double-lens system with apertures is presented.

## 3. Propagation of the flat-topped multi-Gaussian beam through a double-lens system with apertures

_{1}, the position S

_{in}, the front surface of L

_{2}, and the output plane, respectively.

**18**(8), 1897–1904 (2001). [CrossRef]

*x*direction and the

*y*direction respectively, and

_{1}, the aperture of L

_{2}, and the aperture S

_{in}. By using the Huygens-Fresnel diffraction integral and taking the transformation of the thin lens as a phase modulation function

*j = 1, 2, 3*, represent the aperture functions of L

_{1}, S

_{in}, and L

_{2}, respectively. In Eq. (5) and other equations in this paper, the integral limits are all from minus infinity to plus infinity. Applying the integral formula

*N*and

_{j}*M*for different kinds of apertures and different shapes of laser beams. In actual systems, only one or two of these apertures are the main influencing factors generally. Hence, Eq. (8) can be simplified greatly by keeping

_{j}*j*denotes the ignored apertures.

## 4. Results and discussions

_{1}and L

_{2}have the same focal point and the same optic axis, and S

_{in}locates in their common focal plane, the double-lens system shown in Fig. 3 becomes a spatial filter. The spatial filter is the essential component of a high power laser system. It is widely used [19

19. P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. Da Silva, B. J. Macgowan, B. M. Van Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. **37**(12), 2371–2378 (1998). [CrossRef]

22. J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the national ignition facility,” Appl. Opt. **39**(9), 1405–1420 (2000). [CrossRef]

### 4.1 Simplification

### 4.2 Propagation characteristics with different sizes of pinhole

23. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. **46**(16), 3276–3303 (2007). [CrossRef] [PubMed]

*m*and

*n*are the sampling numbers. The Fill factor is defined as the ratio of the average intensity to the maximum intensity. Figure 7(a) illustrates the variations of the FBCs and Fill factors against the sizes of pinholes, which are expressed by the times of the diffraction limit (DL). The FBC decreases with increasing pinhole size gradually, and it exhibits obvious periodicity, which can be explained by the variations of the Fresnel rings with the sizes of the pinholes. Figures 8(a) and 8(b) present the Fresnel rings, corresponding to one of the adjacent pairs of minimum and maximum FBCs, when the pinhole sizes are near the 4 × DL. The Fill factors show the opposite trend. Figure 7(b) shows the variations of the FBCs and Fill factors of different order FMGBs against the distance

*z*when the pinhole size is 10 × DL. The FBCs tend to be bigger with periodic oscillation when the distance

_{4}*z*increases. The Fill factors show the opposite trend. From Fig. 7 and Fig. 8, we can find that the maximum Fill factors and minimum FBCs appear at the same size of pinhole or at the same distance

_{4}*z*.

_{4}### 4.3 Transmission characteristics with different offsets of pinhole

26. Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. **48**(8), 1591–1597 (2009). [CrossRef] [PubMed]

*z*is short. The increase of the Fill factors can be explained by that one side of the higher frequency components is cut off when the offset increases, while the other side of the much higher frequencies passes through the pinhole. The proper higher frequency components can modulate the top of the beam and make it flat. Figure 11 shows that one side of the output FMGB becomes shaper than the other side when the distance

_{4}*z*increases, and Fig. 10(b) can also show that the modulation becomes bigger with increasing distance

_{4}*z*. It is because that one side of the high frequency components is shut off and the other side of the higher frequencies passes through the offset pinhole, which appears with increasing distance

_{4}*z*gradually. Hence, the small offset of the pinhole has little effect on the FBC and Fill factor of the output FMGB when the distance

_{4}*z*is short, but it can induce much bigger modulation with increasing distance

_{4}*z*. It is very dangerous for the high power laser system [20

_{4}20. A. K. Potemkin, T. V. Barmashova, A. V. Kirsanov, M. A. Martyanov, E. A. Khazanov, and A. A. Shaykin, “Spatial filters for high-peak-power multistage laser amplifiers,” Appl. Opt. **46**(20), 4423–4430 (2007). [CrossRef] [PubMed]

23. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. **46**(16), 3276–3303 (2007). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgment

## References and links

1. | H. T. Eyyuboglu, Ç. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express |

2. | X. Du and D. Zhao, “Propagation of the decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A |

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

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

5. | Z. Mei and D. Zhao, “Approximate method for the generalized M |

6. | J. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A |

7. | H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equation of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A |

8. | P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. |

9. | B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A |

10. | B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. |

11. | M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. |

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

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

14. | S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. |

15. | J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. |

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

17. | A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A |

18. | H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. |

19. | P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. Da Silva, B. J. Macgowan, B. M. Van Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. |

20. | A. K. Potemkin, T. V. Barmashova, A. V. Kirsanov, M. A. Martyanov, E. A. Khazanov, and A. A. Shaykin, “Spatial filters for high-peak-power multistage laser amplifiers,” Appl. Opt. |

21. | J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. |

22. | J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the national ignition facility,” Appl. Opt. |

23. | C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. |

24. | Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Acta. Phys. Sin-Ch. Ed. |

25. | Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B |

26. | Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(070.6110) Fourier optics and signal processing : Spatial filtering

(220.1140) Optical design and fabrication : Alignment

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 18, 2009

Revised Manuscript: June 25, 2009

Manuscript Accepted: June 29, 2009

Published: July 13, 2009

**Citation**

Yanqi Gao, Baoqiang Zhu, Daizhong Liu, and Zunqi Lin, "Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures," Opt. Express **17**, 12753-12766 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12753

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### References

- H. T. Eyyuboglu, Ç. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006). [CrossRef] [PubMed]
- X. Du and D. Zhao, “Propagation of the decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23(3), 625–631 (2006). [CrossRef]
- Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27(12), 1007–1009 (2002). [CrossRef]
- Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206(4-6), 225–234 (2002). [CrossRef]
- Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44(8), 1381–1386 (2005). [CrossRef] [PubMed]
- J. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A 24(1), 84–92 (2007). [CrossRef]
- H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equation of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005). [CrossRef]
- P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14(6), 1130–1135 (2005). [CrossRef]
- B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A 17(11), 2001–2004 (2000). [CrossRef]
- B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619–2718 (2001). [CrossRef]
- M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193(1-6), 73–79 (2001). [CrossRef]
- Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6(4), 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(12), 1061–1066 (2004). [CrossRef]
- S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13(3), 201–203 (1988). [CrossRef] [PubMed]
- J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]
- F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994). [CrossRef]
- A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18(8), 1897–1904 (2001). [CrossRef]
- H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008). [CrossRef]
- P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. Da Silva, B. J. Macgowan, B. M. Van Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. 37(12), 2371–2378 (1998). [CrossRef]
- A. K. Potemkin, T. V. Barmashova, A. V. Kirsanov, M. A. Martyanov, E. A. Khazanov, and A. A. Shaykin, “Spatial filters for high-peak-power multistage laser amplifiers,” Appl. Opt. 46(20), 4423–4430 (2007). [CrossRef] [PubMed]
- J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053–2057 (1978). [CrossRef] [PubMed]
- J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the national ignition facility,” Appl. Opt. 39(9), 1405–1420 (2000). [CrossRef]
- C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007). [CrossRef] [PubMed]
- Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Acta. Phys. Sin-Ch. Ed. 57, 6992–6997 (2008).
- Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009). [CrossRef]
- Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009). [CrossRef] [PubMed]

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