## Improvement of Galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size |

Optics Express, Vol. 19, Issue 14, pp. 13105-13117 (2011)

http://dx.doi.org/10.1364/OE.19.013105

Acrobat PDF (1990 KB)

### Abstract

We propose and demonstrate the improvement of conventional Galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size. Based on the detailed study of the refractive beam shaping system, we found that the conventional Galilean beam shaper can only work well for the magnifying beam shaping. Taking the transformation of input beam with Gaussian irradiance distribution into target beam with high order Fermi-Dirac flattop profile as an example, the shaper can only work well at the condition that the size of input and target beam meets *R*_{0}≥1.3*w*_{0}. For the improvement, the shaper is regarded as the combination of magnifying and demagnifying beam shaping system. The surface and phase distributions of the improved Galilean beam shaping system are derived based on Geometric and Fourier Optics. By using the improved Galilean beam shaper, the accurate transformation of input beam with Gaussian irradiance distribution into target beam with flattop irradiance distribution is realized. The irradiance distribution of the output beam is coincident with that of the target beam and the corresponding phase distribution is maintained. The propagation performance of the output beam is greatly improved. Studies of the influences of beam size and beam order on the improved Galilean beam shaping system show that restriction of beam size has been greatly reduced. This improvement can also be used to redistribute the input beam with complicated irradiance distribution into output beam with complicated irradiance distribution.

© 2011 OSA

## 1. Introduction

16. H. T. Ma, Z. J. Liu, P. Zhou, X. L. Wang, Y. X. Ma, and X. J. Xu, “Generation of flat-top beam with phase-only liquid crystal spatial light modulators,” J. Opt. **12**(4), 045704 (2010). [CrossRef]

16. H. T. Ma, Z. J. Liu, P. Zhou, X. L. Wang, Y. X. Ma, and X. J. Xu, “Generation of flat-top beam with phase-only liquid crystal spatial light modulators,” J. Opt. **12**(4), 045704 (2010). [CrossRef]

7. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. **4**(11), 1400–1403 (1965). [CrossRef]

12. S. Zhang, G. Neil, and M. Shinn, “Single-element laser beam shaper for uniform flat-top profiles,” Opt. Express **11**(16), 1942–1948 (2003). [CrossRef] [PubMed]

14. C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express **16**(9), 6675–6682 (2008). [CrossRef] [PubMed]

15. H. T. Ma, P. Zhou, X. L. Wang, Y. X. Ma, F. J. Xi, X. J. Xu, and Z. J. Liu, “Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators,” Opt. Express **18**(8), 8251–8260 (2010). [CrossRef] [PubMed]

16. H. T. Ma, Z. J. Liu, P. Zhou, X. L. Wang, Y. X. Ma, and X. J. Xu, “Generation of flat-top beam with phase-only liquid crystal spatial light modulators,” J. Opt. **12**(4), 045704 (2010). [CrossRef]

14. C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express **16**(9), 6675–6682 (2008). [CrossRef] [PubMed]

14. C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express **16**(9), 6675–6682 (2008). [CrossRef] [PubMed]

## 2. Analysis of the conventional Galilean beam shaping system

2. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. **39**(30), 5488–5499 (2000). [CrossRef] [PubMed]

*z*(

*r*) is the sag of the first aspheric lens at radial position

*r*,

*Z*(

*R*) is the sag of the second aspheric lens at radial position

*R*,

*d*is the distance between two aspheric lenses,

*n*is the refractive index.

*R*=

*h*(

*r*) and

*r*=

*h*

^{−1}(

*R*) are the relationships between the input position at the first aspheric lens and the output position at the second aspheric lens, which can be obtained by using energy conservation principle. According to Fourier Optics, we can get the relationship between the surface and phase distribution in Galilean beam shaping system, which is shown in Eqs. (3) and (4) [15

15. H. T. Ma, P. Zhou, X. L. Wang, Y. X. Ma, F. J. Xi, X. J. Xu, and Z. J. Liu, “Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators,” Opt. Express **18**(8), 8251–8260 (2010). [CrossRef] [PubMed]

**12**(4), 045704 (2010). [CrossRef]

*z*and

_{edge}*Z*denote the edge sags of the first and second aspheric lenses,

_{edge}*λ*represents the wavelength. In the following sections, we will give studies on conventional Galilean beam shaping system. In numerical simulation, the input and target beams are chosen as the Gaussian and high order Fermi-Dirac profile flattop beams respectively. That isandwhere

*w*

_{0}is the beam waist,

*β*is a dimensionless parameter and determines the range over, which the irradiance rolls off exponentially,

*R*

_{0}is the radius at which the irradiance has fallen to half of its value on axis. For the sake of simplicity and consistency, only one parameter is changed in the following discussion. Numerical simulations are based on the MATLAB. In addition, the angular spectrum method combined with MATLAB is used for modeling the propagation of the input beam [18].

### 2.1 Influence of target beam size (*R*_{0})

### 2.2 Influence of the target beam order (*β*)

*w*

_{0}= 3

*mm*,

*R*

_{0}= 3.3

*mm*,

*d =*350

*mm*,

*n*= 1.45, and

*λ*= 1064

*nm*. The changes of phase distributions of dual aspheric lenses along with beam order (

*β*) are shown in Fig. 5 . It is found that the increase of target beam order (

*β*) results in the increase of the difference between center and edge of the dual aspheric lenses. However, increase of the target beam order (

*β*) has little influence on the movement of point of inflexion in phase distribution curve. The corresponding irradiance and phase distributions of the output beam are shown in Figs. 6(a) and 6(b). The change of the shaping error along with the increase of the beam order (

*β*) is shown in Fig. 7 . It can be found that the increase of beam order (

*β*) results in larger difference between output and target beam. The flat area of the output beam decreases along with the increase of the beam order. The reason can be concluded as that the increase of the beam order results in slight decrease of the beam size. Compared with the numerical analysis mentioned in section 2.1, the influence of target beam size on beam shaping is much larger than that of the target beam order, so the shaping error of the conventional Galilean shaping system is mainly caused by the beam size.

## 3. Improvement of the Galilean beam shaping system

*r*on the front surface tend to defocus more than those at larger

*r*, ending at with larger value of

*R*on the rear surface. It can be thought that there must be a point where the input ray passes straight through the optical material and reach the rear surface without any focusing and defocusing. The conventional Galilean beam shaping system is assumed that the relationship between input radial position

*r*and output radial position

*R*is

*R*≥

*r*. The configuration is shown in Fig. 8(a) . In this configuration, only at position

*a*(on axis),

*R*equals to

*r*. We regard this configuration as the magnifying system. The design principle of the conventional Galilean beam shaping system is based on this configuration. However, many applications cannot meet this condition. For example, in the configuration shown in Fig. 8(b), the size of the target beam is smaller than that of the input beam.

*R*equals to

*r*at positions

*a*,

*b*, and

*c.*At the region between

*a*and

*b*or between

*a*and

*c*,

*R*≥

*r*(corresponding to magnifying beam shaping system); at the region between

*b*and the edge or between

*c*and the edge,

*R*≤

*r*(corresponding to demagnifying beam shaping system). We regard this beam shaping system is demagnifying shaping system as a whole. According to the working principle, the design principle of conventional Galilean beam shaping system can only work well for the condition that there is no other straight ray except the center ray on optical axis where

*r*=

*R*= 0.

*h*

_{1}is the position of point of inflexion on the phase distribution curve of the first aspheric lens.

*H*

_{1}is the position of point of inflexion on the phase distribution curve of the second aspheric lens.

*h*

_{1}.

*H*

_{1}. According to Fourier Optics, the relationships between phase distribution and radial position can be shown in Eqs. (10) and (11).andwhere

*z*

_{max}and

*Z*

_{max}are the maximum values of sag distributions of the first and second aspheric lens. Based on the same principle, we can also deduce the surface and phase distribution of more complicated beam shaping system, which contains many magnifying and demagnifying beam shaping systems.

*R*>

*r*is much smaller than the region of

*R*≤

*r*, so the shaping system can be regarded as the combination of only one demagnifying beam shaping system. In this case, the technique proposed in this paper is still valid. The point of the inflexion is proximate the optical axis, which can be represented as

*h*

_{1}=

*H*

_{1}≈0. The detailed analysis will be given in the following paragraphs. In addition, we find that the proposed technique is also valid for solving the surface profile for the magnifying beam shaping system shown in Fig. 8(a). In such a case, the point of inflexion

*h*

_{1}and

*H*

_{1}meets

*h*

_{1}=

*H*

_{1}≥

*h*

_{2}or

*h*

_{1}=

*H*

_{1}≥

*H*

_{2}, so the magnifying beam shaping system shown in Fig. 8(a) and the demagnifying beam shaping system shown in Fig. 9 can be regarded as the two special cases of the improved Galilean beam shaping system shown in Fig. 8(b).

*β*) and smaller target beam size (

*R*

_{0}) result in larger difference between the output beam and target beam, so we choose the target beam with large beam order and small beam size for analysis. The parameters of input and target beams are chosen as

*w*

_{0}= 3

*mm*,

*R*

_{0}= 3

*mm*,

*β*= 20,

*d*= 350

*mm*,

*n*= 1.45,

*λ*= 1064

*nm*. From Eqs. (8)–(11), the surface and phase distributions of dual aspheric lenses are calculated and shown in Figs. 10(a) and 10(b). It is found that the surface and phase distributions of the improved Galilean beam shaping system are not the same as that of the conventional Galilean beam shaping system. The edge’s surface and phase distributions of the improved Galilean beam shaping system are smaller than that of point of inflexion; on the contrary, the edge’s surface and phase distributions of the conventional Galilean beam shaping system are larger than that of point of inflexion. That is the reason that results in the difference between the output beam and the target beam in conventional Galilean beam shaping system. The irradiance and phase distributions of the input Gaussian beam passing through the improved Galilean beam shaping system are shown in Figs. 11(a) and 11(b). The near field irradiance distribution of the output beam is coincident with the target beam and the corresponding phase distribution is maintained. The shaping precision is much larger than that of the conventional Galilean beam shaping system.

*rms*variation of the output beam and conversion efficiency can be given by [2

2. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. **39**(30), 5488–5499 (2000). [CrossRef] [PubMed]

*SE*is the relative

*rms*variation of the output flattop beam.

*a*is the radius of interested region.

*W*is the power of the input beam. From the irradiance distribution shown in Fig. 11(a), we define a region of interest centered on the beam axis and compute the sum, average and variance of the values within the region. By varying the region, we deduce the relationship between variance and the efficiency. Figure 12 shows the relative intensity variation as well as the results of the target flattop beam. From Fig. 12, it is found that approximately 80% of the total power is contained in the flattop region having less than 6%

_{input}*rms*irradiance variation.

*m*is used to focus the output beam. The far field irradiance distribution of the output beam is shown in Fig. 13 . It is found that the far field irradiance of the output beam is coincident with the far field irradiance of the target beam, which shows that output beam is well compensated. The irradiance distributions of the output beam along with the propagation distance are shown in Fig. 14(a) . The corresponding result of the flattop beam generated by conventional Galilean beam shaping system is shown in Fig. 14(b). The intensity profile remains a useful flattop shape without significant diffraction peaks for a propagation distance of more than 1.2

*m*. However, the propagation distance of the flattop beam generated by the conventional Galilean beam shaping system is less than 0.4

*m*. The propagation performance has been greatly improved.

*R*

_{0}) and beam order (

*β*) of the target beam on the improved Galilean beam shaping system. The parameters

*w*

_{0}= 3

*mm*,

*β*= 16,

*d*= 350

*mm*,

*n*= 1.45,

*λ*= 1064

*nm*and the parameters

*w*

_{0}= 3

*mm*,

*R*

_{0}= 3.3

*mm*,

*d*= 350

*mm*,

*n*= 1.45,

*λ*= 1064

*nm*are chosen respectively in the numerical analysis. The changes of the shaping error along with the size and order of the target beam are shown in Figs. 15(a) and 15(b). From Figs. 15(a) and 15(b), we can see that the decrease of the beam size (

*R*

_{0}) and increase of the beam order (

*β*) results in larger shaping error. However, the shaping error of the improved Galilean beam shaping system is much smaller than that of the conventional Galilean beam shaping system. The restriction of beam size on Galilean beam shaping system has been greatly reduced.

*R*

_{0}≤1.8

*mm*, the point of inflexion

*h*

_{1}=

*H*

_{1}≤10

*um*. The magnifying region is much smaller than the demagnifying region. The Galilean beam shaping system can be regarded as the combination of only one demagnifying beam shaping system. The parameters

*w*

_{0}= 3

*mm*,

*R*

_{0}= 0.9

*mm*,

*β*= 16,

*d*= 350

*mm*,

*n*= 1.45, and

*λ*= 1064

*nm*are chosen to investigate the performance of Galilean beam shaping system consisted of only one demagnifying beam shaping system. The irradiance and phase distribution of the output beam are shown in Figs. 16(a) , 16(b) and 16(c). From Figs. 16(a), 16(b) and 16(c), we can see that the flattop beam with plane phase distribution is realized by the improved Galilean beam shaping system. The far field irradiance distribution of the output beam is nearly identical to the target beam. However, there exist difference between the near field irradiance distributions of the output beam and the target beam near the optical axis. It can be concluded that the technique proposed in this paper is suitable for the beam shaping system consisted of only one demagnifying beam shaping system. The error is just become larger with the decrease of the target beam size.

## 5. Conclusions

*rms*power variation. The flattop beam retains a useful flat-top irradiance distribution without significant diffraction peaks for a working distance of more than 1.2

*m*. However, the propagation distance of the flattop beam generated by the conventional Galilean beam shaping system is less than 0.4

*m*. The propagation performance has been greatly improved. The change of the shaping error of the target beam size and beam order on the improved Galilean beam shaping system has been investigated in detail. The restriction of beam size on beam shaping system has been greatly reduced. The transformation of input beam with arbitrary beam size into output beam with arbitrary beam size can be realized by this improved Galilean beam shaping system. This study provides useful guidelines to the improvement of the conventional refractive beam shaping system.

## References and links

1. | F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., |

2. | J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. |

3. | D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. |

4. | P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. |

5. | M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. |

6. | J. J. Kasinski and R. L. Burnham, “Near-diffraction-limited laser beam shaping with diamond-turned aspheric optics,” Opt. Lett. |

7. | B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. |

8. | C. C. Aleksoff, K. K. Ellis, and B. D. Neagle, “Holographic conversion of a Gaussian-beam to a near-field uniform beam,” Opt. Eng. |

9. | J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. |

10. | G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A |

11. | J. M. Auerbach and V. P. Karpenko, “Serrated-aperture apodizers for high-energy laser systems,” Appl. Opt. |

12. | S. Zhang, G. Neil, and M. Shinn, “Single-element laser beam shaper for uniform flat-top profiles,” Opt. Express |

13. | S. Zhang, “A simple bi-convex refractive laser beam shaper,” J. Opt. A, Pure Appl. Opt. |

14. | C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express |

15. | H. T. Ma, P. Zhou, X. L. Wang, Y. X. Ma, F. J. Xi, X. J. Xu, and Z. J. Liu, “Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators,” Opt. Express |

16. | H. T. Ma, Z. J. Liu, P. Zhou, X. L. Wang, Y. X. Ma, and X. J. Xu, “Generation of flat-top beam with phase-only liquid crystal spatial light modulators,” J. Opt. |

17. | J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463 (4 November 1969). |

18. | J. W. Goodman |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

(140.3300) Lasers and laser optics : Laser beam shaping

(220.2740) Optical design and fabrication : Geometric optical design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 21, 2011

Revised Manuscript: May 12, 2011

Manuscript Accepted: June 7, 2011

Published: June 22, 2011

**Citation**

Haotong Ma, Zejin Liu, Pengzhi Jiang, Xiaojun Xu, and Shaojun Du, "Improvement of Galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size," Opt. Express **19**, 13105-13117 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13105

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### References

- F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
- J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000). [CrossRef] [PubMed]
- D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. 45(21), 5118–5131 (2006). [CrossRef] [PubMed]
- P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980). [CrossRef] [PubMed]
- M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. 37(19), 4206–4209 (1998). [CrossRef] [PubMed]
- J. J. Kasinski and R. L. Burnham, “Near-diffraction-limited laser beam shaping with diamond-turned aspheric optics,” Opt. Lett. 22(14), 1062–1064 (1997). [CrossRef] [PubMed]
- B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4(11), 1400–1403 (1965). [CrossRef]
- C. C. Aleksoff, K. K. Ellis, and B. D. Neagle, “Holographic conversion of a Gaussian-beam to a near-field uniform beam,” Opt. Eng. 30(5), 537–543 (1991). [CrossRef]
- J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31(9), 1181–1183 (2006). [CrossRef] [PubMed]
- G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A 18(4), 791–800 (2001). [CrossRef]
- J. M. Auerbach and V. P. Karpenko, “Serrated-aperture apodizers for high-energy laser systems,” Appl. Opt. 33(15), 3179–3183 (1994). [CrossRef] [PubMed]
- S. Zhang, G. Neil, and M. Shinn, “Single-element laser beam shaper for uniform flat-top profiles,” Opt. Express 11(16), 1942–1948 (2003). [CrossRef] [PubMed]
- S. Zhang, “A simple bi-convex refractive laser beam shaper,” J. Opt. A, Pure Appl. Opt. 9(10), 945–950 (2007). [CrossRef]
- C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express 16(9), 6675–6682 (2008). [CrossRef] [PubMed]
- H. T. Ma, P. Zhou, X. L. Wang, Y. X. Ma, F. J. Xi, X. J. Xu, and Z. J. Liu, “Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators,” Opt. Express 18(8), 8251–8260 (2010). [CrossRef] [PubMed]
- H. T. Ma, Z. J. Liu, P. Zhou, X. L. Wang, Y. X. Ma, and X. J. Xu, “Generation of flat-top beam with phase-only liquid crystal spatial light modulators,” J. Opt. 12(4), 045704 (2010). [CrossRef]
- J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463 (4 November 1969).
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005).

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