## Single-element laser beam shaper for uniform flat-top profiles

Optics Express, Vol. 11, Issue 16, pp. 1942-1948 (2003)

http://dx.doi.org/10.1364/OE.11.001942

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### Abstract

A novel design method is presented for a simple laser beam shaper. Unlike earlier reports and designs based on the 2-element model, we prove it is possible to convert a laser beam from a non-uniform profile to a uniform flat-top distribution with one single aspherical lens.

© 2003 Optical Society of America

## 1. Introduction

1. F. M. Dickey and S. C. Holswade, *Laser Beam Shaping-Theory and Techniques* (Marcel Dekker, Inc.2000). [CrossRef]

3. W. Jiang and D. L. Shealy, “Development and testing of a laser beam shaping system,” in Laser Beam Shaping, F. M. Dickey and S. C. Holswade, eds., Proc. SPIE **4095**, 165–175 (2000). [CrossRef]

## 2. Basic concept and analysis

3. W. Jiang and D. L. Shealy, “Development and testing of a laser beam shaping system,” in Laser Beam Shaping, F. M. Dickey and S. C. Holswade, eds., Proc. SPIE **4095**, 165–175 (2000). [CrossRef]

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

1. F. M. Dickey and S. C. Holswade, *Laser Beam Shaping-Theory and Techniques* (Marcel Dekker, Inc.2000). [CrossRef]

3. W. Jiang and D. L. Shealy, “Development and testing of a laser beam shaping system,” in Laser Beam Shaping, F. M. Dickey and S. C. Holswade, eds., Proc. SPIE **4095**, 165–175 (2000). [CrossRef]

*n*

_{1}, as shown in Fig. 2, propagates through a medium (refraction index

*n*

_{0}) and is refracted again at the curved surface of the second lens (refractive index

*n*

_{2}). If the input energy density is

*E*

_{i}(

*r*) and the maximum beam radius is

*r*

_{0}, then it follows from the energy conservation,

*E*

_{o}is the output energy density which is a constant and will be decided by the system magnification factor M=R/r. Applying Snell’s Law at both refractive surfaces and performing ray transform yields the following differential equation,

*z*'=

*dz*/

*dr*,

*γ*

_{1}=

*n*

_{1}/

*n*

_{0}. The constant optical path requires the OPL of any arbitrary ray equals to the OPL of the central ray, which leads to

*Z*'=

*dZ*/

*dr*,

*γ*

_{2}=

*n*

_{0}/

*n*

_{2}. The solutions to Eq. (1) through Eq. (5) will give the exact profiles for the 2 surfaces on the first and the second lens. Equation (2) can be further simplified to a quadratic equation which has following roots,

*n*

_{1}>

*n*

_{0}<

*n*

_{2}holds. This means that rays get defocused at the first surface and recollimated because of the focusing effect at the second refractive surface. This scenario was taken for granted and has been, to the best of our knowledge, the basis for this kind of beam reshaping systems ever since it was proposed.

*n*

_{1}>

*n*

_{0}<

*n*

_{2}has been assumed. But it does represent the right solution in the case of

*n*

_{1}<

*n*

_{0}>

*n*

_{2}. If

*n*

_{1}=

*n*

_{2}=1,

*n*

_{0}>1, then the whole picture will be exactly the same as rays coming from air, entering and refracted by a lens (with refractive index

*n*

_{0}) before exiting into air again. In this case the minus sign in Eq. (4) has to be taken, corresponding to a convex surface which is illustrated in Fig. 2 with dotted lines. Fig. 3 is a clearer picture for the single-lens system that converts the non-uniform input spatial distribution into a uniform flat-top one.

## 3. Calculations and evaluation

*E*

_{i}(

*r*)=exp(-2

*r*

^{2}/

*r*

_{0}is the beam radius where the irradiance falls to

*e*

^{-2}of its maximum. Fused silica is taken as the lens substrate and all other parameters have been listed in the graphs. The calculated lens surface profiles along with the ray trace are shown in Fig. 4. The irradiance distribution before and after the shaping system can be seen in Fig. 5. A non-uniform input beam profile has been converted to a uniform flat-top profile at reduced irradiance which is determined by the beam magnification. The wavelength for all calculations is 527nm.

*h*

^{2}(

*a*

_{s}

*r*/

*r*

_{s}) (

*r*

_{s}is the beam radius at

*e*

^{-2}of maximum,

*a*

_{s}=1.657). When the two input profiles are close to each other (as denoted by GSN2 and Sech2 in Fig. 6), the deviation at the output is very small across the whole aperture. The inputs with large deviation from the desired input profile results in obvious non-uniformity at the output. But even with large variation, the output uniformity within a certain radius is still superior to the beam resulting from a Gaussian profile cut by a hard aperture.

## 4. Summary

5. G. Erdei, G. Szarvas, E. Lorincz, and P. Richer, “Optimization method for the design of beam shaping systems,” Opt. Eng. **41**, 575–591 (2002). [CrossRef]

6. S. R. Jahan and M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. **21**, 27–30 (1987). [CrossRef]

## Acknowledgement

## References

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

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

3. | W. Jiang and D. L. Shealy, “Development and testing of a laser beam shaping system,” in Laser Beam Shaping, F. M. Dickey and S. C. Holswade, eds., Proc. SPIE |

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

5. | G. Erdei, G. Szarvas, E. Lorincz, and P. Richer, “Optimization method for the design of beam shaping systems,” Opt. Eng. |

6. | S. R. Jahan and M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(220.2740) Optical design and fabrication : Geometric optical design

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 27, 2003

Revised Manuscript: July 24, 2003

Published: August 11, 2003

**Citation**

S. Zhang, G. Neil, and M. Shinn, "Single-element laser beam shaper for uniform flat-top profiles," Opt. Express **11**, 1942-1948 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1942

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### References

- F. M. Dickey, S. C. Holswade, Laser Beam Shaping-Theory and Techniques (Marcel Dekker, Inc. 2000). [CrossRef]
- J.A. Hoffnagle and C.M. Johnson, �??Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,�?? Appl. Opt. 39, 5488-5499 (2000). [CrossRef]
- W. Jiang, D. L. Shealy, �??Development and testing of a laser beam shaping system,�?? in Laser BeamShaping, F. M. Dickey and S. C. Holswade, eds., Proc. SPIE 4095, 165-175 (2000). [CrossRef]
- B. R. Frieden, �??Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,�?? Appl. Opt. 4, 1400-1403 (1965). [CrossRef]
- G. Erdei, G. Szarvas, E. Lorincz, P. Richer, �??Optimization method for the design of beam shaping systems,�?? Opt. Eng. 41, 575-591 (2002). [CrossRef]
- S. R. Jahan and M. A. Karim, �??Refracting systems for Gaussian-to-uniform beam transformations,�?? Opt. Laser Technol. 21, 27-30 (1987). [CrossRef]

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