## Study of singular radius and surface boundary constraints in refractive beam shaper design

Optics Express, Vol. 16, Issue 9, pp. 6675-6682 (2008)

http://dx.doi.org/10.1364/OE.16.006675

Acrobat PDF (171 KB)

### Abstract

This paper presents analysis of important issues associated with the design of refractive laser beam shaping systems. The concept of “singular radius” is introduced along with solutions to minimize its adverse effect on shaper performance. In addition, the surface boundary constraint is discussed in detail. This study provides useful guidelines to circumvent possible design errors that would degrade the shaper quality or add undesired complication to the system.

© 2008 Optical Society of America

## 1. Introduction

7. 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**, 5488–5499 (2000). [CrossRef]

9. D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. **45**, 5118–5131 (2006) [CrossRef] [PubMed]

## 2. Shaper design and “Singular Radius” concept

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

12. P. Rhodes and D. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. **19**, 3545–3553 (1980). [CrossRef] [PubMed]

*r*

_{1}, emerges from the second surface with radius

*r*

_{2}and is parallel to Z axis again. The terms

*Z*

_{1}(

*r*) and

*Z*

_{2}(

*r*) represent the first (front) and second (rear) surface profiles, respectively,

*n*is the index of refraction, and

*s*is the distance between the vertex of the two surfaces. These beam shapers must all meet two requirements: (1) energy conservation, the total beam energy remains constant from the input to the output, and (2) equal optical path, all rays passing through the shaper from input pupil to output pupil have zero optical path difference (OPD). These conditions guarantee the shaped beam is perfectly collimated at the output. Applying Snell’s Law at each lens surface, surface profiles can be defined by the following relationship between

*r*

_{1},

*Z*

_{1}(

*r*),

*r*

_{2}and

*Z*

_{2}(

*r*):

*f*(

*r*) and output beam is a super-Gaussian

*g*(

*r*). The relationship between

*r*

_{1}and

*r*

_{2}is determined by the energy conservation condition. To avoid diffraction effects, the shaper radius was designed to be 5.7mm so that the radiance of the input Gaussian beam at the edge of the shaper is e

^{-7}of the peak value. In our design, input beam size was set with w=3mm. The parameter R, a length scale approximately the radius of a flattop for P>2, is 4mm. Here, the dimensionless order parameter P=8 defines the output super-Gaussian radiance profile:

9. D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. **45**, 5118–5131 (2006) [CrossRef] [PubMed]

*r*=3.8mm where, for the R=4mm curve, the first derivative

*dZ*/

*dr*=0. In this case, both the front and rear surfaces are no longer monotonic in terms of optical element smoothness. However, the curve for R=5mm is monotonic and quite normal. The occurrence of a tuning point is highly undesirable because it incurs unnecessary complication, adding particular difficulty in data processing and optical fabrication. Although the turning point is near the edge of the aspherical surface, it is not a location at which either the input or the output radiance may be ignored. For the case R=4mm, the intensity at the turning point (

*r*=3.8mm) of the input beam is about 4% of the peak value, yet over 26.5% for the output beam! For convenience, we define the radius where the turning point appears as the “singular radius” and next discuss its interpretation.

*r*

_{1}on the front surface tend to defocus more than those at larger

*r*

_{1}, ending at with larger value of

*r*

_{2}on the rear surface. This situation is reversed for those rays near the surface edge since the shaped flat-top radiance should be between the center peak and the lower wing of the original Gaussian input radiance. It is easy to imagine there must be a point where the input ray will pass straight through the optical material and reach the rear surface without any focusing or defocusing. In other words, the incident ray from

*r*

_{1}will travel parallel to the optical axis, indicating

*r*

_{1}=

*r*

_{2}. In comparison, rays towards the edge are bent at a larger angle for Type-2 and Type-4 shapers, so there is no other straight ray except the center ray on optical axis where

*r*

_{1}=

*r*

_{2}=0. A straight-forward way to predict the presence of the singular radius is to make a plot like that shown in Fig. 4, where the singular radius is obtained at the intersection point of a line (

*r*

_{1}=

*r*

_{2}) and the curve defined by Eq. (7). Analysis of the relationship between singular radius and other parameters of the shaping system are presented below. We show that it is possible to eliminate the singular radius from Type-1 and Type-3 designs.

## 3. Analysis

### 3.1 Initial beam size (w)

### 3.2 Lens thickness (s)

*s*is clear: singularity occurs when

*r*

_{1}equals

*r*

_{2}, and is solely determined by Eq. (7) which has no dependency on lens thickness.

### 3.3 Index of refraction (n)

*n*won’t make the surface profile get closer to monotonic. With larger

*n*, the sag value does decease, though. This helps to facilitate the optical fabrication.

### 3.4 Output beam parameters P and R

*r*=3.8mm) is somewhat smaller than the length scale parameter (R=4mm) which means the singular radius is along the ramp to the flattop profile. This is true for all design parameters. It is true also that the ramp of the output beam does not move much with order parameter however the slope does change. Consequently, the singular radius does not change significantly with P over a range of interest (P>8 for good uniformity and P<20 for diffraction suppression). However, as shown in Fig. 8, it would be extremely difficult to design a shaper which preserves a small beam while flattening the radiance profile, and difficult to constrict the beam while shaping.

## 4. Surface boundary

*r*

_{1}and

*r*

_{2}of the converging shaper could be big enough to meet the following condition; however, it is not possible for the diverging case. The surface ends as the integrand in Eq. (3) goes to infinity,

*s*the surface terminates at a radius far enough from the optical axis, and it will go further when s increases, which is revealed by Fig. 10.

*s*<15mm, the constraint is fairly rigorous for the design. It is easy to see that more attention needs to be taken when designing Type-4 than Type-3, because a Type-4 shaper inherently has faster bending surface profile. As we know, short spacing

*s*is desirable for ultra short pulse laser beam shaping and a larger aperture is much easier to fabricate and desirable for diffraction suppression. Due to the boundary constraint it is necessary to analyze the compromise between spacing and aperture in order to optimize a specific design.

## 5. Summary

## Acknowledgment

## References and links

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

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

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

4. | W. Jiang, D. L. Shealy, and J. C. Martin, “Design and testing of a refractive reshaping system,” in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer and W. J. Smith, eds., Proc. SPIE |

5. | W. Jiang and D. L. Shealy, “Development and testing of a laser beam shaping system,” Proc. SPIE |

6. | J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. |

7. | 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. |

8. | J. A. Hoffnagle and C. M. Jefferson, “Refractive optical system that converts a laser beam to a collimated flat-top beam,” U.S. patent 6,295,168 (25 September 2001). |

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

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

11. | 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). |

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

13. | C. Wang and D. L. Shealy, “Design of gradient-index lens systems for laser beam reshaping,” Appl. Opt. |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(220.1250) Optical design and fabrication : Aspherics

(220.2740) Optical design and fabrication : Geometric optical design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: February 15, 2008

Revised Manuscript: March 16, 2008

Manuscript Accepted: April 15, 2008

Published: April 25, 2008

**Citation**

C. Liu and S. Zhang, "Study of singular radius and surface boundary constraints in refractive beam shaper design," Opt. Express **16**, 6675-6682 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6675

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### References

- F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005). [CrossRef]
- S. Zhang, G. Neil, and M. Shinn, "Single-element laser beam shaper for uniform flat-top profiles," Opt. Express 11, 1942-1948 (2003). [CrossRef] [PubMed]
- S. Zhang, "A simple bi-convex refractive laser beam shaper," J. Opt. A: Pure Appl. Opt. 9, 945-950 (2007) [CrossRef]
- W. Jiang, D. L. Shealy, and J. C. Martin, "Design and testing of a refractive reshaping system," in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 2000, 64-75 (1993). [CrossRef]
- W. Jiang and D. L. Shealy, "Development and testing of a laser beam shaping system," Proc. SPIE 4095, 165-175 (2000). [CrossRef]
- J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003). [CrossRef]
- 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, 5488-5499 (2000). [CrossRef]
- J. A. Hoffnagle and C. M. Jefferson, "Refractive optical system that converts a laser beam to a collimated flat-top beam," U.S. patent 6,295,168 (25 September 2001).
- D. L. Shealy and J. A. Hoffnagle, "Laser beam shaping profiles and propagation," Appl. Opt. 45, 5118-5131 (2006) [CrossRef] [PubMed]
- B. R. Frieden, "Lossless conversion of a plane laser wave to a plane wave of uniform irradiance," Appl. Opt. 4, 1400-1403 (1965). [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).
- P. Rhodes and D. Shealy, "Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis," Appl. Opt. 19, 3545-3553 (1980). [CrossRef] [PubMed]
- C. Wang and D. L. Shealy, "Design of gradient-index lens systems for laser beam reshaping," Appl. Opt. 32, 4763-4769 (1993). [CrossRef] [PubMed]

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