## Refractive laser beam shaping by means of a functional differential equation based design approach |

Optics Express, Vol. 22, Issue 7, pp. 8001-8011 (2014)

http://dx.doi.org/10.1364/OE.22.008001

Acrobat PDF (989 KB)

### Abstract

Many laser applications require specific irradiance distributions to ensure optimal performance. Geometric optical design methods based on numerical calculation of two plano-aspheric lenses have been thoroughly studied in the past. In this work, we present an alternative new design approach based on functional differential equations that allows direct calculation of the rotational symmetric lens profiles described by two-point Taylor polynomials. The formalism is used to design a Gaussian to flat-top irradiance beam shaping system but also to generate a more complex dark-hollow Gaussian (donut-like) irradiance distribution with zero intensity in the on-axis region. The presented ray tracing results confirm the high accuracy of both calculated solutions and emphasize the potential of this design approach for refractive beam shaping applications.

© 2014 Optical Society of America

## 1. Introduction

1. F. M. Dickey, S. C. Holswade, and D. L. Shealy, *Laser Beam Shaping Applications* (CRC Press, 2005). [CrossRef]

2. D. Palima and J. Glückstad, “Gaussian to uniform intensity shaper based on generalized phase contrast,” Opt. Express **16**, 1507–1516 (2008). [CrossRef] [PubMed]

3. 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]

4. V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A **24**, 3741–3752 (2007). [CrossRef]

5. L. Romero and F. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A **13**, 751–760 (1996). [CrossRef]

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

10. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express **21**, 28693–28701 (2013). [CrossRef]

*r*is refracted at the first aspherical surface to be redirected in parallel direction with radial distance

*R*at the second aspherical surface.

*f*and

*g*described by two-point Taylor series expansions. To further demonstrate the versatility of this new approach, a second design problem is solved in Sec. 3 to generate a more complex dark-hollow Gaussian (donut-like) irradiance profile with zero intensity in the on-axis region. Here, the mapping function is given by a fractional Taylor series which has direct impact on the surface representation as well. The presented ray tracing evaluation confirm the high accuracy of both calculated solutions and emphasize the potential of this design approach for refractive beam shaping applications.

## 2. Analytic solution to convert a Gaussian into a flat-top irradiance distribution

*e*

^{2}width

*w*

_{0}. To avoid diffraction effects in regions of abrupt irradiance change and to increase the distance over which the uniform irradiance distribution can be used [13

13. D. L. Shealy and J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE **8490**, 849003 (2012). [CrossRef]

*q*denotes the beam shape parameter and

*R*is the beam width parameter of the flattened Lorentzian [13

_{FL}13. D. L. Shealy and J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE **8490**, 849003 (2012). [CrossRef]

*r*as and the output encircled energy as a function of

*R*as From conservation of energy between the input and the output plane, it is possible to determine the output radial distance

*R*as a function of the given input coordinate

*r*such that the encircled energies

*A*and

*B*are equal, that is where

*ε*is equal to +1 for a Galilean and −1 for a Keplerian configuration. An explicit mapping function

*R*(

*r*) that transforms a Gaussian into a flattened Lorentzian irradiance distribution is shown as dotted line in Fig. 2(a) for

*ε*= 1,

*w*

_{0}= 2.366mm,

*R*= 3.25mm and

_{FL}*q*= 16.

*R*can be calculated for every input coordinate

*r*, it is possible to derive an integral representation for the sag of each aspherical lens that can be solved numerically following the work of Kreuzer, Shealy and Hoffnagle [13

13. D. L. Shealy and J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE **8490**, 849003 (2012). [CrossRef]

*n*

_{2}. This is shown in Fig. 2(b). Consider an arbitrary light ray path. The ray is emitted from a point

*v⃗*

_{0}on the plane incident wave-front and intersects the first surface at a point

*p⃗*= (

*r*,

*f*(

*r*)), where it is refracted towards (

*R*(

*r*),

*g*(

*R*(

*r*))) on the second surface and refracted again towards the plane outgoing wave-front (parallel to the

*z*-axis). The path of the light rays are governed by Fermat’s principle which states that the optical path length (OPL) between two points is an extremum along the light ray containing these points. The OPL between points

*v⃗*

_{0},

*p⃗*and (

*R*(

*r*),

*g*(

*R*(

*r*))) is given by where

*n⃗*

_{0}denotes the directional vector of the wave-front in

*z*-direction. If the points

*v⃗*

_{0}and (

*R*(

*r*),

*g*(

*R*(

*r*))) are fixed, the only free parameter that can vary to achieve an extremum for

*V*

_{1}is the point

*p⃗*and hence

*r*on the first lens surface. Fermat’s principle thus implies that where the partial derivative indicates that

*R*is held fixed. Furthermore, the slope of the first aspheric surface at point (

*r*,

*f*(

*r*)) must be equal to the slope on the second aspheric surface at point (

*R*,

*g*(

*R*)) as a result of symmetry of the input and output wave-fronts. This can be expressed in a second condition where the partial derivative

*∂*(

_{r}g*r*) is evaluated at position

*R*(

*r*). This specific formulation highlights the functional character of the optical design problem. As the mapping function

*R*is a function of

*r*and appears as an argument in the unknown function

*g*, Eqs. (7) and (8) form a pair of functional differential equations for two unknown functions

*f*(

*r*) and

*g*(

*R*(

*r*)).

*D*= 0 for

_{i}*i*= 1, 2 and provide general solutions for the initial two-point Taylor series coefficients depending upon the single variable

*z*

_{2}. In ascending order, it is now possible to calculate (2

*k*+ 1)

^{th}order Taylor polynomials of

*f*(

*r*) and

*g*(

*r*) by calculating the derivatives and evaluating equations symbolically at points

*r*

_{1}and

*r*

_{2}. For the case

*n*= 0, the initial Taylor series coefficients

*p*

_{0},

*p*

_{1},

*q*

_{0},

*q*

_{1},

*u*

_{0},

*u*

_{1},

*v*

_{0}and

*v*

_{1}as defined in Eqs. (13) solve Eqs. (14).

*n*≥ 1, the set of equations (14) results in four linear algebraic equations for two-point Taylor series coefficients

*p*,

_{i}*q*,

_{i}*u*and

_{i}*v*. By sorting and combining all terms, the equations can be expressed as a compact matrix equation for particular two-point Taylor series coefficients for arbitrary

_{i}*n*≥ 1. The elements of the linear algebraic equations consist of mathematical expressions only dependent on previously calculated Taylor series coefficients. Therefore, all two-point Taylor series coefficients can be calculated as exact symbolic expressions in ascending order - step by step. So far, no approximations have been made. The general solution scheme for

*n*≥ 1 allows to calculate successive Taylor series coefficients of

*f*(

*r*) and

*g*(

*r*) up to a very high order. The inevitable truncation of the infinite sum of terms will be the only approximation made. The radii of convergence for the Taylor expansions

*f*(

*r*) and

*g*(

*r*) are very important, as they indicate the maximum aperture that can be achieved for a given set of initial values. For the examples considered in this work, the radii of convergence are larger than the radial coordinate 0 <

*r*<

*r*of the lenses.

_{max}### 2.1. Evaluation of the analytic solution

*w*

_{0}= 2.366mm,

*R*= 3.25mm and

_{FL}*q*= 16 as before. The two points of the Taylor series are

*r*

_{1}= 0.2mm and

*r*

_{2}= 3.6mm. The chosen distance between

*z*

_{1}= 0mm and

*z*

_{3}= 16mm defines the lens separation. For a chosen refractive index

*n*

_{2}= 1.5, this initial ray path construction determines the OPL. After applying the constant OPL condition for point

*r*

_{2}, there still exists an infinite number of solutions that solve the design problem locally in the neighborhood of

*r*

_{2}. In order to link the two local Taylor expansion solutions at

*r*

_{1}and

*r*

_{2}, the correct relative positioning in

*z*-direction of the two solution parts has to be identified. It is very important to note that both local solutions at the respective points are exact and calculated symbolically, only the relative position of the two parts of the overall solution has to be adjusted numerically. To find the appropriate value of

*z*

_{2}, the implemented calculation of the Taylor coefficients for the lens profiles is defined as a function in

*Mathematica*. After calculation of the Taylor series coefficients up to a certain order

*k*, 10 equidistant rays between 0 mm and 4 mm (corresponds to ≈ 99.7% encircled input energy) are traced and evaluated. The spatial deviation of each ray is then defined by the mapping error

*R′*−

_{i}*R*(

*r*). Where

_{i}*R′*

*is the calculated intersection of the*

_{i}*i*

^{th}ray with the surface profile

*g*and

*R*(

*r*) is the ideal position defined by the mapping function. As a second figure of merit, the variation in optical path length is also calculated for all rays. As one possible overall merit function,

_{i}*m*(

*z*

_{2}) is given by the sum of the two root mean square values of the spatial deviation and variation in OPL of the 10 rays over the full entrance aperture.

*m*(

*z*

_{2}) as a function of variable

*z*

_{2}for values between −0.5 and 0 mm and for a step size of 0.0025 mm. A logarithmic plot is chosen to better cover the wide range of values.

*z*

_{2}using a local optimization in

*Mathematica*.

*r*-direction. This evaluation is performed for 7

^{th}(k=3), 13

^{th}(k=6) and 19

^{th}(k=9) order Taylor polynomials for

*f*and

*g*, respectively. Figure 4(a) shows the variation in optical path length (OPL) and Fig. 4(b) shows the spatial mapping error for 3 calculated solutions of different order. The root mean square values of the variation in OPL are 1.647μm (k=3), 0.617μm (k=6) and 0.058μm (k=9). The root mean square values of the mapping error are 0.559 × 10

^{−2}(k=3), 0.061 × 10

^{−2}(k=6) and 0.009 × 10

^{−2}(k=9), respectively. For both graphs, the variation in OPL and mapping error reduce very well with an increasing order underlining the convergence and achievable high accuracy of the derived solution.

7. J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. **42**, 3090–3099 (2003). [CrossRef]

*r*

_{1}and

*r*

_{2}, respectively.

## 3. Analytic solution for Gaussian to dark-hollow Gaussian refractive beam shapers

*n*denotes the order of the DHG beam and

*H*

_{0}is a constant that is adjusted to conserve the overall energy. For

*n*= 0, Eq. (15) reduces to the fundamental beam with beam waist

*w*

_{1}[15

15. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. **28**, 1084–1086 (2003). [CrossRef] [PubMed]

*R*and input coordinate

*r*such that the encircled energy

*B*(

*R*) at the output is equal to the encircled energy

*A*(

*r*) at the input. The input encircled energy was given in Eq. (3) and the output encircled energy is The solution scheme that was applied in Sec. 2 is not directly applicable in this case, because the mapping function

*R*(

*r*) can not be solved as a closed-form expression. However, it is possible to derive a closed-form expression for the inverse mapping function

*r*(

*R*) =

*R*

^{−1}(

*r*). The mapping function

*R*(

*r*) is then given by applying Lagrange inversion theorem that provides the Taylor series expansion of the inverse function of an analytic function. Both mapping function

*r*(

*R*) and its inverse function

*R*(

*r*) are shown in Fig. 6(a); for parameters

*w*

_{0}= 2.366mm,

*w*

_{1}= 2mm and

*n*= 1.

*R*(

*r*) is then described by a Taylor series expansion with fractional exponents, where the first derivative

*R′*(

*r*) for

*r*→ 0 diverges, as the function intersects the horizontal axis perpendicular at the origin (see dotted line in Fig. 6(a)). When trying to use the same solution scheme for the approximated mapping function

*R*(

*r*) as in the previous section, the problem appears that even for a two-point Taylor series representation of the surfaces the profiles will still not converge for all values of

*r*. This problem is directly linked to the fractional exponents of the mapping function

*R*(

*r*), which will become more clear later.

*v⃗*

_{0},

*p⃗*= (

*R*,

*g*(

*R*)) and (

*r*(

*R*),

*f*(

*r*(

*R*))) is given by and by using the same arguments as before, Fermat’s principle implies that where the partial derivative indicates that

*r*is held fixed. To ensure recollimation of the rays, the second condition is given by where the partial derivative

*∂*

_{R}*f*(

*R*) is evaluated at position

*r*(

*R*). The surface

*g*(

*R*) is defined by a two-point Taylor series expansion of (2

*k*+ 1)

^{th}order By evaluation derivatives of Eqs. (20) and (21) at points

*R*

_{1}and

*R*

_{2}in ascending order it is possible to calculate the Taylor coefficients of

*g*(

*R*) in ascending order up to high orders. However, a similar two-point Taylor polynomial of function

*f*still does not converge for all points

*R*. From Eq. (21) follows that it is also possible to calculate the second surface profile

*f*(

*r*(

*R*)) from the integral equation for known functions

*g*(

*R*) and

*R*(

*r*). By substituting the mapping function

*R*(

*r*) from Eq. (18) as argument in the first derivative

*g′*(

*R*), the unknown function

*f*(

*R*) can be exactly calculated via symbolic integration. The function

*f*(

*R*) is then described by a polynomial function with fractional exponents. This circumstance explains why a conventional Taylor series approach with integer exponents does not work properly. The integration constant

*f*

_{0}is calculated by applying the constant OPL condition at point

*R*

_{2}. Interestingly, the coefficient

*f*

_{1}is almost zero (less than 10

^{−8}) and identical to the slope value

*g′*(0), providing smooth lens shapes in the central on-axis region. Due to the substitution of the mapping function

*R*(

*r*) in

*g′*(

*R*),

*f*(

*R*) easily results in a very high order polynomial. Even though it is not necessary, this function can be truncated to a moderate order in the range of function

*g*(

*R*) without changing the performance.

### 3.1. Evaluation of the analytic solution

*w*

_{0}= 2.366mm,

*w*

_{1}= 2mm and

*n*= 1. The two manually selected points of the Taylor series are

*R*

_{1}= 0.3mm and

*R*

_{2}= 4.2mm. By fixing the distance between

*Z*

_{1}= 0mm and

*Z*

_{2}= 16mm and the refractive index

*n*

_{2}= 1.5, the remaining parameter

*Z*

_{2}that adjusts the relative positioning in

*z*-direction is optimized for an identical merit function as defined in Sec. 2.1. Figure 7(a) shows the variation in OPL and Fig. 7(b) shows the spatial mapping error for 3 different solutions of different order. All calculations have been repeated for 200 rays between 0 and 4 mm along

*r*-direction (which corresponds to ≈ 99.7% encircled input energy). The evaluation is performed for 7

^{th}(k=3), 11

^{th}(k=5) and 15

^{th}(k=7) order Taylor polynomials for

*g*, and

*f*has been truncated to 21

^{th}order.

^{−2}(k=3), 0.776 × 10

^{−2}(k=5) and 0.164 × 10

^{−2}(k=7), respectively. As in the previous design example, the variation in OPL and the mapping error reduce very well with an increasing Taylor series order, confirming the convergence and high accuracy of the derived analytic solution. Furthermore, the maximum collimation error after refraction at the second plane surface is evaluated for the 3 different solutions. The maximum collimation error is 0.744mrad (k=3), 0.063mrad (k=5) and 0.058mrad (k=7). If required, an even higher accuracy can be achieved by increasing the order of the Taylor polynomials and/or by fine-tuning the radial positions

*R*

_{1}and

*R*

_{2}.

## 4. Conclusion

16. F. Duerr, Y. Meuret, and H. Thienpont, “Potential benefits of free-form optics in on-axis imaging applications with high aspect ratio,” Opt. Express **21**, 31072–31081 (2013). [CrossRef]

## Acknowledgments

## References and links

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

2. | D. Palima and J. Glückstad, “Gaussian to uniform intensity shaper based on generalized phase contrast,” Opt. Express |

3. | 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. |

4. | V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A |

5. | L. Romero and F. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A |

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

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

8. | H. Ma, Z. Liu, P. Jiang, X. Xu, and S. Du, “Improvement of Galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size,” Opt. Express |

9. | J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A |

10. | Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express |

11. | F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express |

12. | F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, and H. Thienpont, “Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces,” Opt. Express |

13. | D. L. Shealy and J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE |

14. | R. H. Estes and E. R. Lancaster, “Two-point Taylor series expansions,” NASA TMX-55759 (N67-23965) (1966). |

15. | Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. |

16. | F. Duerr, Y. Meuret, and H. Thienpont, “Potential benefits of free-form optics in on-axis imaging applications with high aspect ratio,” Opt. Express |

17. | F. Duerr, Y. Meuret, and H. Thienpont, “Tailored free-form optics with movement to integrate tracking in concentrating photovoltaics,” Opt. Express |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(080.2740) Geometric optics : Geometric optical design

(080.3620) Geometric optics : Lens system design

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Geometric Optics

**History**

Original Manuscript: January 27, 2014

Revised Manuscript: March 2, 2014

Manuscript Accepted: March 22, 2014

Published: March 28, 2014

**Citation**

Fabian Duerr and Hugo Thienpont, "Refractive laser beam shaping by means of a functional differential equation based design approach," Opt. Express **22**, 8001-8011 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8001

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### References

- F. M. Dickey, S. C. Holswade, D. L. Shealy, Laser Beam Shaping Applications (CRC Press, 2005). [CrossRef]
- D. Palima, J. Glückstad, “Gaussian to uniform intensity shaper based on generalized phase contrast,” Opt. Express 16, 1507–1516 (2008). [CrossRef] [PubMed]
- J. A. Hoffnagle, 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]
- V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A 24, 3741–3752 (2007). [CrossRef]
- L. Romero, F. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751–760 (1996). [CrossRef]
- P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980). [CrossRef] [PubMed]
- J. A. Hoffnagle, C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003). [CrossRef]
- H. Ma, Z. Liu, P. Jiang, X. Xu, S. 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). [CrossRef] [PubMed]
- J. Rubinstein, G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463–469 (2007). [CrossRef]
- Z. Feng, L. Huang, G. Jin, M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21, 28693–28701 (2013). [CrossRef]
- F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012). [CrossRef] [PubMed]
- F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces,” Opt. Express 20, 10839–10846 (2012). [CrossRef] [PubMed]
- D. L. Shealy, J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE 8490, 849003 (2012). [CrossRef]
- R. H. Estes, E. R. Lancaster, “Two-point Taylor series expansions,” NASA TMX-55759 (N67-23965) (1966).
- Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003). [CrossRef] [PubMed]
- F. Duerr, Y. Meuret, H. Thienpont, “Potential benefits of free-form optics in on-axis imaging applications with high aspect ratio,” Opt. Express 21, 31072–31081 (2013). [CrossRef]
- F. Duerr, Y. Meuret, H. Thienpont, “Tailored free-form optics with movement to integrate tracking in concentrating photovoltaics,” Opt. Express 21, A401–A411 (2013). [CrossRef] [PubMed]

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