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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28693–28701
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Designing double freeform optical surfaces for controlling both irradiance and wavefront

Zexin Feng, Lei Huang, Guofan Jin, and Mali Gong  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28693-28701 (2013)
http://dx.doi.org/10.1364/OE.21.028693


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Abstract

We propose an improved double freeform-optical-surface design method for shaping a prescribed irradiance distribution whilst forming a desired wavefront from a given incident beam. This method generalizes our previous work [Opt. Exp. 21, 14728-14735 (2013)] to tackle non-separable beam irradiances. We firstly compute a proper ray mapping using an adaptive mesh method in the framework of the L2 Monge-Kantorovich mass transfer problem. Then, we construct the two freeform optical surfaces according to this mapping using a modified simultaneous point-by-point procedure which is aimed to minimize the surface errors. For the first surface, the modified procedure works by firstly approximating a value to the next point by only using the slope of the current point and then improving it by utilizing both slopes of the two points based on Snell’s law. Its corresponding point on the second surface can be computed using the constant optical path length condition. A design example of producing a challenging irradiance distribution and a non-ideal wavefront demonstrates the effectiveness of the method.

© 2013 Optical Society of America

1. Introduction

The problem we consider here is how to shape a prescribed irradiance distribution whilst forming a desired wavefront (or phase) from a given incident beam in the geometrical optics approximation. Generally, this involves designing at least two reflective or refractive freeform optical surfaces for increasing need of non-rotational symmetry transformations. The designs can play great roles in laser beam shaping applications and many other fields including illumination, solar energy concentration and reflector antennas.

Even the problem of only controlling the irradiance using one or two freeform optical surfaces is quite difficult. A variety of methods has been proposed to tackle this problem theoretically and numerically (see e.g. [1

1. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998). [CrossRef]

13

13. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013). [CrossRef] [PubMed]

].). A second requirement for controlling the wavefront using additional freeform optical surfaces makes the design more demanding. Here, we only consider designing two freeform optical surfaces for the two requirements.

2. Design method

The design problem is shown schematically in Fig. 1
Fig. 1 Sketch of the design problem.
. Consider a laser beam is traveling along the positive z direction. The input beam at z = 0 is supposed to have an irradiance distribution Iin(x,y) and the wavefront close to it is denoted by position vectors S = (x,y,z). The output beam irradiance at z = d is prescribed by Iout(x',y') and the wavefront close to it is denoted by T = (x',y',z'). n1, n0 and n2 are set as the the refractive indices of three mediums divided by the two freeform surfaces. The design goal is to specify P = (x1,y1,z1) and Q = (x2,y2,z2) on the two freeform surfaces in order to realize the above transformations. The two steps of the design are shown in 2.1 and 2.2, respectively.

2.1 Ray mapping computation

The ray mapping from the input beam to the desired output beam is mainly governed by Energy conservation:
Iin(x,y)dxdy=Iout(x',y')dx'dy'
(1)
For separable input and output irradiance distributions, the ray mapping can be simply obtained as x' = f(x) as y' = g(y) [7

7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

]. In a general case for non-separable irradiance distributions, the ray mapping must take the form: x' = f(x, y) and y' = g(x, y). By introducing this ray mapping into the right side of Eq. (1), we can obtain Eq. (2):
Iin(x,y)=Iout(f,g)|J(f,g)|
(2)
wherein J(f, g) is the Jacobian matrix of the ray mapping. Equation (2) alone can’t uniquely specify the ray mapping and need to be intertwined with the geometry of the optical elements for a rigorous solution [14

14. H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE 5876, 587607 (2005). [CrossRef]

, 18

18. V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013). [CrossRef]

]. However, this adds great complexity to the design especially for handling double freeform optical surfaces.

Several methods based on the L2 Monge-Kantorovich problem can help to specify an appropriate ray mapping (see e.g. [21

21. J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math. 84(3), 375–393 (2000). [CrossRef]

23

23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011). [CrossRef]

].). For example, Haker’s method [22

22. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004). [CrossRef]

] was utilized by Bäuerle et al. for approximating an optimum ray mapping in their irradiance tailoring approach [12

12. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012). [CrossRef] [PubMed]

]. Here, our alternative way to derive a proper ray mapping is the parabolic Monge-Ampère (PMA) adaptive mesh method presented by Sulman et al. recently [23

23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011). [CrossRef]

].

The optimal mapping of the L2 Monge-Kantorovich problem can be characterized as the gradient of a convex potential u(x, y) as shown in Eq. (3) [24

24. R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math. 17(3), 497–510 (1966). [CrossRef]

, 25

25. R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J. 80(2), 309–323 (1995). [CrossRef]

]:
(f(x,y),g(x,y))=u(x,y)
(3)
Substitute Eq. (3) into Eq. (2), we can obtain a standard elliptic Monge-Ampère equation shown in Eq. (4):
Iout(u(x,y))det2u(x,y)=Iin(x,y)
(4)
The PMA adaptive mesh method solves Eq. (4) by finding a steady-state solution of the logarithmic parabolic Monge-Ampère equation shown in Eq. (5) [23

23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011). [CrossRef]

]:

ut=log(Iout(u(x,y))det2u(x,y)Iin(x,y))
(5)

Neumann boundary conditions specified by Eq. (4) are enforced at boundary grid points. It is obvious that the steady-state solution u (t→∞) solves Eq. (4). Thus, the ray mapping can be determined by taking the spatial gradient of u.

The PMA adaptive mesh method provides a fairly simple way for solving the approximated optimal ray mapping. It is also convenient for us to solve the inverse ray mapping when the output irradiance distribution is much more complicated than the input one. Once the ray mapping is known, the input ray sequences can be defined by the input wavefront position vectors S and the input ray vectors In, and the output ray sequences can be defined by the output wavefront position vectors T and the output ray vectors Out [19

19. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013). [CrossRef] [PubMed]

].

2.2 Double freeform-surface construction

Now that the input and output ray sequences have been derived, we need to construct the two freeform optical surfaces to achieve that transformation.

3. Design example

To test the functionality of the proposed method, we design a Galilean type dual-lens beam shaper in a setting sketched in Fig. 4
Fig. 4 Sketch of the Galilean type dual-lens beam shaper setting.
. The output irradiance distribution is prescribed as the letters “NIO” on a square background with an 256 × 256 grid (see Fig. 5(a)
Fig. 5 (a) The desired target irradiance distribution, wherein the contrast is set as 4:1:0; (b) The prescribed output wavefront.
). The desired output wavefront is set as an astigmatism aberration with a peak-valley value (PV) of 80λ, wherein the beam wavelength λ = 1064nm (see Fig. 5(b)). The input laser beam is supposed to have a Gaussian irradiance distribution (beam waist: 5mm) and a spherical wavefront with a PV of 100λ.

We implement the PMA method to solve the inverse ray mapping x = f'(x',y') and y = g'(x',y') since the output irradiance distribution is much more complicated than the input one. The final adaptive mesh of the input irradiance is shown in Fig. 6(a)
Fig. 6 The resulting ray mapping for the given design, (a) The final adaptive mesh of the input beam; (b) A uniform Cartesian mesh of the output beam. For better visualization, the 256 × 256 grid was interpolated into an 32 × 32 one.
. A uniform Cartesian mesh of the output irradiance is also presented in Fig. 6(b) for better visualizing the Energy transfer. No surprisingly, the ray mapping is more strongly deformed around the regions corresponding to the edges of the letters where the target irradiance is essentially discontinuous.

According to the ray mapping, we then compute the two required freeform surfaces using the modified simultaneous point-by-point procedure described in 2.2. In the calculation, the wavefronts immediately behind the input plane and before the output plane can be simply obtained by multiplying the input wavefront and the output wavefront by 1/1.5083, respectively. They are directly used in the simultaneous point-by-point procedure for integrating the two freeform optical surfaces. Figure 7
Fig. 7 The (a) first and (b) second resulting freeform surfaces. They were uniformly interpolated into 32 × 32 grids for better visualization.
shows the two resulting freeform optical surfaces. They are fully continuous although the target irradiance is discontinuous. The first freeform surface is neither concave nor convex, which may add difficulties in its fabrication. Fortunately, we may obtain concave or convex freeform optical surfaces by crossing the ray mapping.

The angular spectrum method [27

27. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

] is used to simulate the designed beam shaper with a FFT (Fast Fourier Transform) size of 2048 × 2048, wherein the input light field is interpolated into an 512 × 512 grid for a more accurate simulation. For this case, the two lenses of the beam shaper can be considered as pure phase elements in the simulation, and their phase distributions can be computed mainly based on their thickness functions [27

27. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

]. The resulting output wavefront is derived by unwarping firstly the columns and then the rows of the wrapped phase [28

28. M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf.

]. The performances of the resulting output irradiance IS is quantified by the relative root-mean-square deviation (RRMSD) from the target irradiance, as shown in Eq. (6):
RRMSD=i,j=1256(Is(i,j)Iout(i,j))2/i,j256Iout(i,j)2
(6)
The resulting output wavefront can be assessed by the residual root-mean-square (RMS) error shown in Eq. (7):
RMS=1256×256i,j256(z's(i,j)z'(i,j))2
(7)
wherein z's is the z-coordinate value of the simulated output wavefront.

4. Conclusion

Acknowledgments

References and links

1.

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998). [CrossRef]

2.

W. A. Parkyn, and Jr., “Illuminating lens designed by extrinsic differential geometry,” Teledyne Lighting and Display Products, Inc., US Patent 5924788 (1999).

3.

B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2006). [CrossRef]

4.

W. A. Parkyn and D. G. Pelka, “Free-form lenses for rectangular illumination zones,” Anthony, Inc., US Patent 7674019 (2010).

5.

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

6.

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE 5942, 594207 (2005). [CrossRef]

7.

L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

8.

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]

9.

Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010). [CrossRef] [PubMed]

10.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010). [CrossRef] [PubMed]

11.

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011). [CrossRef] [PubMed]

12.

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012). [CrossRef] [PubMed]

13.

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013). [CrossRef] [PubMed]

14.

H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE 5876, 587607 (2005). [CrossRef]

15.

T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J. 53(5), 1255–1277 (2004). [CrossRef]

16.

V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011). [CrossRef]

17.

J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24(2), 463–469 (2007). [CrossRef] [PubMed]

18.

V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy 3(1), 035599 (2013). [CrossRef]

19.

Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013). [CrossRef] [PubMed]

20.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004). [CrossRef]

21.

J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math. 84(3), 375–393 (2000). [CrossRef]

22.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004). [CrossRef]

23.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011). [CrossRef]

24.

R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math. 17(3), 497–510 (1966). [CrossRef]

25.

R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J. 80(2), 309–323 (1995). [CrossRef]

26.

W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980), Chap.4.

27.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

28.

M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf.

OCIS Codes
(140.3300) Lasers and laser optics : Laser beam shaping
(080.1753) Geometric optics : Computation methods
(080.4225) Geometric optics : Nonspherical lens design
(080.4298) Geometric optics : Nonimaging optics

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: September 13, 2013
Revised Manuscript: October 31, 2013
Manuscript Accepted: November 4, 2013
Published: November 14, 2013

Citation
Zexin Feng, Lei Huang, Guofan Jin, and Mali Gong, "Designing double freeform optical surfaces for controlling both irradiance and wavefront," Opt. Express 21, 28693-28701 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28693


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References

  1. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998). [CrossRef]
  2. W. A. Parkyn, and Jr., “Illuminating lens designed by extrinsic differential geometry,” Teledyne Lighting and Display Products, Inc., US Patent 5924788 (1999).
  3. B. Parkyn and D. Pelka, “Free form lenses designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE6338, 633808 (2006). [CrossRef]
  4. W. A. Parkyn and D. G. Pelka, “Free-form lenses for rectangular illumination zones,” Anthony, Inc., US Patent 7674019 (2010).
  5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19(3), 590–595 (2002). [CrossRef] [PubMed]
  6. V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed illuminance properties,” Proc. SPIE5942, 594207 (2005). [CrossRef]
  7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt.46(18), 3716–3723 (2007). [CrossRef] [PubMed]
  8. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express16(17), 12958–12966 (2008). [CrossRef] [PubMed]
  9. Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express18(9), 9055–9063 (2010). [CrossRef] [PubMed]
  10. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express18(5), 5295–5304 (2010). [CrossRef] [PubMed]
  11. D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett.36(6), 918–920 (2011). [CrossRef] [PubMed]
  12. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express20(13), 14477–14485 (2012). [CrossRef] [PubMed]
  13. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett.38(2), 229–231 (2013). [CrossRef] [PubMed]
  14. H. Ries, “Laser beam shaping by double tailoring,” Proc. SPIE5876, 587607 (2005). [CrossRef]
  15. T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J.53(5), 1255–1277 (2004). [CrossRef]
  16. V. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Ration. Mech. Anal.201(3), 1013–1045 (2011). [CrossRef]
  17. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A24(2), 463–469 (2007). [CrossRef] [PubMed]
  18. V. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. Photonics Energy3(1), 035599 (2013). [CrossRef]
  19. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express21(12), 14728–14735 (2013). [CrossRef] [PubMed]
  20. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43(7), 1489–1502 (2004). [CrossRef]
  21. J. D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numer. Math.84(3), 375–393 (2000). [CrossRef]
  22. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis.60(3), 225–240 (2004). [CrossRef]
  23. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math.61(3), 298–307 (2011). [CrossRef]
  24. R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pac. J. Math.17(3), 497–510 (1966). [CrossRef]
  25. R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J.80(2), 309–323 (1995). [CrossRef]
  26. W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980), Chap.4.
  27. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).
  28. M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” http://www.ljmu.ac.uk/GERI/CEORG_Docs/Two_Dimensional_Phase_Unwrapping_Final.pdf .

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