Abstract

Deformable mirror (DM) is a common-used active freeform optical element. We introduce the concept of Woofer-Tweeter DM system for controlling focal-plane irradiance profiles. We firstly determine a freeform reflective surface for transforming a given incident laser beam into the desired focal-plane irradiance distribution by numerically solving a standard Monge-Ampère equation. Then, we use a low-bandwidth Woofer DM to approximate the required freeform reflective surface and a high-bandwidth Tweeter DM to compensate the residual error. Simulation results show that, compared with single DMs, the Woofer-Tweeter DM system brings the best focal-plane irradiance performances.

©2014 Optical Society of America

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References

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  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]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  14. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  19. P. Yang, Y. Liu, W. Yang, S. Hu, M. Ao, B. Xu, and W. Jiang, “An adaptive laser beam shaping technique based on a genetic algorithm,” Chin. Opt. Lett. 5(9), 497–500 (2007).
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    [Crossref]
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    [Crossref]
  23. S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. 45(12), 2638–2642 (2006).
    [Crossref] [PubMed]
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    [Crossref]
  27. J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010).
    [Crossref]
  28. 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]

2013 (3)

2012 (1)

2011 (3)

2010 (2)

J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010).
[Crossref]

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

2009 (1)

H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009).
[Crossref]

2008 (2)

2007 (2)

2006 (1)

2005 (3)

G. Loeper and F. Rapetti, “Numerical solution of the Monge-Ampére equation by a Newton’s algorithm,” C. R. Math. Acad. Sci. Paris 340(4), 319–324 (2005).
[Crossref]

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

R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005).
[Crossref] [PubMed]

2002 (1)

1999 (1)

Z. Zeng, N. Ling, and W. Jiang, “The investigation of controlling laser focal profile by deformable mirror and wave-front sensor,” J. Mod. Opt. 46(2), 341–348 (1999).
[Crossref]

1998 (1)

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

1997 (2)

1996 (1)

1991 (1)

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

1974 (1)

Aagedal, H.

Ao, M.

Arieli, Y.

Bäuerle, A.

Benamou, J.

J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010).
[Crossref]

Beth, T.

Bi, X.

H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009).
[Crossref]

Bräuer, A.

Bruneton, A.

Bryngdahl, O.

Bulte, H.

Cakmakci, O.

Cassarly, W. J.

Ding, Y.

Egner, S.

Eisenberg, N.

El-Agmy, R.

Feng, Z.

Foroosh, H.

Fournier, F. R.

Froese, B. D.

J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010).
[Crossref]

Fujii, T.

Glaser, I.

Gong, M.

Goto, N.

Greenaway, A. H.

Gu, P. F.

Hou, J.

Hu, S.

Huang, L.

Jiang, W.

Jiang, W. H.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Jin, G.

Kanai, Y. K.

Lewis, A.

Li, H.

Li, X.

H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009).
[Crossref]

Ling, N.

Z. Zeng, N. Ling, and W. Jiang, “The investigation of controlling laser focal profile by deformable mirror and wave-front sensor,” J. Mod. Opt. 46(2), 341–348 (1999).
[Crossref]

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Liu, P.

Liu, X.

Liu, Y.

H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009).
[Crossref]

P. Yang, Y. Liu, W. Yang, S. Hu, M. Ao, B. Xu, and W. Jiang, “An adaptive laser beam shaping technique based on a genetic algorithm,” Chin. Opt. Lett. 5(9), 497–500 (2007).

Loeper, G.

G. Loeper and F. Rapetti, “Numerical solution of the Monge-Ampére equation by a Newton’s algorithm,” C. R. Math. Acad. Sci. Paris 340(4), 319–324 (2005).
[Crossref]

Loosen, P.

Luo, Y.

Michaelis, D.

Moore, B.

Müller-Quade, J.

Muschaweck, J.

Nayuki, T.

Nemoto, K.

Oberman, A. M.

J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010).
[Crossref]

Oliker, V.

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

Parkyn, W. A.

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

Qian, K. Y.

Rao, X. J.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Rapetti, F.

G. Loeper and F. Rapetti, “Numerical solution of the Monge-Ampére equation by a Newton’s algorithm,” C. R. Math. Acad. Sci. Paris 340(4), 319–324 (2005).
[Crossref]

Reid, D.

Ries, H.

Rolland, J. P.

Russell, R. D.

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]

Schmid, M.

Schreiber, P.

Shi, F.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Smilie, P. J.

Stollenwerk, J.

Suleski, T. J.

Sulman, M. M.

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]

Wang, L.

Wester, R.

Williams, J. F.

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]

Wu, J.

Wu, R.

Wyrowski, F.

Xu, B.

Yang, H.

H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009).
[Crossref]

Yang, P.

Yang, W.

Zeng, Z.

Z. Zeng, N. Ling, and W. Jiang, “The investigation of controlling laser focal profile by deformable mirror and wave-front sensor,” J. Mod. Opt. 46(2), 341–348 (1999).
[Crossref]

Zhang, X.

Zhang, Y.

Zheng, Z.

Zheng, Z. R.

Appl. Numer. Math. (1)

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]

Appl. Opt. (3)

C. R. Math. Acad. Sci. Paris (1)

G. Loeper and F. Rapetti, “Numerical solution of the Monge-Ampére equation by a Newton’s algorithm,” C. R. Math. Acad. Sci. Paris 340(4), 319–324 (2005).
[Crossref]

Chin. Opt. Lett. (1)

ESAIM: Math. Model. Numer. Anal. (1)

J. Benamou, B. D. Froese, and A. M. Oberman, “Two numerical methods for the elliptic Monge-Ampére equation,” ESAIM: Math. Model. Numer. Anal. 44(4), 737–758 (2010).
[Crossref]

J. Mod. Opt. (1)

Z. Zeng, N. Ling, and W. Jiang, “The investigation of controlling laser focal profile by deformable mirror and wave-front sensor,” J. Mod. Opt. 46(2), 341–348 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (8)

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008).
[Crossref] [PubMed]

R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005).
[Crossref] [PubMed]

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]

R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013).
[Crossref] [PubMed]

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]

Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013).
[Crossref] [PubMed]

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]

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

Opt. Lett. (3)

Proc. SPIE (4)

H. Yang, Y. Liu, X. Li, and X. Bi, “Technique of adaptive laser beam shaping based on stochastic parallel gradient descent,” Proc. SPIE 7282, 72821W (2009).
[Crossref]

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

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

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

Other (2)

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (CRC, 2005).

F. M. Dickey and H. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).

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Figures (4)

Fig. 1
Fig. 1 Sketch of the Woofer-Tweeter DM system.
Fig. 2
Fig. 2 The (a) wavefront and (b) irradiance of the input laser beam.
Fig. 3
Fig. 3 (a) The required freeform reflective surface and its (b) resulting focal-spot; approximation using the 9-actuator DM: its (c) residual error and (d) focal-spot; Approximation using the 64-actuator DM: its (e) residual error and (f) focal-spot; Approximation using the Woofer-Tweeter DM system: its (g) residual error and (h) resulting focal-spot.
Fig. 4
Fig. 4 Variations of the (a) RRMSD and (b) LEE with respect to the focal spot size; Variations of the (c) RRMSD and (d) LEE with respect to the coupling coefficient.

Tables (2)

Tables Icon

Table 1 System parameters

Tables Icon

Table 2 Performance parameters

Equations (19)

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exp[lnω ( xc /d) α ],
Ω I in (x)dx = T I d (ξ)dξ .
I in (x)= I d ( m(x) )det( m(x) ), xΩ .
E d (ξ)= e ikf iλf e ik(ξξ)/2f Ω E in (x) e ikw(x) e ik(xξ)/f dx ,
I(k)= Ω E in (x) e ikF(x) dx,
F(x)=0.
m(x)=fw(x), xΩ .
I d ( fw(x) )det( D 2 w(x) )= I in (x)/ f 2 , xΩ ,
w(x)n(x)= 1 f m(x)n(x), xΩ ,
w i,j+1 + w i,j1 2 w i,j h 2 w i+1,j + w i1,j 2 w i,j h 2 ( w i+1,j+1 + w i1,j1 w i1,j+1 w i+1,j1 4 h 2 ) 2 = I in,i,j f 2 I d .
w i,j = a 1 + a 2 2 + ( a 1 a 2 2 ) 2 + ( a 3 a 4 4 ) 2 + h 4 4 f 2 I in,i,j I d ,
a 1 = w i+1,j + w i1,j 2 , a 2 = w i,j+1 + w i,j1 2 , a 3 = w i+1,j+1 + w i1,j1 2 , a 4 = w i1,j+1 + w i+1,j1 2 .
w i,j = w i,j +σ[ a 1 + a 2 2 + ( a 1 a 2 2 ) 2 + ( a 3 a 4 4 ) 2 + h 4 4 f 2 I in,i,j I d w i,j ],
s(x) s 1 (x)+ s 2 (x)= p=1 n η p l p (x) + q=1 m τ q h q (x),
Lη=S,
Hτ=SLη.
L=[ l 1 ( x 1 ) l 2 ( x 1 ) l n ( x 1 ) l 1 ( x 2 ) l 2 ( x 2 ) l n ( x 2 ) l 1 ( x N ) l 2 ( x N ) l n ( x N ) ],H=[ h 1 ( x 1 ) h 2 ( x 1 ) h m ( x 1 ) h 1 ( x 2 ) h 2 ( x 2 ) h m ( x 2 ) h 1 ( x N ) h 2 ( x N ) h m ( x N ) ], S= [ s( x 1 ),s( x 2 ),,s( x N ) ] T .
η= ( L T L) 1 L T S,
τ= ( H T H) 1 H T (SLη).

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