Abstract

The model fitting degree of optical freeform surfaces is of utmost design importance. We develop a model with radial basis functions based on the surface slope (RBF-slope) for optical freeform surfaces with asymmetric structures. The RBF-slope model improves the basis-function distribution for circular apertures and establishes a relationship between shape factor and local surface slope, which provides the model with better fitting ability than the conventional RBF model (RBF-direct); fitting experiments for off-axis conic surfaces, “bumpy” paraboloids, and the design of a single mirror magnifier demonstrate the efficacy of our approach. Our method can effectively improve aberration balancing of optical freeform surfaces, resulting in high-quality imaging.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  3. E. Muslimov, E. Hugot, W. Jahn, S. Vives, M. Ferrari, B. Chambion, D. Henry, and C. Gaschet, “Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture,” Opt. Express 25(13), 14598–14610 (2017).
    [Crossref] [PubMed]
  4. 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]
  5. T. Yang, J. Zhu, X. Wu, and G. Jin, “Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method,” Opt. Express 23(8), 10233–10246 (2015).
    [Crossref] [PubMed]
  6. S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25(24), 29627–29641 (2017).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. I. Kaya, K. P. Thompson, and J. P. Rolland, “Comparative assessment of freeform polynomials as optical surface descriptions,” Opt. Express 20(20), 22683–22691 (2012).
    [Crossref] [PubMed]
  13. I. Kaya, K. P. Thompson, and J. P. Rolland, “Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces,” Opt. Express 19(27), 26962–26974 (2011).
    [Crossref] [PubMed]
  14. O. Cakmakci, S. Vo, H. Foroosh, and J. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis magnifier,” Opt. Lett. 33(11), 1237–1239 (2008).
    [Crossref] [PubMed]
  15. 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]
  16. O. Cakmakci, I. Kaya, G. E. Fasshauer, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to represent optical freeform surfaces,” Proc. SPIE 7652, 76520A (2010).
    [Crossref]
  17. M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55(7), 071203 (2016).
    [Crossref]
  18. X. Lin, X. G. Liu, Y. Li, and G. Wei, “A new orbit fitting algorithm of space-borne SAR based on householder transformation,” in Synthetic Aperture Radar, 2009. Apsar 2009. Asian-Pacific Conference (2009), pp. 832–835.
    [Crossref]

2017 (2)

2016 (2)

C. Xu, D. Cheng, J. Chen, and Y. Wang, “Design of all-reflective dual-channel foveated imaging systems based on freeform optics,” Appl. Opt. 55(9), 2353–2362 (2016).
[Crossref] [PubMed]

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55(7), 071203 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (1)

2012 (2)

2011 (2)

2010 (1)

O. Cakmakci, I. Kaya, G. E. Fasshauer, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to represent optical freeform surfaces,” Proc. SPIE 7652, 76520A (2010).
[Crossref]

2008 (2)

2007 (1)

1976 (1)

1934 (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7), 689–704 (1934).
[Crossref]

Benítez, P.

Blen, J.

Cakmakci, O.

Chambion, B.

Chaves, J.

Chen, J.

Cheng, D.

Duerr, F.

Fasshauer, G. E.

O. Cakmakci, I. Kaya, G. E. Fasshauer, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to represent optical freeform surfaces,” Proc. SPIE 7652, 76520A (2010).
[Crossref]

Feng, Z.

Ferrari, M.

Forbes, G. W.

Foroosh, H.

Fuerschbach, K.

Gaschet, C.

Gong, M.

Grabovickic, D.

Henry, D.

Hernández, M.

Hu, X.

Hua, H.

Huang, L.

Hugot, E.

Jahn, W.

Jin, G.

Kaya, I.

Maksimovic, M.

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55(7), 071203 (2016).
[Crossref]

Miñano, J. C.

Mohedano, R.

Moore, B.

Muslimov, E.

Noll, R. J.

Rolland, J.

Rolland, J. P.

Sorgato, S.

Thienpont, H.

Thompson, K. P.

Vives, S.

Vo, S.

Wang, Y.

Wu, X.

Xu, C.

Yang, T.

Zernike, V. F.

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7), 689–704 (1934).
[Crossref]

Zhu, J.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55(7), 071203 (2016).
[Crossref]

Opt. Express (11)

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]

X. Hu and H. Hua, “High-resolution optical see-through multi-focal-plane head-mounted display using freeform optics,” Opt. Express 22(11), 13896–13903 (2014).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref] [PubMed]

G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
[Crossref] [PubMed]

G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20(3), 2483–2499 (2012).
[Crossref] [PubMed]

I. Kaya, K. P. Thompson, and J. P. Rolland, “Comparative assessment of freeform polynomials as optical surface descriptions,” Opt. Express 20(20), 22683–22691 (2012).
[Crossref] [PubMed]

I. Kaya, K. P. Thompson, and J. P. Rolland, “Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces,” Opt. Express 19(27), 26962–26974 (2011).
[Crossref] [PubMed]

E. Muslimov, E. Hugot, W. Jahn, S. Vives, M. Ferrari, B. Chambion, D. Henry, and C. Gaschet, “Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture,” Opt. Express 25(13), 14598–14610 (2017).
[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]

T. Yang, J. Zhu, X. Wu, and G. Jin, “Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method,” Opt. Express 23(8), 10233–10246 (2015).
[Crossref] [PubMed]

S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25(24), 29627–29641 (2017).
[Crossref] [PubMed]

Opt. Lett. (1)

Physica (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7), 689–704 (1934).
[Crossref]

Proc. SPIE (1)

O. Cakmakci, I. Kaya, G. E. Fasshauer, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to represent optical freeform surfaces,” Proc. SPIE 7652, 76520A (2010).
[Crossref]

Other (1)

X. Lin, X. G. Liu, Y. Li, and G. Wei, “A new orbit fitting algorithm of space-borne SAR based on householder transformation,” in Synthetic Aperture Radar, 2009. Apsar 2009. Asian-Pacific Conference (2009), pp. 832–835.
[Crossref]

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Figures (12)

Fig. 1
Fig. 1 (a) The geometry of single mirror magnifier (Ref [16], Fig. 3) (b) Modulation transfer function (MTF) at 23 cycles/mm for the 0-degree (radial) orientation of the single mirror magnifier where the surface was described with a 17 × 17 Gaussian radial basis function (RBF) (Ref [16], Fig. 4) (Text within these Figures has been modified for legibility).
Fig. 2
Fig. 2 Distribution comparison of basis functions between (a) RBF-Direct and (b) RBF-Slope models.
Fig. 3
Fig. 3 (a) Coordinates to characterize off-axis conic, (b) off-axis conic in rectangular aperture, (c) shape factor of RBF-Slope model with number of basis functions = 81 and average shape factor = 0.008.
Fig. 4
Fig. 4 (a) RMS errors and (b) PV errors of fitting for the off-axis conic when the RBF-Direct model and the RBF-Slope model are with 49, 64 and 81 basis functions respectively.
Fig. 5
Fig. 5 (a) Sag of paraboloid with bump, (b) the shape factor of RBF-Slope model when the number of basis functions is 784 and the average shape factor is 5.
Fig. 6
Fig. 6 Sag error in (a) whole aperture and (b) 0.8-aperture as fitted with RBF-Direct model with 784 basis functions, sag error in (c) whole aperture and (d) 0.8-aperture as fitted with RBF-Slope model with 784 basis functions, and sag error in (e) whole aperture and (f) 0.8-aperture fitted with Zernike polynomial with 784 basis functions.
Fig. 7
Fig. 7 Flow diagram of the optical design process.
Fig. 8
Fig. 8 Root mean square (RMS) wavefront error (WFE) in the full field of view (FOV) of the system when RBF-Slope model was applied.
Fig. 9
Fig. 9 (a) Shape factors of basis functions over the whole surface. (b) Surface shape of mirror represented by RBF-Slope model. (c) Surface of the mirror represented by RBF-Slope model in X–Y plane.
Fig. 10
Fig. 10 (a) 2D layout and (b) 3D layouts of single mirror magnifier.
Fig. 11
Fig. 11 Modulation transfer function (MTF) in the full field of view (FOV) of the optical system with application of (a) the RBF-Slope model and (b) the RBF-Direct model.
Fig. 12
Fig. 12 Root mean square (RMS) wavefront error (WFE) in the full field of view (FOV) of the optical system upon applying (a) the RBF-Slope model and (b) the RBF-Direct model.

Tables (4)

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Table 1 Fitting results of parabolic surface with bump

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Table 2 Parameters and achieved values of single mirror magnifier

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Table 3 Specified fields of view (FOVs) and corresponding coordinates

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Table 4 Comparison of optical performance of the two radial basis function models considered in the study

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

z(x,y)= c( x 2 + y 2 ) 1+ 1(1+k) c 2 ( x 2 + y 2 ) + i w i φ i (|| r r i ||) = c( x 2 + y 2 ) 1+ 1(1+k) c 2 ( x 2 + y 2 ) + i w i e ε i 2 ( (x x 0i ) 2 + (y y 0i ) 2 ) ,
ε i =k P V i S i ,
z= ( x 2 + y 2 ) 80 +0.05 e 0.25[ (x7) 2 + (y+6) 2 ] +0.6 e 0.49[ (x+3) 2 + (y2) 2 ] +0.03 e 0.81[ (x5) 2 + (y7) 2 ] ,

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