Abstract

We propose a novel deformable mirror (DM) for adaptive optics in high power laser applications. The mirror is made of a Silicon carbide (SiC) faceplate, and cooling channels are embedded monolithically inside the faceplate with the chemical vapor desposition (CVD) method. The faceplate is 200 mm in diameter and 3 mm in thickness, and is actuated by 137 stack-type piezoelectric transducers arranged in a square grid. We also propose a new actuator influence function optimized for modelling our DM, which has a relatively stiffer faceplate and a higher coupling ratio compared with other DMs having thin faceplates. The cooling capability and optical performance of the DM are verified by simulations and actual experiments with a heat source. The DM is proved to operate at 1 kHz without the coolant flow and 100 Hz with the coolant flow, and the residual errors after compensation are less than 30 nm rms (root-mean-square). This paper presents the design, fabrication, and optical performance of the CVD SiC DM.

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

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References

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  1. H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
    [Crossref]
  2. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Chap. 6.
  3. E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
    [Crossref]
  4. M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
    [Crossref]
  5. G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
    [Crossref]
  6. A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
    [Crossref]
  7. R. H. Freeman and H. R. Garcia, “High-speed deformable mirror system,” Appl. Opt. 21(4), 589–595 (1982).
    [Crossref] [PubMed]
  8. P. Y. Bely, The Design and Construction of Large Optical Telescopes (Springer, 2002), Chap. 4.
  9. K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
    [Crossref]
  10. K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
    [Crossref]
  11. E. Hecht, Optics, 4th ed. (Pearson, 2014), Chap. 13.
  12. C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
    [Crossref]
  13. W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
    [Crossref]
  14. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. 32(13), 2431–2438 (1993).
    [Crossref] [PubMed]
  15. B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
    [Crossref]
  16. M. A. van Dam, D. Le Mignant, and B. A. Macintosh, “Performance of the Keck Observatory adaptive-optics system,” Appl. Opt. 43(29), 5458–5467 (2004).
    [Crossref] [PubMed]
  17. L. Huang, C. Rao, and W. Jiang, “Modified Gaussian influence function of deformable mirror actuators,” Opt. Express 16(1), 108–114 (2008).
    [Crossref] [PubMed]
  18. S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
    [Crossref]
  19. H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
    [Crossref]
  20. R. R. Shannon and J. C. Wyant, Applied Optics and Optical Engineering (Academic, 1992).
  21. J. Hermann, “Least-squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980).
    [Crossref]
  22. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980).
    [Crossref]
  23. S. Hyun, K. H. Kim, J. Y. Bae, and H. C. Kang, “Development of a water-cooling system for the 100 mm diameter deformable mirror,” in Proceeding of KSPE 2015 Autumn Conference (Korean Society for Precision Engineering, 2015), pp. 106–106.
  24. A. F. Mills, Basic Heat and Mass Transfer (IRWIN, 1995), Chap. 6.

2016 (1)

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

2015 (1)

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

2014 (1)

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

2013 (1)

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

2008 (1)

2006 (1)

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

2005 (1)

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

2004 (1)

1997 (1)

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

1995 (1)

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

1993 (1)

1992 (1)

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

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]

1982 (1)

1980 (2)

1978 (1)

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

1953 (1)

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
[Crossref]

Ahn, K.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

Alda, J.

Babcock, H. W.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
[Crossref]

Boreman, G. D.

Chalfant, A. I.

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

Dekany, R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Ealey, M.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Ealey, M. A.

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

Freeman, R. H.

Garcia, H. R.

Hermann, J.

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]

Kihm, H.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

Kudryashov, A. V.

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

Le Mignant, D.

Lee, H. J.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

Lee, J. H.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

Li, S.

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

Ling, N.

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

Luis, C. J.

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

Macintosh, B. A.

Oppenheimer, B. R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Palmer, D.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Price, T.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Puertas, I.

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

Rabczuk, G.

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

Rao, C.

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]

Rhee, H. G.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

Samarkin, V. V.

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

Sawczak, M.

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

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]

Sivaramakrishnan, A.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Southwell, W. H.

Szetela, E. J.

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

van Dam, M. A.

Villa, G.

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

Wellman, J. A.

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

Yang, H. S.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

Yang, H.-S.

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

Zhang, S.

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

Appl. Opt. (3)

J. Korean Phys. Soc. (1)

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

J. Mat. Proc. Technol. (1)

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

J. Opt. Soc. Am. (2)

Korean J. Opt. Photonics (1)

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

Opt. Commun. (1)

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

Opt. Eng. (1)

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

Opt. Express (1)

Opto-Electron. Rev. (1)

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

Proc. SPIE (3)

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

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

Publ. Astron. Soc. Pac. (1)

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
[Crossref]

Res. Astron. Astrophys. (1)

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

Thermochim. Acta (1)

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

Other (6)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Chap. 6.

P. Y. Bely, The Design and Construction of Large Optical Telescopes (Springer, 2002), Chap. 4.

E. Hecht, Optics, 4th ed. (Pearson, 2014), Chap. 13.

R. R. Shannon and J. C. Wyant, Applied Optics and Optical Engineering (Academic, 1992).

S. Hyun, K. H. Kim, J. Y. Bae, and H. C. Kang, “Development of a water-cooling system for the 100 mm diameter deformable mirror,” in Proceeding of KSPE 2015 Autumn Conference (Korean Society for Precision Engineering, 2015), pp. 106–106.

A. F. Mills, Basic Heat and Mass Transfer (IRWIN, 1995), Chap. 6.

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Figures (19)

Fig. 1
Fig. 1 Design of the CVD SiC faceplate with monolithic cooling channels. Small circles indicate the positions of 137 actuators. Blue and red arrows represent inlets and outlets respectively.

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Fig. 2
Fig. 2 The CFD simulation results; (a) Steady state water temperature, (b) faceplate temperature with cooling, and (c) faceplate displacement due to the steady state temperature gradient.

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Fig. 3
Fig. 3 A CVD SiC faceplate with monolithic cooling channels; (a) external view, (b) ultrasonic scan image showing the internal shape of the cooling channels.

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Fig. 4
Fig. 4 An assembled CVD SiC DM with 5-axis mount for the experiments.

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Fig. 5
Fig. 5 Comparison of an actual IF obtained from the CVD SiC DM and its fitted analytical models. (a) Actual IF measured with a 300-mm Fizeau interferometer for flatness tests, (b) the Gaussian IF, (c) the MGIF, and (d) the proposed OGIF

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Fig. 6
Fig. 6 Residual error images (left) and their cross-sectional plots (right) of the analytical IF models after fitting with the actual IF. (a) The Gaussian IF, (b) the MGIF, and (c) the OGIF

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Fig. 7
Fig. 7 A comparison of (a) the IFs simulated by using FEA and (b) the IFs measured by a Zygo interferometer. All IFs are downsized and placed at the corresponding actuator locations. IFs in the second quadrant are displayed due to the symmetry.

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Fig. 8
Fig. 8 The wavefront reconstruction experimental setup. (a) is a block diagram of the experimental setup and (b) is the actual implementation

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Fig. 9
Fig. 9 (a) The initial surface error of the faceplate before flattening, and (b) the surface error of the faceplate after flattening. The faceplate has the clear aperture of 160 mm in diameter which is 80% of faceplate.

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Fig. 10
Fig. 10 The Zernike modes measured with the interferometer (experiment) and their residual errors with respect to theoretical ones. The generated Zernike modes have the PV value of 2 waves within the clear aperture which is 80% of the physical diameter.

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Fig. 11
Fig. 11 The residual surface errors (PV and rms) of the reconstructed Zernike modes. The amplitude of the input Zernike modes is 2 waves in PV (λ = 632.8 nm).

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Fig. 12
Fig. 12 (a) Block diagram, and (b) actual implementation of the closed-loop AO system used for wavefront compensation experiments.

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Fig. 13
Fig. 13 History of the wavefront rms in the closed-loop AO system without coolant flow.

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Fig. 14
Fig. 14 (a) Simulation model and (b) schematic diagram of the CVD SiC DM irradiated by a heat source. The heat source is 250 mm apart from the faceplate and the angle is 45° between the heat source and the faceplate.

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Fig. 15
Fig. 15 2D view of the surface temperature variations (left) and the surface errors (right) from the CFD simulation results. (a) is the results without the coolant flow, and (b) is the results with the coolant flow.

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Fig. 16
Fig. 16 (a) Block diagram of the closed-loop AO system with the cooling system and a heat source, (b) the CVD SiC DM with a heat source and an IR camera.

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Fig. 17
Fig. 17 (a) A steady-state thermal image of the faceplate when irradiated by a heat source without cooling, and (b) with cooling.

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Fig. 18
Fig. 18 The temperature variation of the faceplate measured by 4 thermocouples. The positions of the thermocouples are indicated inside the figure.

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Fig. 19
Fig. 19 History of the wavefront rms in the closed-loop AO system when the DM is irradiated by a heat source and the coolant flows.

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Tables (1)

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Table 1 Faceplate material’s figure of merit

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Equations (8)

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

I(r)= I 0 e −2 r 2 / w 0 2
I(r)=exp[ ln(ω) ( r d ) α ]
I(r,θ)=exp[ ln(ω) { r[1+τcos(4θ)] d ¯ } α ]+βexp[ − { (r− d ¯ ) γ } 2 ]
I(r,θ)=(1−β)exp[ ln(ω) { r[1+τcos(4(θ− π 4 ))] d ¯ } α ]+βexp[ ln(ω) ( r d ) 2 ]
s(x,y)= ∑ i=1 n v i ϕ i (x,y)+ s 0
S=VΦ+ S 0
V= Φ −1 ( S d − S 0 )
d Q ˙ 12 = Id A 1 cos θ 1 d A 2 cos θ 2 R 2

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