Abstract

It is shown that the degree of polarization analysis is useful to find objective spectra of exoplanets immersed in noisy stellar spectra. We report the laboratory experiment of polarization differential objective spectroscopy with a four-quadrant polarization mask coronagraph, where partially polarized planetary signal is expected to be discerned from unpolarized stellar noise. The detection of the planet signal is impeded by the stellar noise remained after subtracting mutually orthogonally polarized components of light. We distinguish clearly the planetary spectrum by use of the degree of polarization. We also show the refinement of the spectrum of the planet model.

© 2007 Optical Society of America

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References

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  1. http://vo.obspm.fr/exoplanets/encyclo/encycl.html
  2. N. Woolf and J. R. Angel, “Astronomical searches for Earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
    [CrossRef]
  3. C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication 99–3 Pasadena, CA, 1999).
  4. E. Serabyn, J. K. Wallace, G. J. Hardy, E. G. H. Schmidtlin, and H. T. Nguyen, “Deep nulling of visible laser light,” Appl. Opt. 38, 7128–7132 (1999).
    [CrossRef]
  5. N. Baba, N. Murakami, and T. Ishigaki, “Nulling interferometry by use of geometric phase,” Opt. Lett. 26, 1167–1169 (2001).
    [CrossRef]
  6. A. Tavrov, R. Bohr, M. Totzeck, H. Tiziani, and M. Takeda, “Achromatic nulling interferometer based on a geometric spin-redirection phase,” Opt. Lett. 27, 2070–2072 (2002).
    [CrossRef]
  7. F. Roddier and C. Roddier, “Stellar coronagraph with phase mask,” Publ. Astron. Soc. Pac. 109, 815–820 (1997).
    [CrossRef]
  8. P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraphy,” Astron. Astrophys. 141, 319–329 (2000).
  9. N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27, 1373–1375 (2002).
    [CrossRef]
  10. O. Guyon and F. Roddier, “A nulling wide field imager for exoplanets detection and general astrophysics,” Astron. Astrophys. 391, 379–395 (2002).
    [CrossRef]
  11. G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
    [CrossRef]
  12. D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
    [CrossRef]
  13. R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
    [CrossRef]
  14. J. R. Kuhn, D. Potter, and B. Parise, “Imaging polarimetric observations of a new circumsteller disk system”, Astrophys. J. 553, L189–L191 (2001).
    [CrossRef]
  15. N. Baba and N. Murakami, “A method to image extrasolar planets with polarized light,” Publ. Astron. Soc. Pac. 115, 1363–1366 (2003).
    [CrossRef]
  16. O. Guyon. “Synchronous interferometric speckle subtraction,” Astrophys. J. 615, 562–572 (2004).
    [CrossRef]
  17. C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).
  18. D. M. Stam, J. W. Hovenier, and L. B. F. M. Waters, “Using polarimetry to detect and characterize Jupiterlike extrasolar planets,” Astron. Astrophys. 428, 663–672 (2004).
    [CrossRef]
  19. N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
    [CrossRef]
  20. D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
    [CrossRef]
  21. J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004).
    [CrossRef]
  22. J. B. Breckinridge, “Image Formation in high contrast optical systems: the role of polarization,” Proc. SPIE 5487, 1337–1345 (2004).
    [CrossRef]
  23. J. Hong, “Precision compensation for polarization anisotropies in metal reflectors,” Opt. Eng. 43, 1276–1277 (2004).
    [CrossRef]

2006 (2)

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

2005 (1)

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
[CrossRef]

2004 (5)

D. M. Stam, J. W. Hovenier, and L. B. F. M. Waters, “Using polarimetry to detect and characterize Jupiterlike extrasolar planets,” Astron. Astrophys. 428, 663–672 (2004).
[CrossRef]

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004).
[CrossRef]

J. B. Breckinridge, “Image Formation in high contrast optical systems: the role of polarization,” Proc. SPIE 5487, 1337–1345 (2004).
[CrossRef]

J. Hong, “Precision compensation for polarization anisotropies in metal reflectors,” Opt. Eng. 43, 1276–1277 (2004).
[CrossRef]

O. Guyon. “Synchronous interferometric speckle subtraction,” Astrophys. J. 615, 562–572 (2004).
[CrossRef]

2003 (1)

N. Baba and N. Murakami, “A method to image extrasolar planets with polarized light,” Publ. Astron. Soc. Pac. 115, 1363–1366 (2003).
[CrossRef]

2002 (3)

2001 (3)

2000 (2)

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraphy,” Astron. Astrophys. 141, 319–329 (2000).

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

1999 (2)

E. Serabyn, J. K. Wallace, G. J. Hardy, E. G. H. Schmidtlin, and H. T. Nguyen, “Deep nulling of visible laser light,” Appl. Opt. 38, 7128–7132 (1999).
[CrossRef]

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

1998 (1)

N. Woolf and J. R. Angel, “Astronomical searches for Earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

1997 (1)

F. Roddier and C. Roddier, “Stellar coronagraph with phase mask,” Publ. Astron. Soc. Pac. 109, 815–820 (1997).
[CrossRef]

Absil, O.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
[CrossRef]

Angel, J. R.

N. Woolf and J. R. Angel, “Astronomical searches for Earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

Baba, N.

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

N. Baba and N. Murakami, “A method to image extrasolar planets with polarized light,” Publ. Astron. Soc. Pac. 115, 1363–1366 (2003).
[CrossRef]

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27, 1373–1375 (2002).
[CrossRef]

N. Baba, N. Murakami, and T. Ishigaki, “Nulling interferometry by use of geometric phase,” Opt. Lett. 26, 1167–1169 (2001).
[CrossRef]

Baudoz, P.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraphy,” Astron. Astrophys. 141, 319–329 (2000).

Beichman, C. A.

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication 99–3 Pasadena, CA, 1999).

Boccaletti, A.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

Bohr, R.

Breckinridge, J. B.

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004).
[CrossRef]

J. B. Breckinridge, “Image Formation in high contrast optical systems: the role of polarization,” Proc. SPIE 5487, 1337–1345 (2004).
[CrossRef]

Clenet, Y.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

Doyon, R.

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

Gay, J.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraphy,” Astron. Astrophys. 141, 319–329 (2000).

Guyon, O.

O. Guyon and F. Roddier, “A nulling wide field imager for exoplanets detection and general astrophysics,” Astron. Astrophys. 391, 379–395 (2002).
[CrossRef]

Guyon., O.

O. Guyon. “Synchronous interferometric speckle subtraction,” Astrophys. J. 615, 562–572 (2004).
[CrossRef]

Hardy, G. J.

Hashimoto, N.

Hong, J.

J. Hong, “Precision compensation for polarization anisotropies in metal reflectors,” Opt. Eng. 43, 1276–1277 (2004).
[CrossRef]

Hovenier, J. W.

D. M. Stam, J. W. Hovenier, and L. B. F. M. Waters, “Using polarimetry to detect and characterize Jupiterlike extrasolar planets,” Astron. Astrophys. 428, 663–672 (2004).
[CrossRef]

Ishigaki, T.

Kuhn, J. R.

J. R. Kuhn, D. Potter, and B. Parise, “Imaging polarimetric observations of a new circumsteller disk system”, Astrophys. J. 553, L189–L191 (2001).
[CrossRef]

Labeyrie, A.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

Lafrenière, D.

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

Lindensmith, C. A.

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication 99–3 Pasadena, CA, 1999).

Macintosh, B.

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

Marois, C.

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

Mawet, D.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
[CrossRef]

Murakami, N.

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

N. Baba and N. Murakami, “A method to image extrasolar planets with polarized light,” Publ. Astron. Soc. Pac. 115, 1363–1366 (2003).
[CrossRef]

N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, “Polarization interferometric stellar coronagraph,” Opt. Lett. 27, 1373–1375 (2002).
[CrossRef]

N. Baba, N. Murakami, and T. Ishigaki, “Nulling interferometry by use of geometric phase,” Opt. Lett. 26, 1167–1169 (2001).
[CrossRef]

Nadeau, D.

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

Nguyen, H. T.

Oppenheimer, B. R.

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004).
[CrossRef]

Parise, B.

J. R. Kuhn, D. Potter, and B. Parise, “Imaging polarimetric observations of a new circumsteller disk system”, Astrophys. J. 553, L189–L191 (2001).
[CrossRef]

Potter, D.

J. R. Kuhn, D. Potter, and B. Parise, “Imaging polarimetric observations of a new circumsteller disk system”, Astrophys. J. 553, L189–L191 (2001).
[CrossRef]

Rabbia, Y.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraphy,” Astron. Astrophys. 141, 319–329 (2000).

Racine, R.

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

Riaud, P.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
[CrossRef]

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

Roddier, C.

F. Roddier and C. Roddier, “Stellar coronagraph with phase mask,” Publ. Astron. Soc. Pac. 109, 815–820 (1997).
[CrossRef]

Roddier, F.

O. Guyon and F. Roddier, “A nulling wide field imager for exoplanets detection and general astrophysics,” Astron. Astrophys. 391, 379–395 (2002).
[CrossRef]

F. Roddier and C. Roddier, “Stellar coronagraph with phase mask,” Publ. Astron. Soc. Pac. 109, 815–820 (1997).
[CrossRef]

Rouan, D.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

Sato, Y.

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

Schmidtlin, E. G. H.

Serabyn, E.

Stam, D. M.

D. M. Stam, J. W. Hovenier, and L. B. F. M. Waters, “Using polarimetry to detect and characterize Jupiterlike extrasolar planets,” Astron. Astrophys. 428, 663–672 (2004).
[CrossRef]

Surdej, J.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
[CrossRef]

Swartzlander, G. A.

Takeda, M.

Tamura, M.

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

Tate, Y.

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

Tavrov, A.

Tiziani, H.

Totzeck, M.

Walker, G.

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

Wallace, J. K.

Waters, L. B. F. M.

D. M. Stam, J. W. Hovenier, and L. B. F. M. Waters, “Using polarimetry to detect and characterize Jupiterlike extrasolar planets,” Astron. Astrophys. 428, 663–672 (2004).
[CrossRef]

Woolf, N.

N. Woolf and J. R. Angel, “Astronomical searches for Earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

Woolf, N. J.

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication 99–3 Pasadena, CA, 1999).

Angular differential imaging: a powerful high-contrast imaging technique (1)

C. Marois, D. Lafrenière, R. Doyon, B. Macintosh, and D. Nadeau, “Angular differential imaging: a powerful high-contrast imaging technique,” Astrophys. J. 641, 556–564 (2006).

Annu. Rev. Astron. Astrophys. (1)

N. Woolf and J. R. Angel, “Astronomical searches for Earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (3)

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interfero coronagraphy,” Astron. Astrophys. 141, 319–329 (2000).

O. Guyon and F. Roddier, “A nulling wide field imager for exoplanets detection and general astrophysics,” Astron. Astrophys. 391, 379–395 (2002).
[CrossRef]

D. M. Stam, J. W. Hovenier, and L. B. F. M. Waters, “Using polarimetry to detect and characterize Jupiterlike extrasolar planets,” Astron. Astrophys. 428, 663–672 (2004).
[CrossRef]

Astrophys. J. (4)

J. R. Kuhn, D. Potter, and B. Parise, “Imaging polarimetric observations of a new circumsteller disk system”, Astrophys. J. 553, L189–L191 (2001).
[CrossRef]

J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004).
[CrossRef]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. 633, 1191–1200 (2005).
[CrossRef]

O. Guyon. “Synchronous interferometric speckle subtraction,” Astrophys. J. 615, 562–572 (2004).
[CrossRef]

Opt. Eng. (1)

J. Hong, “Precision compensation for polarization anisotropies in metal reflectors,” Opt. Eng. 43, 1276–1277 (2004).
[CrossRef]

Opt. Lett. (4)

Proc. SPIE (1)

J. B. Breckinridge, “Image Formation in high contrast optical systems: the role of polarization,” Proc. SPIE 5487, 1337–1345 (2004).
[CrossRef]

Publ. Astron. Soc. Pac. (5)

N. Baba and N. Murakami, “A method to image extrasolar planets with polarized light,” Publ. Astron. Soc. Pac. 115, 1363–1366 (2003).
[CrossRef]

N. Murakami, N. Baba, Y. Tate, Y. Sato, and M. Tamura, “Polarization differential objective spectroscopy with nulling coronagraph,” Publ. Astron. Soc. Pac. 118, 774–779 (2006).
[CrossRef]

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph I. Principle,” Publ. Astron. Soc. Pac. 112, 1479–1486 (2000).
[CrossRef]

R. Racine, G. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999).
[CrossRef]

F. Roddier and C. Roddier, “Stellar coronagraph with phase mask,” Publ. Astron. Soc. Pac. 109, 815–820 (1997).
[CrossRef]

Other (2)

http://vo.obspm.fr/exoplanets/encyclo/encycl.html

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication 99–3 Pasadena, CA, 1999).

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Figures (6)

Fig. 1.
Fig. 1.

Optical setup of polarization differential spectroscopy with an FQPoM coronagraph.

Fig. 2.
Fig. 2.

Coronagraphic images with the s- (a) and p- (b) polarized components of light and their differential image (c). The arrows indicate the planet position.

Fig. 3.
Fig. 3.

Objective spectra with s- (a) and p-polarized components (b) and their differential objective spectra (c). The arrows indicate the position of the planetary spectrum.

Fig. 4.
Fig. 4.

The degree of polarization of the differential spectra with a constant (a) and a linear (b) scaling factors. The blue color is used for displaying negative values and the red one for positive ones. The arrows indicate the location of the planetary spectrum.

Fig. 5.
Fig. 5.

(a). Spectral transmittance of our coronagraphic optics with retardations of the LCVR in the states of a wave plate, Twp (green) and a half-wave plate, Thwp (blue). (b) Plot of Twp/Thwp with respect to wavelength. The line shows the fitted one in the range of 580–680 nm.

Fig. 6.
Fig. 6.

(a). Comparison of the planetary spectrum extracted from the differential spectrum (red) and the spectrum of the planet without the LCVR and starlight (green). (b) Spectral transmittance of LCVR: a wave plate mode (upper), a half-wave plate mode (lower). (c) Refined planetary spectrum (blue) and the planetary spectrum without the LCVR and starlight (green).

Equations (3)

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q ij = I ij wp I ij hwp I ij wp + I ij hwp ,
Q ij = I ij wp k j × I ij hwp .
q ij = I ij wp k j × I ij hwp I ij wp + k j × I ij hwp .

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