Abstract

We theoretically investigate atomic cooling using a spatially varying AC Stark shift to compensate for the changing Doppler shift of an unchirped cooling beam. An integrated approach using waveguide-based atom photonics is ideal to achieve the required spatial tailoring of the AC Stark beam intensity. We present two examples of the design procedure to cool sodium atoms in hollow-core antiresonant reflecting optical waveguides over tens of centimeters resulting in final velocities comparable to a Zeeman slower. The methods presented here are applicable to other experimental arrangements and atomic species.

© 2014 Optical Society of America

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  17. H. Ito, T. Nakata, K. Sakaki, and M. Ohtsu, Phys. Rev. Lett. 76, 4500 (1996).
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  20. V. Venkataraman, P. Londero, A. R. Bhagwat, A. D. Slepkovm, and A. L. Gaeta, Opt. Lett. 35, 2287 (2010).
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  21. P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
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  22. W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, Nat. Photonics 1, 331 (2007).
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  23. B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
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  24. L. Stern, B. Desiatov, I. Goykhman, and U. Levy, Nat. Commun. 4, 1548 (2012).
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  26. W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).
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  27. M. A. Joffee, W. Ketterle, A. Martin, and D. E. Pritchard, J. Opt. Soc. Am. B 10, 2257 (1993).
    [CrossRef]
  28. Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).
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2012 (2)

L. Stern, B. Desiatov, I. Goykhman, and U. Levy, Nat. Commun. 4, 1548 (2012).
[CrossRef]

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

2010 (3)

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
[CrossRef]

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Phys. Rev. A 81, 053825 (2010).
[CrossRef]

V. Venkataraman, P. Londero, A. R. Bhagwat, A. D. Slepkovm, and A. L. Gaeta, Opt. Lett. 35, 2287 (2010).
[CrossRef]

2009 (2)

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
[CrossRef]

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

2008 (1)

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

2007 (3)

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, Opt. Lett. 32, 1323 (2007).
[CrossRef]

T. Takekoshi and R. J. Knize, Phys. Rev. Lett. 98, 210404 (2007).
[CrossRef]

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, Nat. Photonics 1, 331 (2007).
[CrossRef]

2004 (2)

D. Yin, J. Barber, A. R. Hawkins, D. W. Deamer, and H. Schmidt, Appl. Phys. Lett. 85, 3477 (2004).
[CrossRef]

C. J. Dedman, J. Nes, T. M. Hanna, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, Rev. Sci. Instrum. 75, 12 (2004).
[CrossRef]

2000 (1)

R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000).

1998 (1)

W. D. Phillips, Rev. Mod. Phys. 70, 721 (1998).
[CrossRef]

1997 (1)

C. S. Adams and E. Riss, Prog. Quantum Electron. 21, 1 (1997).
[CrossRef]

1996 (2)

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, Phys. Rev. A 53, R648 (1996).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, and M. Ohtsu, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef]

1995 (1)

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 75, 3253 (1995).
[CrossRef]

1993 (2)

M. A. Ol’Shanii, Y. B. Ovchinnikov, and V. S. Letokhov, Opt. Commun. 98, 77 (1993).
[CrossRef]

M. A. Joffee, W. Ketterle, A. Martin, and D. E. Pritchard, J. Opt. Soc. Am. B 10, 2257 (1993).
[CrossRef]

1985 (2)

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef]

J. Prodan, A. Migdall, and W. D. Phillips, Phys. Rev. Lett. 54, 992 (1985).
[CrossRef]

1982 (1)

W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

1975 (1)

T. W. Hänsch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

1970 (2)

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 25, 1321 (1970).
[CrossRef]

Adams, C. S.

C. S. Adams and E. Riss, Prog. Quantum Electron. 21, 1 (1997).
[CrossRef]

Anderson, D. Z.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, Phys. Rev. A 53, R648 (1996).
[CrossRef]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 75, 3253 (1995).
[CrossRef]

Ashkin, A.

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 25, 1321 (1970).
[CrossRef]

Bajcsy, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

Baldwin, K. G. H.

C. J. Dedman, J. Nes, T. M. Hanna, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, Rev. Sci. Instrum. 75, 12 (2004).
[CrossRef]

Balic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

Barber, J.

D. Yin, J. Barber, A. R. Hawkins, D. W. Deamer, and H. Schmidt, Appl. Phys. Lett. 85, 3477 (2004).
[CrossRef]

Benabid, F.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
[CrossRef]

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, Opt. Lett. 32, 1323 (2007).
[CrossRef]

Bhagwat, A. R.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Phys. Rev. A 81, 053825 (2010).
[CrossRef]

V. Venkataraman, P. Londero, A. R. Bhagwat, A. D. Slepkovm, and A. L. Gaeta, Opt. Lett. 35, 2287 (2010).
[CrossRef]

Bird, D. M.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
[CrossRef]

Blatt, R.

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef]

Christensen, C. A.

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

Conkey, D. B.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, Nat. Photonics 1, 331 (2007).
[CrossRef]

Cornell, E. A.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, Phys. Rev. A 53, R648 (1996).
[CrossRef]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 75, 3253 (1995).
[CrossRef]

Couny, F.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
[CrossRef]

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, Opt. Lett. 32, 1323 (2007).
[CrossRef]

Dall, R. G.

C. J. Dedman, J. Nes, T. M. Hanna, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, Rev. Sci. Instrum. 75, 12 (2004).
[CrossRef]

Deamer, D. W.

D. Yin, J. Barber, A. R. Hawkins, D. W. Deamer, and H. Schmidt, Appl. Phys. Lett. 85, 3477 (2004).
[CrossRef]

Dedman, C. J.

C. J. Dedman, J. Nes, T. M. Hanna, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, Rev. Sci. Instrum. 75, 12 (2004).
[CrossRef]

Desiatov, B.

L. Stern, B. Desiatov, I. Goykhman, and U. Levy, Nat. Commun. 4, 1548 (2012).
[CrossRef]

Donley, E. A.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, Phys. Rev. A 53, R648 (1996).
[CrossRef]

Ertmer, W.

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef]

Gaeta, A. L.

V. Venkataraman, P. Londero, A. R. Bhagwat, A. D. Slepkovm, and A. L. Gaeta, Opt. Lett. 35, 2287 (2010).
[CrossRef]

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Phys. Rev. A 81, 053825 (2010).
[CrossRef]

Goykhman, I.

L. Stern, B. Desiatov, I. Goykhman, and U. Levy, Nat. Commun. 4, 1548 (2012).
[CrossRef]

Grimm, R.

R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000).

Gulbransen, D.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

Hafezi, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

Hall, J. L.

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef]

Hanna, T. M.

C. J. Dedman, J. Nes, T. M. Hanna, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, Rev. Sci. Instrum. 75, 12 (2004).
[CrossRef]

Hänsch, T. W.

T. W. Hänsch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

Hawkins, A. R.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
[CrossRef]

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, Nat. Photonics 1, 331 (2007).
[CrossRef]

D. Yin, J. Barber, A. R. Hawkins, D. W. Deamer, and H. Schmidt, Appl. Phys. Lett. 85, 3477 (2004).
[CrossRef]

Hofferberth, S.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

Hulbert, J. F.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
[CrossRef]

Hurd, K.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
[CrossRef]

Ito, H.

H. Ito, T. Nakata, K. Sakaki, and M. Ohtsu, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef]

Jo, G.

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

Joffee, M. A.

Ketterle, W.

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

M. A. Joffee, W. Ketterle, A. Martin, and D. E. Pritchard, J. Opt. Soc. Am. B 10, 2257 (1993).
[CrossRef]

Knize, R. J.

T. Takekoshi and R. J. Knize, Phys. Rev. Lett. 98, 210404 (2007).
[CrossRef]

Letokhov, V. S.

M. A. Ol’Shanii, Y. B. Ovchinnikov, and V. S. Letokhov, Opt. Commun. 98, 77 (1993).
[CrossRef]

Levy, U.

L. Stern, B. Desiatov, I. Goykhman, and U. Levy, Nat. Commun. 4, 1548 (2012).
[CrossRef]

Light, P. S.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
[CrossRef]

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, Opt. Lett. 32, 1323 (2007).
[CrossRef]

Londero, P.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Phys. Rev. A 81, 053825 (2010).
[CrossRef]

V. Venkataraman, P. Londero, A. R. Bhagwat, A. D. Slepkovm, and A. L. Gaeta, Opt. Lett. 35, 2287 (2010).
[CrossRef]

Luiten, A. N.

Lukin, M. D.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

Lunt, E. J.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
[CrossRef]

Maric, M.

Martin, A.

Measor, P.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

Metcalf, H.

W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Migdall, A.

J. Prodan, A. Migdall, and W. D. Phillips, Phys. Rev. Lett. 54, 992 (1985).
[CrossRef]

Montgomery, D.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 75, 3253 (1995).
[CrossRef]

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, and M. Ohtsu, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef]

Nes, J.

C. J. Dedman, J. Nes, T. M. Hanna, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, Rev. Sci. Instrum. 75, 12 (2004).
[CrossRef]

Ohtsu, M.

H. Ito, T. Nakata, K. Sakaki, and M. Ohtsu, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef]

Ol’Shanii, M. A.

M. A. Ol’Shanii, Y. B. Ovchinnikov, and V. S. Letokhov, Opt. Commun. 98, 77 (1993).
[CrossRef]

Ovchinnikov, Y. B.

R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000).

M. A. Ol’Shanii, Y. B. Ovchinnikov, and V. S. Letokhov, Opt. Commun. 98, 77 (1993).
[CrossRef]

Ozcelik, D.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

Parks, J.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

Pearce, G. J.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, Appl. Phys. Lett. 94, 141103 (2009).
[CrossRef]

Peyronel, T.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. 102, 203902 (2009).
[CrossRef]

Phillips, B.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

Phillips, W. D.

W. D. Phillips, Rev. Mod. Phys. 70, 721 (1998).
[CrossRef]

J. Prodan, A. Migdall, and W. D. Phillips, Phys. Rev. Lett. 54, 992 (1985).
[CrossRef]

W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Pritchard, D.

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

Pritchard, D. E.

Prodan, J.

J. Prodan, A. Migdall, and W. D. Phillips, Phys. Rev. Lett. 54, 992 (1985).
[CrossRef]

Renn, M. J.

M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, Phys. Rev. A 53, R648 (1996).
[CrossRef]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 75, 3253 (1995).
[CrossRef]

Riss, E.

C. S. Adams and E. Riss, Prog. Quantum Electron. 21, 1 (1997).
[CrossRef]

Saba, M.

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

Sakaki, K.

H. Ito, T. Nakata, K. Sakaki, and M. Ohtsu, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef]

Schawlow, A. L.

T. W. Hänsch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

Schmidt, H.

Y. Zhao, B. Phillips, D. Ozcelik, J. Parks, P. Measor, D. Gulbransen, H. Schmidt, and A. R. Hawkins, J. Biophoton. 5, 703 (2012).

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, Nat. Photonics 4, 776 (2010).
[CrossRef]

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, Nat. Photonics 1, 331 (2007).
[CrossRef]

D. Yin, J. Barber, A. R. Hawkins, D. W. Deamer, and H. Schmidt, Appl. Phys. Lett. 85, 3477 (2004).
[CrossRef]

Shin, Y.

C. A. Christensen, S. Will, M. Saba, G. Jo, Y. Shin, W. Ketterle, and D. Pritchard, Phys. Rev. A 78, 033429 (2008).
[CrossRef]

Slepkov, A. D.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Phys. Rev. A 81, 053825 (2010).
[CrossRef]

Slepkovm, A. D.

Stern, L.

L. Stern, B. Desiatov, I. Goykhman, and U. Levy, Nat. Commun. 4, 1548 (2012).
[CrossRef]

Takekoshi, T.

T. Takekoshi and R. J. Knize, Phys. Rev. Lett. 98, 210404 (2007).
[CrossRef]

Truscott, A. G.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Example beam geometry. Doppler “cool” beam counterpropagates while an “AC” Stark beam co-propagates with the atom. (b) The relevant hyperfine transitions for Na cooling.

Fig. 2.
Fig. 2.

(a) Cross section of 12-layer HC-ARROW. (b) A(w) and α(w) for a HC-ARROW with 10 μm core thickness. (c) w(z) for To=30°C, 50°C, 70°C, 90°C, 110°C, and 130°C (blue, green, red, light blue, purple, and gold, respectively). Arrow indicates increasing To, and the inset shows a top-down view of the spatially varying hollow core width for To=50°C.

Fig. 3.
Fig. 3.

(a) α(z) for To=30°C, 50°C, 70°C, 90°C, 110°C, and 130°C (colors and arrow same as in Fig. 3). (b) dTO(z) for varying To. (c) v(z) for varying To.

Equations (3)

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

ΔEAC(z)=πc2Γ2ωo3(1PgFmFΔ1,F)I(z),
ΔED(z)=hλv(z)=hλvo2+2az,
α(z)=(1z)ln(πc2ΓλPin(1PgFmFΔ1,F)2ωo3Aohvo22az).

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