Abstract

We present a coherent-population-trapping-based scheme to attain subnanoscale resolution for microscopy. We use three-level atoms coupled to an amplitude modulated probe field and a spatially dependent (standing-wave or Laguerre–Gaussian) coupling field. The probe field modulation allows us to tap into the steep dispersion normally associated with electromagnetically induced transparency and offers subnanometer resolution using optical fields.

© 2010 Optical Society of America

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  1. J. M. Vigoureux and D. Courjon, Appl. Opt. 31, 3170(1992).
    [CrossRef] [PubMed]
  2. C. Girard and A. Dereux, Rep. Prog. Phys. 59, 657(1996).
    [CrossRef]
  3. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
    [CrossRef] [PubMed]
  4. S. W. Hell, Science 316, 1153 (2007).
    [CrossRef] [PubMed]
  5. E. Paspalakis and P. L. Knight, Phys. Rev. A 63, 065802 (2001).
    [CrossRef]
  6. E. Paspalakis and P. L. Knight, J. Mod. Opt. 52, 1685 (2005).
    [CrossRef]
  7. G. S. Agarwal and K. T. Kapale, J. Phys. B 39, 3437(2006).
    [CrossRef]
  8. M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
    [CrossRef] [PubMed]
  9. H. S. Park, S. K. Lee, and J. Y. Lee, Opt. Express 16, 21982 (2008).
    [CrossRef] [PubMed]
  10. H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
    [CrossRef]
  11. M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
    [CrossRef]
  12. K. T. Kapale and M. S. Zubairy, Phys. Rev. A 73, 023813 (2006).
    [CrossRef]
  13. M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
    [CrossRef]
  14. H. G. de Chatellus and J.-P. Pique, Opt. Lett. 34, 755 (2009).
    [CrossRef] [PubMed]
  15. A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
    [CrossRef] [PubMed]
  16. E. Arimondo, in Progress in Optics, E.Wolf, ed. (Elsevier Science B. V., 1996), Vol. XXXV, pp. 257–354.
    [CrossRef]
  17. R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
    [CrossRef]
  18. D. D. Yavuz and N. A. Proite, Phys. Rev. A 76, 041802(R) (2007).
  19. J. Mompart, V. Ahufinger, and G. Birkl, Phys. Rev. A 79, 053638 (2009).
    [CrossRef]
  20. D. D. Yavuz, N. A. Proite, and J. T. Green, Phys. Rev. A 79, 055401 (2009).
    [CrossRef]
  21. L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
    [CrossRef]
  22. K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
    [CrossRef]
  23. S. E. Harris, Phys. Today 50, 36 (1997).
    [CrossRef]
  24. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
    [CrossRef]

2009 (4)

J. Mompart, V. Ahufinger, and G. Birkl, Phys. Rev. A 79, 053638 (2009).
[CrossRef]

D. D. Yavuz, N. A. Proite, and J. T. Green, Phys. Rev. A 79, 055401 (2009).
[CrossRef]

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

H. G. de Chatellus and J.-P. Pique, Opt. Lett. 34, 755 (2009).
[CrossRef] [PubMed]

2008 (4)

H. S. Park, S. K. Lee, and J. Y. Lee, Opt. Express 16, 21982 (2008).
[CrossRef] [PubMed]

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
[CrossRef] [PubMed]

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

2007 (4)

S. W. Hell, Science 316, 1153 (2007).
[CrossRef] [PubMed]

D. D. Yavuz and N. A. Proite, Phys. Rev. A 76, 041802(R) (2007).

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
[CrossRef]

2006 (2)

K. T. Kapale and M. S. Zubairy, Phys. Rev. A 73, 023813 (2006).
[CrossRef]

G. S. Agarwal and K. T. Kapale, J. Phys. B 39, 3437(2006).
[CrossRef]

2005 (3)

E. Paspalakis and P. L. Knight, J. Mod. Opt. 52, 1685 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
[CrossRef]

2001 (1)

E. Paspalakis and P. L. Knight, Phys. Rev. A 63, 065802 (2001).
[CrossRef]

2000 (1)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

1997 (1)

S. E. Harris, Phys. Today 50, 36 (1997).
[CrossRef]

1996 (1)

C. Girard and A. Dereux, Rep. Prog. Phys. 59, 657(1996).
[CrossRef]

1992 (1)

1976 (1)

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal and K. T. Kapale, J. Phys. B 39, 3437(2006).
[CrossRef]

Ahufinger, V.

J. Mompart, V. Ahufinger, and G. Birkl, Phys. Rev. A 79, 053638 (2009).
[CrossRef]

Arimondo, E.

E. Arimondo, in Progress in Optics, E.Wolf, ed. (Elsevier Science B. V., 1996), Vol. XXXV, pp. 257–354.
[CrossRef]

Birkl, G.

J. Mompart, V. Ahufinger, and G. Birkl, Phys. Rev. A 79, 053638 (2009).
[CrossRef]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Courjon, D.

de Chatellus, H. G.

Dereux, A.

C. Girard and A. Dereux, Rep. Prog. Phys. 59, 657(1996).
[CrossRef]

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Evers, J.

M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
[CrossRef] [PubMed]

M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Fujita, K.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

Girard, C.

C. Girard and A. Dereux, Rep. Prog. Phys. 59, 657(1996).
[CrossRef]

Gong, S.

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

Gorshkov, A. V.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

Green, J. T.

D. D. Yavuz, N. A. Proite, and J. T. Green, Phys. Rev. A 79, 055401 (2009).
[CrossRef]

Greiner, M.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris, Phys. Today 50, 36 (1997).
[CrossRef]

Hell, S. W.

S. W. Hell, Science 316, 1153 (2007).
[CrossRef] [PubMed]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Jiang, L.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

Jin, L.

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

Jin, S.

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

Kapale, K. T.

K. T. Kapale and M. S. Zubairy, Phys. Rev. A 73, 023813 (2006).
[CrossRef]

G. S. Agarwal and K. T. Kapale, J. Phys. B 39, 3437(2006).
[CrossRef]

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Kash, M. M.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Kawano, S.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

Kawata, S.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

Keitel, C. H.

M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
[CrossRef]

Kiffner, M.

M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
[CrossRef] [PubMed]

Knight, P. L.

E. Paspalakis and P. L. Knight, J. Mod. Opt. 52, 1685 (2005).
[CrossRef]

E. Paspalakis and P. L. Knight, Phys. Rev. A 63, 065802 (2001).
[CrossRef]

Kobayashi, M.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Lee, J. Y.

Lee, S. K.

Li, H.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Lukin, M. D.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

Macovei, M.

M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
[CrossRef]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Mompart, J.

J. Mompart, V. Ahufinger, and G. Birkl, Phys. Rev. A 79, 053638 (2009).
[CrossRef]

Niu, Y.

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

Park, H. S.

Paspalakis, E.

E. Paspalakis and P. L. Knight, J. Mod. Opt. 52, 1685 (2005).
[CrossRef]

E. Paspalakis and P. L. Knight, Phys. Rev. A 63, 065802 (2001).
[CrossRef]

Pique, J.-P.

Proite, N. A.

D. D. Yavuz, N. A. Proite, and J. T. Green, Phys. Rev. A 79, 055401 (2009).
[CrossRef]

D. D. Yavuz and N. A. Proite, Phys. Rev. A 76, 041802(R) (2007).

Rostovtsev, Y. V.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Sahrai, M.

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Sautenkov, V. A.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Scully, M. O.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Sokolov, A. V.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Stroud, C. R.

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Sun, H.

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

Tajalli, H.

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Vigoureux, J. M.

Welch, G. R.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

Whitley, R. M.

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Yamanaka, M.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

Yavuz, D. D.

D. D. Yavuz, N. A. Proite, and J. T. Green, Phys. Rev. A 79, 055401 (2009).
[CrossRef]

D. D. Yavuz and N. A. Proite, Phys. Rev. A 76, 041802(R) (2007).

Zoller, P.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

Zubairy, M. S.

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
[CrossRef] [PubMed]

M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
[CrossRef]

K. T. Kapale and M. S. Zubairy, Phys. Rev. A 73, 023813 (2006).
[CrossRef]

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (2)

E. Paspalakis and P. L. Knight, J. Mod. Opt. 52, 1685 (2005).
[CrossRef]

L. Jin, H. Sun, Y. Niu, S. Jin, and S. Gong, J. Mod. Opt. 56, 805 (2009).
[CrossRef]

J. Phys. B (1)

G. S. Agarwal and K. T. Kapale, J. Phys. B 39, 3437(2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (9)

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 (2008).
[CrossRef]

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 (2005).
[CrossRef]

K. T. Kapale and M. S. Zubairy, Phys. Rev. A 73, 023813 (2006).
[CrossRef]

M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, Phys. Rev. A 75, 033801 (2007).
[CrossRef]

E. Paspalakis and P. L. Knight, Phys. Rev. A 63, 065802 (2001).
[CrossRef]

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

D. D. Yavuz and N. A. Proite, Phys. Rev. A 76, 041802(R) (2007).

J. Mompart, V. Ahufinger, and G. Birkl, Phys. Rev. A 79, 053638 (2009).
[CrossRef]

D. D. Yavuz, N. A. Proite, and J. T. Green, Phys. Rev. A 79, 055401 (2009).
[CrossRef]

Phys. Rev. Lett. (4)

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, Phys. Rev. Lett. 99, 228105 (2007).
[CrossRef]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
[CrossRef] [PubMed]

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef] [PubMed]

Phys. Today (1)

S. E. Harris, Phys. Today 50, 36 (1997).
[CrossRef]

Rep. Prog. Phys. (1)

C. Girard and A. Dereux, Rep. Prog. Phys. 59, 657(1996).
[CrossRef]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Science (1)

S. W. Hell, Science 316, 1153 (2007).
[CrossRef] [PubMed]

Other (1)

E. Arimondo, in Progress in Optics, E.Wolf, ed. (Elsevier Science B. V., 1996), Vol. XXXV, pp. 257–354.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Atomic level scheme for CPT. (b) Comparison of the measurable feature size within the Rayleigh limit (white circle) and with the CPT scheme (small red circle in the center) for R = 1000 in the x y plane. At this value of R , the improvement in resolution is by a factor of 50.

Fig. 2
Fig. 2

Fully modulated CPT results with standing-wave coupling field profile. (a) Geometry; (b) point-spread functions f i , Eq. (5), versus k x ; (c) 2D FWHMs of the point-spread functions against the background of the density plot of the 2D standing wave ( sin 2 ( k x ) + sin 2 ( k y ) ) near the node. Legend: red solid curve, unmodulated CPT result; blue dashed curve, DC term ( f 0 ); green dotted curve, coefficient of cos 2 ν t ( f 2 ); black dotted–dashed curve, coefficient of cos 4 ν t ( f 4 ); thin black solid line, guide to judge the FWHM in (a). The green and black plots are shifted vertically for easy comparison of their FWHM. The resolution improvement from the Rayleigh limit of λ / 2 to the black circle is by a factor of 1000, and the resolution improvement from the outer red circle (unmodulated CPT result) to the inner black circle (coefficient of the second term in the Fourier expansion) is by a factor of about 23.

Fig. 3
Fig. 3

Fully modulated CPT results for a transverse LG coupling profile. (a) Geometry; (b) point-spread functions f i , Eq. (5), versus k x ; (c) 2D version with the density plot of LG beam in the background. Legend and resolution improvement are the same as in Fig. 2.

Equations (5)

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

H = ( Ω p | 3 1 | + Ω c ( x ) | 2 1 | ) e i Δ t + H . c .
ρ ˙ = i [ H , ρ ] i = 2 , 3 γ 1 i 2 ( | 1 1 | ρ 2 ρ 11 | i i | + ρ | 1 1 | ) ,
ρ 22 ( t ) = | Ω p ( x ) | 2 cos 2 ν t | Ω p ( x ) | 2 cos 2 ν t + | Ω c ( x ) | 2 = l = 1 f 2 l cos ( 2 l ν t ) ,
f 2 l = m = ( 1 ) m 1 ( Ω p ( x ) Ω c ( x ) ) 2 m ( 2 m m ) 2 2 2 m ( 2 m m l ) .
f 0 = ρ 22 ( 0 ) = 1 1 1 + R , f 2 = ρ 22 ( 1 ) = 2 + 4 R 1 + R 4 R , f 4 = ρ 22 ( 2 ) = 2 [ 1 + 8 R ( 1 + R ) ] 1 + R + 8 R ( 1 + 2 R ) ,

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