Abstract

Superpositions of coaxial Laguerre–Gaussian modes with common waists and the same Gaussian envelopes are known to form beams whose transverse structures experience the self-similar transformation during free beam propagation: the beam shape remains the same except for the divergence and rotation around the propagation axis. We show that under certain conditions this transformation can be represented as if every point of the beam cross section performs a centrifugal straight-line fly off.

© 2006 Optical Society of America

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  1. M. V. Berry, in Physics of Defects (Les Houches Lecture Series Session XXXV), R.Balian, M.Kléman, and J.-P.Poirier, eds. (North-Holland, 1981), p. 453.
  2. M. S. Soskin and M. V. Vasnetsov, Prog. Opt. 42, 219 (2001).
    [CrossRef]
  3. V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).
  4. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
    [CrossRef]
  5. L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
    [CrossRef]
  6. I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
    [CrossRef]
  7. M. Berry, in Proc. SPIE 3487, 6 (1998).
    [CrossRef]
  8. G. Molina-Terriza, E. M. Wright, and L. Torner, Opt. Lett. 26, 163 (2001).
    [CrossRef]
  9. F. S. Roux, Opt. Commun. 236, 433 (2004).
    [CrossRef]
  10. F. S. Roux, Opt. Commun. 242, 45 (2004).
    [CrossRef]
  11. F. S. Roux, J. Opt. Soc. Am. B 21, 664 (2004).
    [CrossRef]
  12. M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
    [CrossRef]
  13. A. Ya. Bekshaev, in Proc. SPIE 3904, 131 (1999).
    [CrossRef]
  14. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  15. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. Soc. Am. A 20, 1635 (2003).
    [CrossRef]
  16. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. A Pure Appl. Opt. 6, S170 (2004).
    [CrossRef]
  17. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 249, 367 (2005).
    [CrossRef]
  18. C. N. Alexeyev and M. A. Yavorsky, J. Opt. A Pure Appl. Opt. 7, 416 (2005).
    [CrossRef]
  19. V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, Pis'ma Zh. Tekh. Fiz. 23, 1 (1997) (in Russian).
    [CrossRef]
  20. D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
    [CrossRef]
  21. A. Desyatnikov, C. Denz, and Y. Kivshar, J. Opt. A Pure Appl. Opt. 6, S209 (2004).
    [CrossRef]
  22. D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., J. Opt. Soc. Am. B 14, 3054 (1997).
    [CrossRef]
  23. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
    [CrossRef]
  24. G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
    [CrossRef]
  25. A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).
  26. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
    [CrossRef] [PubMed]
  27. A. V. Volyar, V. G. Shvedov, and T. A. Fadeeva, Tech. Phys. Lett. 25, 891 (1999).
    [CrossRef]
  28. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
    [CrossRef]

2005

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 249, 367 (2005).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, J. Opt. A Pure Appl. Opt. 7, 416 (2005).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
[CrossRef]

2004

A. Desyatnikov, C. Denz, and Y. Kivshar, J. Opt. A Pure Appl. Opt. 6, S209 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. A Pure Appl. Opt. 6, S170 (2004).
[CrossRef]

F. S. Roux, Opt. Commun. 236, 433 (2004).
[CrossRef]

F. S. Roux, Opt. Commun. 242, 45 (2004).
[CrossRef]

F. S. Roux, J. Opt. Soc. Am. B 21, 664 (2004).
[CrossRef]

2003

2002

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

2001

1999

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

A. Ya. Bekshaev, in Proc. SPIE 3904, 131 (1999).
[CrossRef]

A. V. Volyar, V. G. Shvedov, and T. A. Fadeeva, Tech. Phys. Lett. 25, 891 (1999).
[CrossRef]

1998

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

M. Berry, in Proc. SPIE 3487, 6 (1998).
[CrossRef]

1997

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, Pis'ma Zh. Tekh. Fiz. 23, 1 (1997) (in Russian).
[CrossRef]

D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., J. Opt. Soc. Am. B 14, 3054 (1997).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

1995

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

1993

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

1990

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Alexeyev, C. N.

C. N. Alexeyev and M. A. Yavorsky, J. Opt. A Pure Appl. Opt. 7, 416 (2005).
[CrossRef]

Allen, L.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
[CrossRef] [PubMed]

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 249, 367 (2005).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. A Pure Appl. Opt. 6, S170 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. Soc. Am. A 20, 1635 (2003).
[CrossRef]

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
[CrossRef] [PubMed]

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

A. Ya. Bekshaev, in Proc. SPIE 3904, 131 (1999).
[CrossRef]

Berry, M.

M. Berry, in Proc. SPIE 3487, 6 (1998).
[CrossRef]

Berry, M. V.

M. V. Berry, in Physics of Defects (Les Houches Lecture Series Session XXXV), R.Balian, M.Kléman, and J.-P.Poirier, eds. (North-Holland, 1981), p. 453.

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
[CrossRef]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
[CrossRef]

Denz, C.

A. Desyatnikov, C. Denz, and Y. Kivshar, J. Opt. A Pure Appl. Opt. 6, S209 (2004).
[CrossRef]

Desyatnikov, A.

A. Desyatnikov, C. Denz, and Y. Kivshar, J. Opt. A Pure Appl. Opt. 6, S209 (2004).
[CrossRef]

Fadeeva, T. A.

A. V. Volyar, V. G. Shvedov, and T. A. Fadeeva, Tech. Phys. Lett. 25, 891 (1999).
[CrossRef]

Firth, W. J.

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

Freund, I.

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Indebetouw, G.

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, Pis'ma Zh. Tekh. Fiz. 23, 1 (1997) (in Russian).
[CrossRef]

Kivshar, Y.

A. Desyatnikov, C. Denz, and Y. Kivshar, J. Opt. A Pure Appl. Opt. 6, S209 (2004).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, Pis'ma Zh. Tekh. Fiz. 23, 1 (1997) (in Russian).
[CrossRef]

Law, C. T.

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
[CrossRef]

MacVicar, I.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Molina-Terriza, G.

O'Neil, A. T.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Padgett, M. J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
[CrossRef]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

Roux, F. S.

F. S. Roux, Opt. Commun. 236, 433 (2004).
[CrossRef]

F. S. Roux, Opt. Commun. 242, 45 (2004).
[CrossRef]

F. S. Roux, J. Opt. Soc. Am. B 21, 664 (2004).
[CrossRef]

Rozas, D.

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

Shvedov, V. G.

A. V. Volyar, V. G. Shvedov, and T. A. Fadeeva, Tech. Phys. Lett. 25, 891 (1999).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

Slyusar, V. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
[CrossRef] [PubMed]

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

Soifer, V. A.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, Pis'ma Zh. Tekh. Fiz. 23, 1 (1997) (in Russian).
[CrossRef]

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 249, 367 (2005).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. A Pure Appl. Opt. 6, S170 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. Soc. Am. A 20, 1635 (2003).
[CrossRef]

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
[CrossRef] [PubMed]

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

M. S. Soskin and M. V. Vasnetsov, Prog. Opt. 42, 219 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Swartzlander, G. A.

Torner, L.

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 249, 367 (2005).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. A Pure Appl. Opt. 6, S170 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. Soc. Am. A 20, 1635 (2003).
[CrossRef]

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
[CrossRef] [PubMed]

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

M. S. Soskin and M. V. Vasnetsov, Prog. Opt. 42, 219 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Volyar, A. V.

A. V. Volyar, V. G. Shvedov, and T. A. Fadeeva, Tech. Phys. Lett. 25, 891 (1999).
[CrossRef]

Wright, E. M.

Yavorsky, M. A.

C. N. Alexeyev and M. A. Yavorsky, J. Opt. A Pure Appl. Opt. 7, 416 (2005).
[CrossRef]

J. Mod. Opt.

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

J. Opt. A Pure Appl. Opt.

A. Desyatnikov, C. Denz, and Y. Kivshar, J. Opt. A Pure Appl. Opt. 6, S209 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. A Pure Appl. Opt. 6, S170 (2004).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, J. Opt. A Pure Appl. Opt. 7, 416 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

JETP Lett.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

New J. Phys.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005).
[CrossRef]

Opt. Commun.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

F. S. Roux, Opt. Commun. 236, 433 (2004).
[CrossRef]

F. S. Roux, Opt. Commun. 242, 45 (2004).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 249, 367 (2005).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Phys. Rev. E

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

Phys. Rev. Lett.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Pis'ma Zh. Tekh. Fiz.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, Pis'ma Zh. Tekh. Fiz. 23, 1 (1997) (in Russian).
[CrossRef]

Proc. SPIE

A. Ya. Bekshaev, in Proc. SPIE 3904, 131 (1999).
[CrossRef]

M. Berry, in Proc. SPIE 3487, 6 (1998).
[CrossRef]

Prog. Opt.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, Prog. Opt. 42, 219 (2001).
[CrossRef]

Tech. Phys. Lett.

A. V. Volyar, V. G. Shvedov, and T. A. Fadeeva, Tech. Phys. Lett. 25, 891 (1999).
[CrossRef]

Ukr. J. Phys.

A. Ya. Bekshaev, I. V. Basistiy, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, Ukr. J. Phys. 47, 1035 (2002).

Other

M. V. Berry, in Physics of Defects (Les Houches Lecture Series Session XXXV), R.Balian, M.Kléman, and J.-P.Poirier, eds. (North-Holland, 1981), p. 453.

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Figures (1)

Fig. 1
Fig. 1

Evolution of the transverse profile for a two-term CB of the form of Eq. (11) with a positive sign in the last exponent and nonzero coefficients C 00 = 1 , C 03 = 1.22 . The central inset presents the waist plane, main area plane z = z R . In the initial (waist) cross-section point r 0 is chosen that belongs to the contour (gray) of equal intensity at the level 20% of maximum; its current image (point r) lies on the corresponding contour (black) of the current transverse profile.

Equations (11)

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

u ( r , ϕ , z ) = p = 0 l = C p l u p l ( r , ϕ , z ) ,
u p l ( r , ϕ , z ) = 1 b p ! π ( p + l ) ! ( r b ) l L p l ( r 2 b 2 ) exp ( r 2 2 b 2 + i k r 2 2 R ) exp { i [ l ϕ ( 2 p + l + 1 ) ψ ] }
R ( z ) = z R 2 + z 2 z , b 2 ( z ) = z R 2 + z 2 k z R , ψ ( z ) = arctan ( z z R )
B = 2 ( p p ) + l l l l = const
ϕ ( z ) = ϕ 0 + B ψ ( z ) .
r ( z ) = r 0 b ( z ) b 0 .
ϕ ( r ) = ϕ 0 + B arccos ( r 0 r ) .
B = 1 .
ϕ ( r ) = ϕ 0 ± arccos ( r 0 r ) .
d r r 0 d z = d d z r 0 tan ψ = r 0 z R .
u ( r , ϕ , z ) = 1 b exp ( r 2 2 b 2 + i k r 2 2 R i ψ ) n = 0 C 0 n π n ! [ r e i ( ± ϕ ψ ) b ] n ,

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