Abstract

Simplified versions of the communication modes in the Fresnel domain are derived when the system apertures are large. The approximate modes, which are in the form of spherical waves and sinc functions with a spherical curvature, give physical insight into the communication modes approach and the basic limits of free-space optical communication systems. They also show that Gabor’s information theory is readily derived from the communication modes.

© 2007 Optical Society of America

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References

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  1. D. Gabor, in Progress in Optics, E.Wolf, ed. (North-Holland, 1961), Vol. I, p. 109.
    [CrossRef]
  2. B. R. Frieden, in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. VIII, p. 311.
    [CrossRef]
  3. R. Pierri and F. Soldovieri, Inverse Probl. 14, 321 (1998).
    [CrossRef]
  4. D. A. B. Miller, Appl. Opt. 39, 1681 (2000).
    [CrossRef]
  5. A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
    [CrossRef]
  6. R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, J. Opt. Soc. Am. A 18, 352 (2001).
    [CrossRef]
  7. A. Burvall, P. Martinsson, and A. T. Friberg, Opt. Express 12, 377 (2004).
    [CrossRef] [PubMed]
  8. Preliminary results of this work were presented in A. Burvall, P. Martinsson, and A. T. Friberg, Proceedings of the Sixth International Conference on Laser and Fiber-Optical Networks Modeling (LFNM 2004), (IEEE/LEOS Ukraine Chapter, 2004), p. 112.
  9. W. Streifer, J. Opt. Soc. Am. 55, 868 (1965).
    [CrossRef]
  10. D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications (Cambridge U. Press, 1990).
  11. F. Gori and R. Grella, Opt. Commun. 45, 5 (1983).
    [CrossRef]
  12. B. Saleh, Photoelectron Statistics (Springer, 1978).
  13. Since the eigenvalues are equal, suitable linear combinations of terms in Eq. with n and −n yield approximations to the PSWFs, which alternately are symmetric (cosines) or antisymmetric (sines).
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

2004 (1)

2003 (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
[CrossRef]

2001 (1)

2000 (1)

1998 (1)

R. Pierri and F. Soldovieri, Inverse Probl. 14, 321 (1998).
[CrossRef]

1983 (1)

F. Gori and R. Grella, Opt. Commun. 45, 5 (1983).
[CrossRef]

1965 (1)

Burvall, A.

A. Burvall, P. Martinsson, and A. T. Friberg, Opt. Express 12, 377 (2004).
[CrossRef] [PubMed]

Preliminary results of this work were presented in A. Burvall, P. Martinsson, and A. T. Friberg, Proceedings of the Sixth International Conference on Laser and Fiber-Optical Networks Modeling (LFNM 2004), (IEEE/LEOS Ukraine Chapter, 2004), p. 112.

Friberg, A. T.

A. Burvall, P. Martinsson, and A. T. Friberg, Opt. Express 12, 377 (2004).
[CrossRef] [PubMed]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
[CrossRef]

Preliminary results of this work were presented in A. Burvall, P. Martinsson, and A. T. Friberg, Proceedings of the Sixth International Conference on Laser and Fiber-Optical Networks Modeling (LFNM 2004), (IEEE/LEOS Ukraine Chapter, 2004), p. 112.

Frieden, B. R.

B. R. Frieden, in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. VIII, p. 311.
[CrossRef]

Gabor, D.

D. Gabor, in Progress in Optics, E.Wolf, ed. (North-Holland, 1961), Vol. I, p. 109.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gori, F.

F. Gori and R. Grella, Opt. Commun. 45, 5 (1983).
[CrossRef]

Grella, R.

F. Gori and R. Grella, Opt. Commun. 45, 5 (1983).
[CrossRef]

Karelin, M.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
[CrossRef]

Liseno, A.

Martinsson, P.

A. Burvall, P. Martinsson, and A. T. Friberg, Opt. Express 12, 377 (2004).
[CrossRef] [PubMed]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
[CrossRef]

Preliminary results of this work were presented in A. Burvall, P. Martinsson, and A. T. Friberg, Proceedings of the Sixth International Conference on Laser and Fiber-Optical Networks Modeling (LFNM 2004), (IEEE/LEOS Ukraine Chapter, 2004), p. 112.

Miller, D. A. B.

Pierri, R.

Porter, D.

D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications (Cambridge U. Press, 1990).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, 1978).

Soldovieri, F.

Solimene, R.

Stirling, D. S. G.

D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications (Cambridge U. Press, 1990).

Streifer, W.

Thaning, A.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
[CrossRef]

Appl. Opt. (1)

Inverse Probl. (1)

R. Pierri and F. Soldovieri, Inverse Probl. 14, 321 (1998).
[CrossRef]

J. Opt. A (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, J. Opt. A 5, 153 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

F. Gori and R. Grella, Opt. Commun. 45, 5 (1983).
[CrossRef]

Opt. Express (1)

Other (7)

Preliminary results of this work were presented in A. Burvall, P. Martinsson, and A. T. Friberg, Proceedings of the Sixth International Conference on Laser and Fiber-Optical Networks Modeling (LFNM 2004), (IEEE/LEOS Ukraine Chapter, 2004), p. 112.

D. Porter and D. S. G. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications (Cambridge U. Press, 1990).

D. Gabor, in Progress in Optics, E.Wolf, ed. (North-Holland, 1961), Vol. I, p. 109.
[CrossRef]

B. R. Frieden, in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. VIII, p. 311.
[CrossRef]

B. Saleh, Photoelectron Statistics (Springer, 1978).

Since the eigenvalues are equal, suitable linear combinations of terms in Eq. with n and −n yield approximations to the PSWFs, which alternately are symmetric (cosines) or antisymmetric (sines).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (2)

Fig. 1
Fig. 1

Illustration of the geometry and notations. The Fresnel domain communication modes a n ( x T ) and b n ( x R ) are normalized PSWFs with quadratic phase factors in both apertures.

Fig. 2
Fig. 2

The approximate communication modes a a n ( x T ) are converging spherical waves, and they create the modes b a n ( x R ) that are sinc functions with quadratic phase factors in the receiving domain.

Equations (20)

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U R ( x R ) = Δ x T Δ x T G ( x R , x T ) U T ( x T ) d x T ,
G ( x R , x T ) = exp ( i k z ) i λ z exp [ i k ( x R x T ) 2 2 z ] ,
a ( x T ) = U T ( x T ) exp ( i k x T 2 2 z ) ,
b ( x R ) = U R ( x R ) exp ( i k x R 2 2 z ) ,
b ( x R ) = Δ x T Δ x T g ( x R , x T ) a ( x T ) d x T ,
g ( x R , x T ) = exp ( i k z ) i λ z exp ( i k x R x T z ) .
g ( x R , x T ) = n = 0 N g n β n ( x R ) α n * ( x T ) ,
Δ x T Δ x T g ( x R , x T ) α n ( x T ) d x T = g n β n ( x R ) ,
Δ x R Δ x R g * ( x R , x T ) β n ( x R ) d x R = g n * α n ( x T ) ,
g n 2 α n ( x T ) = Δ x T Δ x T sin [ Ω T ( x T x T ) ] π ( x T x T ) α n ( x T ) d x T ,
a ( x T ) = n a ̃ n α n ( x T ) ,
a ̃ n = Δ x T Δ x T a ( x T ) α n * ( x T ) d x T ,
U T ( x T ) = n a ̃ n a n ( x T ) = n a ̃ n α n ( x T ) exp ( i k x T 2 2 z ) ,
U R ( x R ) = n b ̃ n b n ( x R ) = n g n a ̃ n β n ( x R ) exp ( i k x R 2 2 z ) ,
α a n ( x T ) = 1 2 Δ x T exp ( i n π x T Δ x T ) ,
g a n = 1 , when n n max = 2 Δ x T Δ x R λ z ,
β a n ( x R ) = exp ( i k z ) i λ z 2 Δ x T sinc [ k Δ x T z ( x R n λ z 2 Δ x T ) ] ,
a a n ( x T ) = α a n ( x T ) exp ( i k x T 2 2 z ) ,
b a n ( x R ) = β a n ( x R ) exp ( i k x R 2 2 z ) .
N = 2 Δ x T 2 Δ x R λ z

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