Abstract

The communication modes, which constitute a convenient method for the propagation and information analysis of optical fields, are formulated in the generalized axicon geometry. The transmitting region is the axicon’s annular aperture, and the observation domain is the optical axis containing the focal line segment. We show that in rotational symmetry one may employ the prolate spheroidal wave functions to represent the communication modes. Further, in usual circumstances the modes can be approximated by quadratic waves in the aperture domain and by sinc functions in the image domain. Both the exact communication modes and the approximate technique are confirmed numerically, with linear axicons as examples.

© 2004 Optical Society of America

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References

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    [CrossRef]
  2. L.M. Soroko, Meso-Optics �?? Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2 and references therein.
  3. Z. Jaroszewicz, Axicons: Design and Propagation Properties, Research & Development Treatises, Vol. 5 (SPIE Polish Chapter, Warsaw, 1997).
  4. J.A. Davis, E. Carcole, and D.M. Cottrell, �??Range-finding by triangulation with nondifracting beams,�?? Appl. Opt. 35, 2159�??2161 (1996).
    [CrossRef] [PubMed]
  5. G. Haüsler and W. Heckel, �??Light sectioning with large depth and high resolution,�?? Appl. Opt. 27, 5165�??5169 (1988).
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  6. V.E. Peet and R.V. Tusbin, �??Third-harmonic generation and multiphoton ionization in Bessel beams,�?? Phys. Rev. A 56, 1613�??1620 (1997).
    [CrossRef]
  7. J. Arlt, T. Hitomi, and K. Dholakia, �??Atom guiding along Laguerre-Gaussian and Bessel light beams,�?? Appl. Phys. B 71, 549�??554 (2000).
    [CrossRef]
  8. Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, �??High resolution optical coherence tomography over a large depth range with an axicon lens,�?? Opt. Lett. 27, 243�??245 (2002).
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  9. F. Gori and R. Grella, �??The converging prolate spheroidal functions and their use in Fresnel optics,�?? Opt. Commun. 45, 5�??10 (1983).
    [CrossRef]
  10. R. Pierri and F. Soldovieri, �??On the information content of the radiated fields in the near zone over bounded domains,�?? Inverse Problems 14, 321-337 (1998).
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    [CrossRef]
  13. A. Thaning, P. Martinsson, M. Karelin, and A.T. Friberg, �??Limits of diffractive optics by communication modes,�?? J. Opt. A: Pure Appl. Opt. 5 153�??158 (2003).
    [CrossRef]
  14. R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, �??In-depth resolution for a strip source in the Fresnel zone,�?? J. Opt. Soc. Am. A 18, 352-359 (2001).
    [CrossRef]
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  16. D. Slepian and H.O. Pollak, �??Prolate spheroidal wave functions, Fourier analysis and uncertainty �?? I,�?? Bell Syst. Techn. J. 40, 43�??63 (1961).
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Appl. Opt (2)

J.A. Davis, E. Carcole, and D.M. Cottrell, �??Range-finding by triangulation with nondifracting beams,�?? Appl. Opt. 35, 2159�??2161 (1996).
[CrossRef] [PubMed]

G. Haüsler and W. Heckel, �??Light sectioning with large depth and high resolution,�?? Appl. Opt. 27, 5165�??5169 (1988).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

J. Arlt, T. Hitomi, and K. Dholakia, �??Atom guiding along Laguerre-Gaussian and Bessel light beams,�?? Appl. Phys. B 71, 549�??554 (2000).
[CrossRef]

Bell Syst. Techn. J. (2)

D. Slepian and H.O. Pollak, �??Prolate spheroidal wave functions, Fourier analysis and uncertainty �?? I,�?? Bell Syst. Techn. J. 40, 43�??63 (1961).

H.J. Landau and H.O. Pollak, �??Prolate spheroidal wave functions, Fourier analysis and uncertainty �?? II,�?? Bell Syst. Techn. J. 40, 65�??84 (1961).

Inverse Problems (1)

R. Pierri and F. Soldovieri, �??On the information content of the radiated fields in the near zone over bounded domains,�?? Inverse Problems 14, 321-337 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

A. Thaning, P. Martinsson, M. Karelin, and A.T. Friberg, �??Limits of diffractive optics by communication modes,�?? J. Opt. A: Pure Appl. Opt. 5 153�??158 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

F. Gori and R. Grella, �??The converging prolate spheroidal functions and their use in Fresnel optics,�?? Opt. Commun. 45, 5�??10 (1983).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

V.E. Peet and R.V. Tusbin, �??Third-harmonic generation and multiphoton ionization in Bessel beams,�?? Phys. Rev. A 56, 1613�??1620 (1997).
[CrossRef]

Progress in Optics (1)

B.R. Frieden, �??Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,�?? in Progress in Optics, Vol. IX, E. Wolf, editor (North-Holland, Amsterdam, 1971), pp. 311�??407.
[CrossRef]

Other (3)

B. Saleh, Photoelectron Statistics (Springer-Verlag, Heidelberg, 1978), chap. 2.4.4.

L.M. Soroko, Meso-Optics �?? Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2 and references therein.

Z. Jaroszewicz, Axicons: Design and Propagation Properties, Research & Development Treatises, Vol. 5 (SPIE Polish Chapter, Warsaw, 1997).

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Figures (4)

Fig. 1.
Fig. 1.

Geometry and notations related to annular axicon imaging.

Fig. 2.
Fig. 2.

On-axis intensity distribution produced by a linear axicon, calculated using the communication modes. The system parameters are γ=0.025, R 1=5 mm, R 2=10 mm, d 1=150 mm and d 2=450 mm, and the wavelength is λ=600 nm. The number of modes is N=278.

Fig. 3.
Fig. 3.

On-axis intensity profile produced by a linear axicon, calculated using both the exact and the approximate communication modes technique. The two graphs overlap almost completely. The parameters are the same as in Fig. 2.

Fig. 4.
Fig. 4.

Relative error of the approximation in Eqs. (18)(21), as a function of the number of degrees of freedom. The geometry and the axicon phase function are the same as in Figs. 2 and 3. The wavelength is changed to alter N.

Equations (23)

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E out ( z ) = k exp ( i kz ) i z R 1 R 2 E in ( ρ ) ρ exp ( i k ρ 2 2 z ) d ρ .
u = ρ 2 ( R 1 2 + R 2 2 ) / 2 ,
E out ( z ) = k exp ( i kz ) 2 i z exp ( i k 2 z R 1 2 + R 2 2 2 ) U U E ˜ in ( u ) exp ( i k 2 z u ) d u ,
G ( z , u ) = k exp ( i kz ) 2 i z exp ( i k 2 z R 1 2 + R 2 2 2 ) exp ( i k 2 z u ) ,
g n 2 a n ( u ) = U U K ( u , u ) a n ( u ) d u ,
K ( u , u ) = d 1 d 2 G * ( z , u ) G ( z , u ) d z ,
g n 2 α n ( u ) = k π U U sin [ k D diff ( u u ) / 2 ] π ( u u ) α n ( u ) d u .
N = k U D diff π = 1 2 λ ( R 2 2 R 1 2 ) ( 1 d 1 1 d 2 ) .
g n b n ( z ) = U U G ( z , u ) a n ( u ) d u .
a n ( u ) = exp ( i k D sum 2 u ) α n ( u ) ,
b n ( z ) = 1 z exp ( i kz ) exp ( i k 2 z R 1 2 + R 2 2 2 ) β n ( 1 z D sum ) ,
g n = i ( n 1 ) k π · λ n ,
G ( z , u ) n = 1 N g n a n * ( u ) b n ( z ) ,
E out ( z ) = n = 1 N g n A n b n ( z ) ,
A n = U U a n * ( u ) E ˜ in ( u ) d u ,
f z = 1 Δ z = 1 z ( R 2 2 R 1 2 2 λ z 1 ) .
N 2 .
a n ( u ) = exp ( i k D sum 2 u ) 1 2 U exp [ i ( n N 2 ) π U u ] ,
g n = { i k π : n N , 0 : n > N .
g n b n ( z ) = k exp ( i kz ) 2 i z 2 U exp ( i k 2 z R 1 2 + R 2 2 2 )
× U U exp { i [ k U 2 ( 1 z D sum ) ( n N 2 ) π ] u U } d u .
b n ( z ) = kU 2 π 1 z exp ( i kz ) exp ( i k 2 z R 1 2 + R 2 2 2 ) sin [ kU ( 1 / z D sum ) / 2 ( n N / 2 ) π ] kU ( 1 / z D sum ) / 2 ( n N / 2 ) π .
ε = d 1 d 2 I cm ( z ) I appr ( z ) 2 dz / d 1 d 2 I cm ( z ) 2 dz ,

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