Abstract

This work describes the use of a generalized modal scattering matrix theory as a fast, efficient approach to the analysis of incoherent Doppler lidars. The new technique uses Bessel beams, a type of optical vortices, as the basic modal expansion characterizing optical signals. The tactic allows solving both multilayered reflections problems and spatial diffraction phenomena using scattering parameters associated with the transmitted and reflected spectrum of vortices. Here, we will show the capabilities of the technique by considering realistic incoherent Doppler systems based on Fabry-Perot etalons.

© 2006 Optical Society of America

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References

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  1. J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design (Artech House, Boston, 1991).
  2. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966).
  3. K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans.Microwave Theory Tech. 13, 194-202 (1965).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).
  5. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4,651-654 (1986).
    [CrossRef]
  6. J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  7. A. Yariv, Introduction to Optical Electronics (Holt, Reinhart and Winston, New York, 1985).
  8. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (John Wiley & Sons, Inc., New York, 1991).
  9. V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974).
    [CrossRef]
  10. D. M. Pozart, Microwave Engineering (Wiley, New York, 2004).
  11. G. Hernandez, Fabry-Perot Interferometers (Cambridge U.Press, Cambridge, UK, 1988).
  12. J.A. McKay, "Single and tandem Fabry-Perot Etalons as solar background filters for lidar," Appl. Opt. 38, 5851-5857 (1999).
    [CrossRef]

1999 (1)

1987 (1)

J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1986 (1)

1974 (1)

V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974).
[CrossRef]

1965 (1)

K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans.Microwave Theory Tech. 13, 194-202 (1965).
[CrossRef]

Durnin, J.

J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4,651-654 (1986).
[CrossRef]

Eberly, J.H.

J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Kurokawa, K.

K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans.Microwave Theory Tech. 13, 194-202 (1965).
[CrossRef]

McKay, J.A.

Miceli, J.J.

J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Monaco, V.A.

V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974).
[CrossRef]

Tiberio, P.

V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (1)

V.A. Monaco, P. Tiberio, "Computer-Aided Analysis of Microwave Circuits," IEEE Trans. Microwave Theory Tech. 22, 249-263 (1974).
[CrossRef]

IEEE Trans.Microwave Theory Tech. (1)

K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans.Microwave Theory Tech. 13, 194-202 (1965).
[CrossRef]

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

Phys. Rev. Lett. (1)

J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Other (7)

A. Yariv, Introduction to Optical Electronics (Holt, Reinhart and Winston, New York, 1985).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (John Wiley & Sons, Inc., New York, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Boston, 1996).

J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design (Artech House, Boston, 1991).

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, New York, 1966).

D. M. Pozart, Microwave Engineering (Wiley, New York, 2004).

G. Hernandez, Fabry-Perot Interferometers (Cambridge U.Press, Cambridge, UK, 1988).

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Figures (4)

Fig. 1
Fig. 1

Outgoing and incoming waves in two-port optical network. We denote the incoming and outgoing waves at ports i by the L=(2M+1)×(N+1)-component vectors ai, and bi, respectively

Fig. 2.
Fig. 2.

A Fabry-Perot interferometer can be analyzed as a multiport optical system composed of three different two-port elements. Waves b2 transmitted by the first reflective layer propagated a distance d to reach the second reflective layer as incoming waves a5 . By properly defining the scattering matrix of any of these three elements, our modal approach allows to describes any interference and diffractive problem characterizing the Fabry-Perot etalon behavior.

Fig. 3.
Fig. 3.

Flow chart for a Fabry-Perot interferometer. In bold line, the signal path for an input wave that propagates to the output of the etalon. To find the transfer function of the etalon system represented by this block diagram we use Mason’s gain rule.

Fig. 4.
Fig. 4.

Flow chart (up) for two etalons with a attenuator used to reduce reflections between etalons. In bold line, it is market the signal path for an input wave that propagate to the output of the second etalon. Transmittance for the dual-etalon solar filter without absorber, with 5% absorber, and isolated etalons omitting reflections (down).

Equations (6)

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U r φ = m = M M n = 0 N f mn J m ( v m n r ) e j m φ ,
S ji , mn = b j , n a i , m | a k , v = 0 , k i , v m
( b 1 b 2 ) = S ( a 1 a 2 ) = ( R 11 T 12 T 21 R 22 ) ( a 1 a 2 )
S 11 ( v m n ) = S 22 ( v m n ) = R + R exp [ jkd 1 ( λ v m n ) 2 ] 1 R exp [ jkd 1 ( λ v m n ) 2 ]
S 21 ( v m n ) = S 12 ( v m n ) = 1 R 2 exp [ jkd 1 ( λ v m n ) 2 ] 1 R exp [ jkd 1 ( λ v m n ) 2 ]
T ( ρ l ) = S 21 ( ρ l ) = T 1 τ 21 T 2 1 τ 21 R 2 τ 12 R 1

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