Abstract

We propose, analyze, and demonstrate the use of a holographic method for cohering the output of a fiber tapped delay line (FTDL) that enables the use of fiber-remote optical modulators in coherent optical processing systems. We perform a theoretical examination of the phase-cohering process and show experimental results for a radio frequency (RF) spectrum analyzer that uses a lens to spatially Fourier transform the output of a holographically phase-cohered FTDL providing 50 MHz resolution and bandwidths approaching 3 GHz. Substantial improvements in bandwidth should be achievable with better fiber length-trimming accuracy and improvements in resolution can be obtained with longer fiber delay lines. We also analyze and demonstrate the use of a parallel holographic technique that compensates for polarization state scrambling induced by propagation through an array of single-mode fibers. Both the phase-cohering holography and the polarization fluctuation compensation can operate on hundreds of fibers in parallel, enabling both coherent optical signal processing with FTDLs and coherent fiber remoting of optically modulated RF signals from antenna arrays.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Ramaswami, K. N. Sivarajan, Optical Networks: a Practical Perspective (Morgan Kaufmann, 1998).
  2. J. T. Mayhan, A. J. Simmons, W. C. Cummings, “Wideband adaptive antenna nulling using tapped delay-lines,” IEEE Trans. Antennas Propag. AP-29, 923–936 (1981).
    [CrossRef]
  3. A. K. Ghosh, J. Trepka, “Design of fiber optic adaline neural networks,” Opt. Eng. 36, 843–848 (1997).
    [CrossRef]
  4. P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).
    [CrossRef] [PubMed]
  5. T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
    [CrossRef]
  6. K. Wilner, A. P. van den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807 (1976).
    [CrossRef]
  7. K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
    [CrossRef]
  8. E. Anemogiannis, R. P. Kenan, “Integrated optical architectures for tapped delay-lines,” J. Lightwave Technol. 8, 1167–1176 (1990).
    [CrossRef]
  9. D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Appl. Opt. 23, 3475–3481 (1984).
    [CrossRef] [PubMed]
  10. G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).
    [CrossRef]
  11. I. C. Chang, D. L. Hecht, “Characteristics of acousto-optic devices for signal processors,” Opt. Eng. 21, 76–81 (1982).
    [CrossRef]
  12. R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
    [CrossRef]
  13. T. W. Bristol, P. J. Hagon, “Programmable surface acoustic-wave tapped delay-lines,” IEEE Trans. Sonics Ultrason. SU-19, 414 (1972).
  14. J. W. Goodman, “Fan-in and fan-out with optical interconnections,” Opt. Acta 32, 1489–1496 (1985).
    [CrossRef]
  15. R. van Dijk, J. D. Bregman, A. Roodnat, F. E. van Vliet, “Photonic true time delay beamformer demonstration for a radio astronomical array antenna,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2000), pp. 78–80.
  16. T. Turpin, “Spectrum analysis using optical processing.” Proc. IEEE 69, 80–92 (1981).
    [CrossRef]
  17. D. E. N. Davies, G. W. James, “Fiber-optic tapped delay line filter employing coherent optical processing,” Electron. Lett. 20, 95–97 (1984).
    [CrossRef]
  18. K. P. Jackson, G. Xiao, H. J. Shaw, “Coherent optical fibre delay-line processor,” Electron. Lett. 22, 1335–1337 (1986).
    [CrossRef]
  19. M. Shadaram, J. Medrano, S. A. Pappert, M. H. Berry, D. M. Gookin, “Technique for stabilizing the phase of the reference signals in analog fiber-optic links,” Appl. Opt. 34, 8283–8288 (1995).
    [CrossRef] [PubMed]
  20. R. T. Weverka, K. Wagner, A. Sarto, “Photorefractive processing for large adaptive phased arrays,” Appl. Opt. 35, 1344–1366 (1996).
    [CrossRef] [PubMed]
  21. J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
    [CrossRef]
  22. A. Kiruluta, G. S. Pati, G. Kriehn, P. E. X. Silveira, A. W. Sarto, K. Wagner, “Spatio-temporal operator formalism for holographic recording and diffraction in a photorefractive-based true-time-delay phased-array processor,” Appl. Opt. 42, 5334–5350 (2003).
    [CrossRef] [PubMed]
  23. K. Noguchi, H. Miyazawa, O. Mitomi, “Frequency-dependent propagation characteristics of coplanar waveguide electrode on 100 GHz Ti:LiNbO3 optical modulator,” Electron. Lett. 34, 661–663 (1998).
    [CrossRef]
  24. D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
    [CrossRef]
  25. W. T. Rhodes, “Acousto-optic signal processing: convolution and correlation,” Proc. IEEE 69, 65–79 (1981).
    [CrossRef]
  26. A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).
    [CrossRef] [PubMed]
  27. A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
    [CrossRef]
  28. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, 1993).
  29. L. G. Kazovsky, “Phase- and polarization-diversity coherent optical techniques,” J. Lightwave Technol. 7, 279–292 (1989).
    [CrossRef]
  30. A. E. Willner, S. M. R. M. Nezam, L. S. Yan, Z. Q. Pan, M. C. Hauer, “Monitoring and control of polarization-related impairments in optical fiber systems,” J. Lightwave Technol. 22, 106–125 (2004).
    [CrossRef]
  31. A. P. Goutzoulis, D. K. Davies, “Hardware-compressive 2-D fiber optic delay line architecture for time steering of phase-array antennas,” Appl. Opt. 29, 5353–5359 (1990).
    [CrossRef] [PubMed]
  32. B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
    [CrossRef]
  33. T. M. Turpin, F. F. Froehlich, D. B. Nichols, “Optical tapped delay line,” U.S. Patent6,608,721 (19August2003).
  34. P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optoelectronic implementation of a 256-channel sonar adaptive-array processor,” Appl. Opt. 43, 6421–6439 (2004).
    [CrossRef] [PubMed]

2004

2003

2002

2000

1998

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

K. Noguchi, H. Miyazawa, O. Mitomi, “Frequency-dependent propagation characteristics of coplanar waveguide electrode on 100 GHz Ti:LiNbO3 optical modulator,” Electron. Lett. 34, 661–663 (1998).
[CrossRef]

1997

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

A. K. Ghosh, J. Trepka, “Design of fiber optic adaline neural networks,” Opt. Eng. 36, 843–848 (1997).
[CrossRef]

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

1996

1995

1990

A. P. Goutzoulis, D. K. Davies, “Hardware-compressive 2-D fiber optic delay line architecture for time steering of phase-array antennas,” Appl. Opt. 29, 5353–5359 (1990).
[CrossRef] [PubMed]

E. Anemogiannis, R. P. Kenan, “Integrated optical architectures for tapped delay-lines,” J. Lightwave Technol. 8, 1167–1176 (1990).
[CrossRef]

1989

L. G. Kazovsky, “Phase- and polarization-diversity coherent optical techniques,” J. Lightwave Technol. 7, 279–292 (1989).
[CrossRef]

1986

K. P. Jackson, G. Xiao, H. J. Shaw, “Coherent optical fibre delay-line processor,” Electron. Lett. 22, 1335–1337 (1986).
[CrossRef]

1985

J. W. Goodman, “Fan-in and fan-out with optical interconnections,” Opt. Acta 32, 1489–1496 (1985).
[CrossRef]

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

1984

D. E. N. Davies, G. W. James, “Fiber-optic tapped delay line filter employing coherent optical processing,” Electron. Lett. 20, 95–97 (1984).
[CrossRef]

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Appl. Opt. 23, 3475–3481 (1984).
[CrossRef] [PubMed]

1983

R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
[CrossRef]

1982

I. C. Chang, D. L. Hecht, “Characteristics of acousto-optic devices for signal processors,” Opt. Eng. 21, 76–81 (1982).
[CrossRef]

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

1981

T. Turpin, “Spectrum analysis using optical processing.” Proc. IEEE 69, 80–92 (1981).
[CrossRef]

J. T. Mayhan, A. J. Simmons, W. C. Cummings, “Wideband adaptive antenna nulling using tapped delay-lines,” IEEE Trans. Antennas Propag. AP-29, 923–936 (1981).
[CrossRef]

W. T. Rhodes, “Acousto-optic signal processing: convolution and correlation,” Proc. IEEE 69, 65–79 (1981).
[CrossRef]

1976

K. Wilner, A. P. van den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807 (1976).
[CrossRef]

1972

T. W. Bristol, P. J. Hagon, “Programmable surface acoustic-wave tapped delay-lines,” IEEE Trans. Sonics Ultrason. SU-19, 414 (1972).

Anderson, A. C.

R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
[CrossRef]

Anemogiannis, E.

E. Anemogiannis, R. P. Kenan, “Integrated optical architectures for tapped delay-lines,” J. Lightwave Technol. 8, 1167–1176 (1990).
[CrossRef]

Berry, M. H.

Bregman, J. D.

R. van Dijk, J. D. Bregman, A. Roodnat, F. E. van Vliet, “Photonic true time delay beamformer demonstration for a radio astronomical array antenna,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2000), pp. 78–80.

Brignon, A.

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

Bristol, T. W.

T. W. Bristol, P. J. Hagon, “Programmable surface acoustic-wave tapped delay-lines,” IEEE Trans. Sonics Ultrason. SU-19, 414 (1972).

Bucaro, J. A.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Chang, I. C.

I. C. Chang, D. L. Hecht, “Characteristics of acousto-optic devices for signal processors,” Opt. Eng. 21, 76–81 (1982).
[CrossRef]

Chen, A. T.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Chen, D. T.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Cole, J. H.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Cummings, W. C.

J. T. Mayhan, A. J. Simmons, W. C. Cummings, “Wideband adaptive antenna nulling using tapped delay-lines,” IEEE Trans. Antennas Propag. AP-29, 923–936 (1981).
[CrossRef]

Cutler, C. C.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

Dalton, L. R.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Dandridge, A.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Davies, D. E. N.

D. E. N. Davies, G. W. James, “Fiber-optic tapped delay line filter employing coherent optical processing,” Electron. Lett. 20, 95–97 (1984).
[CrossRef]

Davies, D. K.

Esman, R. D.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Fetterman, H. R.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Froehlich, F. F.

T. M. Turpin, F. F. Froehlich, D. B. Nichols, “Optical tapped delay line,” U.S. Patent6,608,721 (19August2003).

Garrett, M. H.

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

Geffroy, R.

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

Ghosh, A. K.

A. K. Ghosh, J. Trepka, “Design of fiber optic adaline neural networks,” Opt. Eng. 36, 843–848 (1997).
[CrossRef]

Giallorenzi, T. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Goodman, J. W.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

J. W. Goodman, “Fan-in and fan-out with optical interconnections,” Opt. Acta 32, 1489–1496 (1985).
[CrossRef]

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Gookin, D. M.

Goutzoulis, A. P.

Griffiths, L.

Hagon, P. J.

T. W. Bristol, P. J. Hagon, “Programmable surface acoustic-wave tapped delay-lines,” IEEE Trans. Sonics Ultrason. SU-19, 414 (1972).

Hauer, M. C.

Hecht, D. L.

I. C. Chang, D. L. Hecht, “Characteristics of acousto-optic devices for signal processors,” Opt. Eng. 21, 76–81 (1982).
[CrossRef]

Hong, J.

Huignard, J.-P.

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

Jackson, K. P.

K. P. Jackson, G. Xiao, H. J. Shaw, “Coherent optical fibre delay-line processor,” Electron. Lett. 22, 1335–1337 (1986).
[CrossRef]

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

James, G. W.

D. E. N. Davies, G. W. James, “Fiber-optic tapped delay line filter employing coherent optical processing,” Electron. Lett. 20, 95–97 (1984).
[CrossRef]

Kazovsky, L. G.

L. G. Kazovsky, “Phase- and polarization-diversity coherent optical techniques,” J. Lightwave Technol. 7, 279–292 (1989).
[CrossRef]

Kenan, R. P.

E. Anemogiannis, R. P. Kenan, “Integrated optical architectures for tapped delay-lines,” J. Lightwave Technol. 8, 1167–1176 (1990).
[CrossRef]

Kiruluta, A.

Kraut, S.

Kriehn, G.

Livingston, M.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Mayhan, J. T.

J. T. Mayhan, A. J. Simmons, W. C. Cummings, “Wideband adaptive antenna nulling using tapped delay-lines,” IEEE Trans. Antennas Propag. AP-29, 923–936 (1981).
[CrossRef]

Medrano, J.

Mitomi, O.

K. Noguchi, H. Miyazawa, O. Mitomi, “Frequency-dependent propagation characteristics of coplanar waveguide electrode on 100 GHz Ti:LiNbO3 optical modulator,” Electron. Lett. 34, 661–663 (1998).
[CrossRef]

Miyazawa, H.

K. Noguchi, H. Miyazawa, O. Mitomi, “Frequency-dependent propagation characteristics of coplanar waveguide electrode on 100 GHz Ti:LiNbO3 optical modulator,” Electron. Lett. 34, 661–663 (1998).
[CrossRef]

Mnushkina, I.

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

Moslehi, B.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Newton, S. A.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

Nezam, S. M. R. M.

Nichols, D. B.

T. M. Turpin, F. F. Froehlich, D. B. Nichols, “Optical tapped delay line,” U.S. Patent6,608,721 (19August2003).

Nichols, L. T.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Noguchi, K.

K. Noguchi, H. Miyazawa, O. Mitomi, “Frequency-dependent propagation characteristics of coplanar waveguide electrode on 100 GHz Ti:LiNbO3 optical modulator,” Electron. Lett. 34, 661–663 (1998).
[CrossRef]

Pan, Z. Q.

Pappert, S. A.

Parent, M. G.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Pati, G. S.

Priest, R. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Psaltis, D.

Ramaswami, R.

R. Ramaswami, K. N. Sivarajan, Optical Networks: a Practical Perspective (Morgan Kaufmann, 1998).

Rashleigh, S. C.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Reible, S. A.

R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, “Acousto-optic signal processing: convolution and correlation,” Proc. IEEE 69, 65–79 (1981).
[CrossRef]

Roman, J. E.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Roodnat, A.

R. van Dijk, J. D. Bregman, A. Roodnat, F. E. van Vliet, “Photonic true time delay beamformer demonstration for a radio astronomical array antenna,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2000), pp. 78–80.

Sarto, A.

Sarto, A. W.

Shadaram, M.

Shaw, H. J.

K. P. Jackson, G. Xiao, H. J. Shaw, “Coherent optical fibre delay-line processor,” Electron. Lett. 22, 1335–1337 (1986).
[CrossRef]

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Shi, Y. Q.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Sigel, G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Silveira, P. E. X.

Simmons, A. J.

J. T. Mayhan, A. J. Simmons, W. C. Cummings, “Wideband adaptive antenna nulling using tapped delay-lines,” IEEE Trans. Antennas Propag. AP-29, 923–936 (1981).
[CrossRef]

Sivarajan, K. N.

R. Ramaswami, K. N. Sivarajan, Optical Networks: a Practical Perspective (Morgan Kaufmann, 1998).

Steier, W. H.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Tavik, G. C.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Trepka, J.

A. K. Ghosh, J. Trepka, “Design of fiber optic adaline neural networks,” Opt. Eng. 36, 843–848 (1997).
[CrossRef]

Tur, M.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Turpin, T.

T. Turpin, “Spectrum analysis using optical processing.” Proc. IEEE 69, 80–92 (1981).
[CrossRef]

Turpin, T. M.

T. M. Turpin, F. F. Froehlich, D. B. Nichols, “Optical tapped delay line,” U.S. Patent6,608,721 (19August2003).

van den Heuvel, A. P.

K. Wilner, A. P. van den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807 (1976).
[CrossRef]

van Dijk, R.

R. van Dijk, J. D. Bregman, A. Roodnat, F. E. van Vliet, “Photonic true time delay beamformer demonstration for a radio astronomical array antenna,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2000), pp. 78–80.

van Vliet, F. E.

R. van Dijk, J. D. Bregman, A. Roodnat, F. E. van Vliet, “Photonic true time delay beamformer demonstration for a radio astronomical array antenna,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2000), pp. 78–80.

Wagner, K.

Wagner, K. H.

Walge, E. K.

Wang, W. S.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Weaver, S.

Weverka, R. T.

Williams, K. J.

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

Willner, A. E.

Wilner, K.

K. Wilner, A. P. van den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807 (1976).
[CrossRef]

Withers, R. S.

R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
[CrossRef]

Wright, P. V.

R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
[CrossRef]

Xiao, G.

K. P. Jackson, G. Xiao, H. J. Shaw, “Coherent optical fibre delay-line processor,” Electron. Lett. 22, 1335–1337 (1986).
[CrossRef]

Yan, L. S.

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, 1993).

Appl. Opt.

D. Psaltis, J. Hong, “Adaptive acoustooptic filter,” Appl. Opt. 23, 3475–3481 (1984).
[CrossRef] [PubMed]

A. P. Goutzoulis, D. K. Davies, “Hardware-compressive 2-D fiber optic delay line architecture for time steering of phase-array antennas,” Appl. Opt. 29, 5353–5359 (1990).
[CrossRef] [PubMed]

M. Shadaram, J. Medrano, S. A. Pappert, M. H. Berry, D. M. Gookin, “Technique for stabilizing the phase of the reference signals in analog fiber-optic links,” Appl. Opt. 34, 8283–8288 (1995).
[CrossRef] [PubMed]

R. T. Weverka, K. Wagner, A. Sarto, “Photorefractive processing for large adaptive phased arrays,” Appl. Opt. 35, 1344–1366 (1996).
[CrossRef] [PubMed]

A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).
[CrossRef] [PubMed]

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).
[CrossRef]

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).
[CrossRef] [PubMed]

A. Kiruluta, G. S. Pati, G. Kriehn, P. E. X. Silveira, A. W. Sarto, K. Wagner, “Spatio-temporal operator formalism for holographic recording and diffraction in a photorefractive-based true-time-delay phased-array processor,” Appl. Opt. 42, 5334–5350 (2003).
[CrossRef] [PubMed]

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optoelectronic implementation of a 256-channel sonar adaptive-array processor,” Appl. Opt. 43, 6421–6439 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett.

D. T. Chen, H. R. Fetterman, A. T. Chen, W. H. Steier, L. R. Dalton, W. S. Wang, Y. Q. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70, 3335–3337 (1997).
[CrossRef]

Electron. Lett.

D. E. N. Davies, G. W. James, “Fiber-optic tapped delay line filter employing coherent optical processing,” Electron. Lett. 20, 95–97 (1984).
[CrossRef]

K. P. Jackson, G. Xiao, H. J. Shaw, “Coherent optical fibre delay-line processor,” Electron. Lett. 22, 1335–1337 (1986).
[CrossRef]

K. Noguchi, H. Miyazawa, O. Mitomi, “Frequency-dependent propagation characteristics of coplanar waveguide electrode on 100 GHz Ti:LiNbO3 optical modulator,” Electron. Lett. 34, 661–663 (1998).
[CrossRef]

IEEE J. Quantum Electron.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

IEEE Trans. Antennas Propag.

J. T. Mayhan, A. J. Simmons, W. C. Cummings, “Wideband adaptive antenna nulling using tapped delay-lines,” IEEE Trans. Antennas Propag. AP-29, 923–936 (1981).
[CrossRef]

IEEE Trans. Magn.

R. S. Withers, A. C. Anderson, P. V. Wright, S. A. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn. M-19, 480–484 (1983).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–210 (1985).
[CrossRef]

J. E. Roman, L. T. Nichols, K. J. Williams, R. D. Esman, G. C. Tavik, M. Livingston, M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microwave Theory Tech. 46, 2317–2323 (1998).
[CrossRef]

IEEE Trans. Sonics Ultrason

T. W. Bristol, P. J. Hagon, “Programmable surface acoustic-wave tapped delay-lines,” IEEE Trans. Sonics Ultrason. SU-19, 414 (1972).

J. Lightwave Technol.

E. Anemogiannis, R. P. Kenan, “Integrated optical architectures for tapped delay-lines,” J. Lightwave Technol. 8, 1167–1176 (1990).
[CrossRef]

L. G. Kazovsky, “Phase- and polarization-diversity coherent optical techniques,” J. Lightwave Technol. 7, 279–292 (1989).
[CrossRef]

A. E. Willner, S. M. R. M. Nezam, L. S. Yan, Z. Q. Pan, M. C. Hauer, “Monitoring and control of polarization-related impairments in optical fiber systems,” J. Lightwave Technol. 22, 106–125 (2004).
[CrossRef]

Opt. Acta

J. W. Goodman, “Fan-in and fan-out with optical interconnections,” Opt. Acta 32, 1489–1496 (1985).
[CrossRef]

Opt. Commun.

A. Brignon, R. Geffroy, J.-P. Huignard, M. H. Garrett, I. Mnushkina, “Experimental investigations of the photorefractive properties of rhodium-doped BaTiO3 at 1.06 μm,” Opt. Commun. 137, 311–316 (1997).
[CrossRef]

Opt. Eng.

A. K. Ghosh, J. Trepka, “Design of fiber optic adaline neural networks,” Opt. Eng. 36, 843–848 (1997).
[CrossRef]

I. C. Chang, D. L. Hecht, “Characteristics of acousto-optic devices for signal processors,” Opt. Eng. 21, 76–81 (1982).
[CrossRef]

Proc. IEEE

T. Turpin, “Spectrum analysis using optical processing.” Proc. IEEE 69, 80–92 (1981).
[CrossRef]

W. T. Rhodes, “Acousto-optic signal processing: convolution and correlation,” Proc. IEEE 69, 65–79 (1981).
[CrossRef]

K. Wilner, A. P. van den Heuvel, “Fiber-optic delay lines for microwave signal processing,” Proc. IEEE 64, 805–807 (1976).
[CrossRef]

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fibre-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Other

T. M. Turpin, F. F. Froehlich, D. B. Nichols, “Optical tapped delay line,” U.S. Patent6,608,721 (19August2003).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, 1993).

R. van Dijk, J. D. Bregman, A. Roodnat, F. E. van Vliet, “Photonic true time delay beamformer demonstration for a radio astronomical array antenna,” in IEEE International Topical Meeting on Microwave Photonics (IEEE, 2000), pp. 78–80.

R. Ramaswami, K. N. Sivarajan, Optical Networks: a Practical Perspective (Morgan Kaufmann, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Schematic representation of a holographically cohered, diffracted FTDL. A beam splitter produces both a phase-cohering reference beam and a signal beam from light generated by a laser. The signal beam is coupled into an EOM that is controlled by a RF input signal. The modulated signal beam exits the EOM and propagates through a FTDL composed of a concatenated set of 3 dB couplers and a set of time delay loops, giving delay that increments linearly in space as shown. The beams exit the FTDL and interfere with the reference beam inside a photorefractive crystal, where they produce a set of phase-cohering holographic gratings. The holographic gratings diffract a cohered set of signal beams that are then Fourier transformed with a lens to produce a spatial version of the signal spectra.

Fig. 2
Fig. 2

Schematic representation of phase-cohering holography incorporating polarization fluctuation compensation. The light in the fiber remains polarized until it enters the FTDL, at which point propagation through the SM fiber in the FTDL causes the polarization to wander in a random fashion. Upon exiting the FTDL, the light is resolved into ŝ and p ^-polarization states with a Wollaston prism. The undesired ŝ state is rotated into p ^ polarization with a half-wave plate oriented at 45°. The two beams from each fiber, now both extraordinarily polarized, interfere with the phase-cohering reference beam at spatially multiplexed locations to write a sequential pair of holographic gratings in the volume of the photorefractive crystal. The diffraction of the modulated sidebands off these gratings combine to produce an output independent of the input polarization, which is then Fourier transformed to produce the coherent summations of the signal spectra at each position.

Fig. 3
Fig. 3

Polarization fluctuation compensation sequential beam interaction geometry. Both gratings are in a single photorefractive crystal. The reference beam, A2(0), writes the first grating with the residual carrier from the FTDL as projected into the ordinary and extraordinary polarization states and rotated into the extraordinary polarization state [A1(0)] and A1′(0)]. The modulated sidebands associated with each polarization state projection (A3(0) and A3′(0) diffract off of these gratings to produce a phase-flat output, A4′(L), which is relatively immune to polarization state fluctuations in the FTDL output.

Fig. 4
Fig. 4

Diffracted field amplitude for a FTDL without polarization compensation with γL = 1.0 and ρ = 1.0. The radius represents the diffracted field amplitude, and the elevation and azimuth angles correspond to the polarization state of the field emerging from the FTDL as would be plotted on the Poincaré sphere. We use the standard Poincare sphere coordinates, with left-hand circular (LHC) and right-hand circular (RHC) polarization states at the poles and linear polarization states, including horizontal (H) and vertical (V), along the equator.

Fig. 5
Fig. 5

Diffracted field amplitudes for a FTDL with polarization compensation for different values of γL and ρ. As in Fig. 4, the radius represents the diffracted field amplitude and the elevation and azimuth angles correspond to the polarization state of the field as would be plotted on the Poincaré sphere. The radius of the Poincaré sphere is 1.25 times the maximum diffracted field amplitude for a FTDL without polarization compensation for corresponding values of ρL and ρ. For ρ = 1.0, the maximum diffracted field amplitude actually increases with polarization compensation due to the effective increase in grating length.

Fig. 6
Fig. 6

Complete phase-cohering holography experimental setup. The output from a Nd:YAG laser is split with a polarizing beam splitter (PBS) to provide a phase-cohering reference beam and signal beam. The signal beam travels through an EOM and then propagates through a FTDL. The FTDL’s tapped outputs are arrayed out of the plane of the picture. The fiber outputs interfere with the reference beam in the volume of a photorefractive crystal, producing a hologram that diffracts the phase-cohered version of the fiber outputs. A lens Fourier transforms this diffracted output, producing the signal spectrum, which is detected by a CCD.

Fig. 7
Fig. 7

Three different topologies for FTDLs. (a) The tree-based FTDL and (b) the binary divide-and-conquer FTDL both use sets of concatenated couplers and strategically placed fiber delay loops to produce a set of time-delayed outputs. (c) The mandrel-based FTDL generates time delay by use of a single fiber wound around a mandrel, with each loop separated by a propagation distance approximately equal to group index ng times the circumference of the mandrel.32 The fiber can be polished, kinked, or notched (depending on whether the fiber is SM) to produce a set of time-delayed outputs. A phase-cohering operation transforms these phase-aberrated outputs into phase-flat outputs for each type of FTDL while preserving the RF time delay. (d) FTDLs can also be used in conjunction with a tilted etalon to implement fine time delay.

Fig. 8
Fig. 8

Measured values for total time delay (left) and incremental time delay (right) τ for each fiber tap. The target value for incremental time delay was 1.25 ns, and the fit was 1.25 ± 0.04 ns. The error bars correspond to the PNA’s measurement accuracy of ±0.02 ns.

Fig. 9
Fig. 9

Simulated lower diffracted sidelobe position map for various RF phase modulation frequencies for both SSB (left) and DSB (right) modulation by use of the measured time delay errors as shown in Fig. 8. The time delay errors cause spurious sidelobes to appear at 5 GHz when the tap length mismatch becomes a larger fraction of the RF wavelength.

Fig. 10
Fig. 10

Unmodulated Fourier plane intensity profile. The V groove is a comb of fibers, each with its own Gaussian amplitude profile, apodized by a rectangle function. The Fourier plane of the V groove will be a comb apodized by a broad Gaussian window with each comb order broadened by a sinc function. The presence of the dc order and the two peaks indicate that the FTDL outputs have been phase cohered. If the phase-cohering operation had been unsuccessful, the intensity profile would have been a speckle pattern with an expectation mean given by the Fourier transform of the Gaussian output from a single fiber.

Fig. 11
Fig. 11

Intensity profiles for −1 sidelobes associated with the dc order and +1 sidelobes associated with the −1 comb order. Note the Gaussian apodization of the outlying sidelobes that are due to the Fourier transform of the lenslet beams. The peak that is due to 400 MHz modulation is significantly larger than the adjacent peaks because the two diffracted sidelobes overlap as a result of the free spectral range overlap at this modulation frequency. Fixed background data have been subtracted away.

Fig. 12
Fig. 12

Experimental result for the lower half of the Fourier plane. The sidelobes begin to disappear at a modulation frequency of approximately 2.5 GHz as the uncorrected rms phase errors in the RF modulation term exceed λ/20. Peaks at 2.8 and 3.6 GHz are still visible because of the coherent addition of the overlapping sidelobes, as in Fig. 11. Background data have been subtracted away.

Equations (19)

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

A ( x , y , t ) = A ( y V + t - τ 0 ) m = 1 M g ( x , y - m D ) ,
E f ( x , y , t ) = E 0 M m = 1 M 1 2 [ 1 + η 0 s ˜ ( t - m τ RF - e m υ g ) + c . c . ] × exp [ i ω l ( t - m τ - e m υ p ) ] g ( x , y - m D ) ,
s ˜ ( t - m τ RF - e m v g ) = S ˜ ( ω RF ) × exp [ i ω RF ( t - m τ - e m v g ) ] d ω RF .
g h ( x , y , t ) = η H - t | E 0 M m = 1 M g ( x , y - m D ) × exp [ i ω l ( t - m τ - e m v p ) ] × 1 2 { η 0 S ˜ exp [ i ω RF ( t - m τ RF - e m v g ) ] × d ω RF + 1 + c . c . } + E r exp [ i ( ω l t + k x x ) ] | 2 × exp [ - ( t - τ ) / τ pr ] d t ,
g m ( x , y , t ) E r * E 0 M m = 1 M g ( x , y - m D ) × cos [ k x x - ω l ( m τ RF + e m v g ) ] .
E diff ( x , y , t ) = E r E 0 2 M exp [ i ( ω l t + k x x ) ] m = 1 M g 2 ( x , y - m D ) 1 2 [ 1 + η 0 S ˜ ( ω RF ) exp [ i ω RF ( t - m τ RF - e m v g ) ] d ω RF + c . c . ] .
E diff ( x , y , t ) exp [ i ( ω l t + k x x ) ] m = 1 M g 2 ( x , y - m D ) × [ 1 + s ( t + y V + e m v g ) ] .
I det ( x 1 , y 1 ) | G 2 ( x 1 λ F , y 1 λ F ) m = 1 M exp ( i m D y 1 λ F ) * sinc ( M D y 1 λ F ) * { δ ( y 1 λ F ) + V S ( V y 1 λ F ) × exp [ i V y 1 λ F ( t + e m v g ) ] } | 2 ,
d d z A 1 = - 1 2 Γ ( A 1 A 2 * ) A 2 / I 0 ,
d d z A 2 = 1 2 Γ * ( A 1 * A 2 ) A 1 / I 0 ,
d d z A 3 = - 1 2 Γ ( A 1 A 2 * ) A 4 exp ( i Δ K z ) / I 0 ,
d d z A 4 = 1 2 Γ * ( A 1 * A 2 ) A 3 exp ( - i Δ K z ) / I 0 ,
G ( z ) = i A 1 A 2 * I 0 Γ 2 = γ 4 cosh ( γ z - 2 ln r 2 ) exp ( i ψ ) ,
u ( z ) = tan - 1 [ 1 r exp ( γ z / 2 ) ] - tan - 1 ( 1 r ) .
[ A 1 ( z ) A 2 ( z ) ] = [ cos u - exp ( i ψ ) sin u exp ( i ψ ) sin u cos u ] [ A 1 ( 0 ) A 2 ( 0 ) ] ,
[ A 3 ( z ) A 4 ( z ) ] = [ cos u - exp ( i ψ ) sin u exp ( i ψ ) sin u cos u ] [ A 3 ( 0 ) A 4 ( 0 ) ] .
A diff = A RF ( α cos { cot - 1 [ r exp ( - γ L / 2 ) ] } × sin { cot - 1 [ r exp ( - γ L / 2 ) ] } + β sin { cot - 1 [ r exp ( - γ L / 2 ) ] } ) .
A diff = A RF 1 + exp ( γ L ) ρ 2 β 2 ( α sin { α ρ cot - 1 [ α ρ exp ( γ L / 2 ) ] } + 1 ρ exp ( γ L / 2 ) ) .
A diff 1 ρ ( 1 + γ L 2 ) A RF .

Metrics