Abstract

Beam cleanup via stimulated Raman scattering in multimode fibers is modeled by numerically considering the competition between the Stokes modes of graded-index and step-index fibers. The relative gain of each Stokes mode is calculated by considering the overlap the various pump and Stokes modes of the fibers. Mode competition in a graded-index fiber favors the LP01 Stokes mode while mode competition in a step-index fiber does not favor the LP01 Stokes mode. This model explains why beam cleanup has only been reported for graded-index fibers and not for step-index fibers.

©2007 Optical Society of America

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References

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  1. K. S. Chiang, “Stimulated Raman scattering in a multimode optical fiber: evolution of modes in Stokes waves,” Opt. Lett. 17, 352 (1992).
    [Crossref] [PubMed]
  2. T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
    [Crossref]
  3. P. L. Baldeck, F. Raccah, and R. R. Alfano, “Observation of self-focusing in optical fibers with picosecond pulses,” Opt. Lett. 12, 588 (1987).
    [Crossref] [PubMed]
  4. Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).
  5. B. M. Flusche, T. G. Alley, T. H. Russell, and Won B. Roh, “Multi-port beam combination and cleanup in large multimode fiber using stimulated Raman scattering,” Opt. Express 14, 11748, (2006).
    [Crossref] [PubMed]
  6. T.Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
    [Crossref]
  7. S. H. Baek and W. B. Roh, “Single-mode Raman fiber laser based on a multimode fiber,” Opt. Lett. 29, 153 (2004).
    [Crossref] [PubMed]
  8. N. B. Terry, K. T. Engel, T. G. Alley, and T. H. Russell, “Use of a continuous wave Raman fiber laser in graded-index multimode fiber for SRS beam combination,” Opt. Express 15, 602–7 (2007).
    [Crossref] [PubMed]
  9. A. Polley and S. E. Ralph, “Raman amplification in multimode Fiber,” IEEE Photon. Technol. Lett. 19, 218 (2007).
    [Crossref]
  10. M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
    [Crossref]
  11. R. G. Smith, “Optical power handing capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2490 (1972).
    [Crossref]
  12. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed., (Academic Press, San Diego, 2001), Chap. 8.
  13. R. H. Stolen, “Fiber Raman lasers,” Fiber and Integr. Opt. 3, 21–51 (1980).
    [Crossref]
  14. E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
    [Crossref]
  15. T. H. Russell, B. W. Grime, T. G. Alley, and W. B. Roh, “Stimulated Brillouin scattering beam cleanup and Combining in optical Fiber”, to be published in Nonlinear Optics and Recent Advances in Optics, Research Signpost (2007).
  16. A. W. Synder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, New York, 1983).
  17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley and Sons, New York, 1991), Chap. 8.
    [Crossref]

2007 (2)

2006 (1)

2005 (1)

T.Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[Crossref]

2004 (1)

2002 (1)

T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
[Crossref]

2000 (1)

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

1994 (1)

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

1992 (1)

1987 (1)

1981 (1)

Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).

1980 (1)

R. H. Stolen, “Fiber Raman lasers,” Fiber and Integr. Opt. 3, 21–51 (1980).
[Crossref]

1972 (1)

R. G. Smith, “Optical power handing capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2490 (1972).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed., (Academic Press, San Diego, 2001), Chap. 8.

Aleksandrov, I. V.

Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).

Alfano, R. R.

Alley, T. G.

Baek, S. H.

Baldeck, P. L.

Bubnov, M. M.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Bufetov, I. A.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Butler, D. L.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Chiang, K. S.

Crookston, M. B.

T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
[Crossref]

Deliso, E. M.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Dianov, E. M.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Engel, K. T.

Fan, T.Y.

T.Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[Crossref]

Flusche, B. M.

Grekov, M. V.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Grime, B. W.

T. H. Russell, B. W. Grime, T. G. Alley, and W. B. Roh, “Stimulated Brillouin scattering beam cleanup and Combining in optical Fiber”, to be published in Nonlinear Optics and Recent Advances in Optics, Research Signpost (2007).

Guryanov, A. N.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Khopin, V. F.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Love, J. D.

A. W. Synder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, New York, 1983).

Medvedkov, O. I.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Nesterova, Z. V.

Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).

Polley, A.

A. Polley and S. E. Ralph, “Raman amplification in multimode Fiber,” IEEE Photon. Technol. Lett. 19, 218 (2007).
[Crossref]

Polnitskii, A. A.

Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).

Posey, R.

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

Raccah, F.

Ralph, S. E.

A. Polley and S. E. Ralph, “Raman amplification in multimode Fiber,” IEEE Photon. Technol. Lett. 19, 218 (2007).
[Crossref]

Roh, W. B.

S. H. Baek and W. B. Roh, “Single-mode Raman fiber laser based on a multimode fiber,” Opt. Lett. 29, 153 (2004).
[Crossref] [PubMed]

T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
[Crossref]

T. H. Russell, B. W. Grime, T. G. Alley, and W. B. Roh, “Stimulated Brillouin scattering beam cleanup and Combining in optical Fiber”, to be published in Nonlinear Optics and Recent Advances in Optics, Research Signpost (2007).

Roh, Won B.

Russell, T. H.

N. B. Terry, K. T. Engel, T. G. Alley, and T. H. Russell, “Use of a continuous wave Raman fiber laser in graded-index multimode fiber for SRS beam combination,” Opt. Express 15, 602–7 (2007).
[Crossref] [PubMed]

B. M. Flusche, T. G. Alley, T. H. Russell, and Won B. Roh, “Multi-port beam combination and cleanup in large multimode fiber using stimulated Raman scattering,” Opt. Express 14, 11748, (2006).
[Crossref] [PubMed]

T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
[Crossref]

T. H. Russell, B. W. Grime, T. G. Alley, and W. B. Roh, “Stimulated Brillouin scattering beam cleanup and Combining in optical Fiber”, to be published in Nonlinear Optics and Recent Advances in Optics, Research Signpost (2007).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley and Sons, New York, 1991), Chap. 8.
[Crossref]

Sasiliev, S. A.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Sattarov, D. K.

Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).

Sharma, M. D.

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

Shubin, A.V.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Smith, R. G.

R. G. Smith, “Optical power handing capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2490 (1972).
[Crossref]

Stolen, R. H.

R. H. Stolen, “Fiber Raman lasers,” Fiber and Integr. Opt. 3, 21–51 (1980).
[Crossref]

Synder, A. W.

A. W. Synder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, New York, 1983).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley and Sons, New York, 1991), Chap. 8.
[Crossref]

Terry, N. B.

Venkateswarlu, P.

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

Williams, A.

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

Willis, S. M.

T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
[Crossref]

Wu, Z. Q.

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

Yashkov, M. V.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Appl. Opt. (1)

R. G. Smith, “Optical power handing capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2490 (1972).
[Crossref]

Fiber and Integr. Opt. (1)

R. H. Stolen, “Fiber Raman lasers,” Fiber and Integr. Opt. 3, 21–51 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T.Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Polley and S. E. Ralph, “Raman amplification in multimode Fiber,” IEEE Photon. Technol. Lett. 19, 218 (2007).
[Crossref]

J. Nonlinear Opt. Phys. Mater. (1)

T. H. Russell, S. M. Willis, M. B. Crookston, and W. B. Roh, “Stimulated Raman scattering in multimode fibers and its application to beam cleanup and combining,” J. Nonlinear Opt. Phys. Mater. 11, 303–316 (2002).
[Crossref]

JETP Lett. (1)

Z. V. Nesterova, I. V. Aleksandrov, A. A. Polnitskii, and D. K. Sattarov, “Propagation characteristics of high power ultrashort light pulses in optical fibers,” JETP Lett. 34, 371 (1981).

Opt. Commun. (1)

M. D. Sharma, Z. Q. Wu, R. Posey, A. Williams, and P. Venkateswarlu, “Stimulated Raman scattering in a multimode optical fiber with bend-induced loss,” Opt. Commun. 111, 127 (1994).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Proc. SPIE (1)

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Sasiliev, O. I. Medvedkov, A.V. Shubin, A. N. Guryanov, V. F. Khopin, M. V. Yashkov, E. M. Deliso, and D. L. Butler, “1.3 µm Raman fiber amplifier,” Proc. SPIE 4083, 101–109, (2000).
[Crossref]

Other (4)

T. H. Russell, B. W. Grime, T. G. Alley, and W. B. Roh, “Stimulated Brillouin scattering beam cleanup and Combining in optical Fiber”, to be published in Nonlinear Optics and Recent Advances in Optics, Research Signpost (2007).

A. W. Synder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, New York, 1983).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley and Sons, New York, 1991), Chap. 8.
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed., (Academic Press, San Diego, 2001), Chap. 8.

Supplementary Material (2)

» Media 1: MOV (700 KB)     
» Media 2: MOV (990 KB)     

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Figures (5)

Fig. 1.
Fig. 1. Probability that a given Stokes mode of a multimode fiber has the greatest Raman gain given random pump launching conditions. In a graded-index fiber, the LP01 mode tends to have much higher gain than the higher-order Stokes modes. In a step-index fiber, the LP01 mode is not dominant; this explains why beam cleanup does not occur in a step-index fiber.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Average relative gain of various Stokes modes in a multimode fiber calculated for the case of unseeded SRS given 50,000 different sets of random launching conditions of the pump. The LP01 Stokes mode of the graded-index fiber has more gain than the higher-order Stokes modes, while the LP01 Stokes mode does not does not have the greatest gain in a step-index fiber. This explains why beam cleanup requires a graded-index fiber.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. A comparison of mode competition in a graded-index fiber and a step-index fiber. In a notional graded-index fiber, the pump configuration shown in (a) resulted in the Stokes output shown in (b). In a notional step-index fiber, the pump configuration shown in (c) resulted in the Stokes output shown in (d).

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. An animation depicting SRS beam cleanup in a graded-index fiber. The Stokes beam generated by spontaneous Raman scattering in a graded-index fiber is shown in the first frame. Subsequent frames show the evolution of the Stokes beam as each Stokes mode is amplified by grelative.. (MOV video, Figure4.mov, 0.7 Mbytes). [Media 1]

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. An animation depicting the Stokes evolution associated with SRS in a step-index fiber. The Stokes beam generated by spontaneous Raman scattering in a step-index fiber is shown in the first frame. Subsequent frames show the evolution of the Stokes beam as each Stokes mode is amplified by grelative.. (MOV video, Figure5.mov, 1.0 Mbytes). [Media 2]

Download Full Size | PPT Slide | PDF

Tables (2)

Tables Icon

Table 1. This table of the normalized overlap integrals of a graded-index fiber shows that the overlap of the LP01 Stokes mode with the LP01 pump mode (highlighted in blue) is larger than any other overlap integral. Other selfoverlap integrals are highlighted in bold.

View Table | View all tables in this article
Tables Icon

Table 2. This table of the normalized overlap integrals of a step-index fiber shows that the overlap of the LP12 Stokes mode with the LP12 pump mode (highlighted in blue) is larger than any other overlap integral. The overlap of the LP01 Stokes mode with the LP01 pump mode has a lower value than several of the self-overlap integrals highlighted in bold, suggesting that higher-order Stokes modes will dominate the Stokes output.

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Equations (22)

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

d I p ( r , θ , z , t ) d z = − ω p ω s g R I p ( r , θ , z , t ) I s ( r , θ , z , t ) − α p I p ( r , θ , z , t )
d I s ( r , θ , z , t ) d z = g R I p ( r , θ , z , t ) I s ( r , θ , z , t ) − α s I s ( r , θ , z , t ) .
d P p ( z , t ) d z = − g ( z , t ) ω p ω s P p ( z , t ) P s ( z , t ) − α p P p ( z , t )
d P s ( z , t ) d z = g ( z , t ) P p ( z , t ) P s ( z , t ) − α s P s ( z , t ) ,
g ( z , t ) = g R ∬ I p ( r , θ , z , t ) I s ( r , θ , z , t ) r d r d θ P p ( z ) P s ( z ) .
I p , s ( r , θ , z , t ) = 2 ε 0 c n p , s ( r ) 〈 ∣ E ⇀ p , s ( r , θ , z , t ) ∣ 2 〉
E ⇀ p , s ( r , θ , z , t ) = ∑ q A p , s q ψ p , s q ( r , θ ) [ cos ⁡ ( k ⇀ p , s q · z ̂ − ω p , s t + φ p , s q ) ] e ̂ p , s ,
ψ lm = cos ⁡ ( 1 φ ) R 1 L m − 1 ( 1 ) ( V R 2 ) exp ( − 1 2 V R 2 ) ,
ψ lm = J l ( U lm R ) J l − 1 ( U lm ) cos ⁡ ( 1 θ ) ,
U lm J l + 1 ( U lm ) J l ( U lm ) = W lm K l + 1 ( W lm ) K l ( W lm ) ,
g ( z ) = g R [ ε 0 c n ( r ) ] 2 ∑ j A s j ∑ ν A s ν ∑ m A p m ∑ q A p q γ j ν mq P s ( z ) P p ( z ) ,
γ j ν m q = ∫ 0 a ∫ 0 2 π [ ψ s j ψ s ν ψ p m ψ p q cos ⁡ ( Δ k ⇀ s j ν · z ̂ + Δ φ s j ν ) ×
cos ⁡ ( Δ ⇀ k p mq · z ̂ + Δ φ p mq ) r d r d θ .
Δ k ⇀ p , s nl ≡ k ⇀ p , s n − k ⇀ p , s 1
Δ φ p , s nl ≡ φ p , s n − φ p , s 1
A s j A s ν A p m A p q γ j ν mq = g j ν mq ,
g ( z ) ∝
g 1111 ( z ) + g 1112 ( z ) + g 1121 ( z ) + g 1122 ( z ) +
g 2211 ( z ) + g 2212 ( z ) + g 2221 ( z ) + g 2222 ( z ) +
g 1211 ( z ) + g 1212 ( z ) + g 1221 ( z ) + g 1222 ( z ) +
g 2111 ( z ) + g 2112 ( z ) + g 2121 ( z ) + g 2122 ( z ) .
g j ( z ) = g R [ ε 0 c n ( r ) ] 2 A s j ∑ ν A s ν ∑ m A p m ∑ q A p q γ j ν mq P s ( z ) P q ( z ) .

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