Abstract

Selection of modes containing different dark regions was studied in resonators with conical reflectors. The possibility of selecting whole subgroups of such modes was shown in circularly symmetric resonators. To handle single-mode selection employing extra intracavity spatial filters, modified integral equations and a numerical method of their analysis are proposed. Usage of the filter symmetry reduces the size of the four-dimensional matrices corresponding to the equation kernels, and they are analyzed by algorithms for two-dimensional matrices with the best convergence. The optimum resonator parameters for effective selection of different dark modes are found.

© 2011 Optical Society of America

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References

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  1. Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
    [CrossRef]
  2. B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
    [CrossRef] [PubMed]
  3. A. P. Kol’chenko, A. G. Nikitenko, and Y. Troitski, “Control of the structure of transverse modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980).
    [CrossRef]
  4. R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
    [CrossRef]
  5. Y. Ananev, Laser Resonators and the Beam Divergence Problem, Adam Hilger Series on Optics and Optoelectronics (Adam Hilger, 1992).
  6. P. A. Belanger, R. L. Lachance, and C. Pare, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1992).
    [CrossRef] [PubMed]
  7. Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Two regions of mode selection in resonators with biprism-like elements,” Appl. Opt. 44, 2546–2552 (2005).
    [CrossRef] [PubMed]
  8. Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with bilens mirror,” IEEE J. Quantum Electron. 46, 478–483 (2010).
    [CrossRef]
  9. Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Laser mode selection by a combination of biprism-like reflectors with narrow amplitude masks,” Appl. Opt. 45, 2761–2765 (2006).
    [CrossRef] [PubMed]
  10. Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in diffraction coupled semi-concentric resonators by means of a biprism-like element,” Opt. Eng. 11, 114201-1–114201-7(2007).
  11. Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with conical reflectors,” IEEE J. Quantum Electron. 44, 456–461 (2008).
    [CrossRef]
  12. M. Rioux, M. Cornier, and P. A. Belanger, “High order circular mode selection in a conical resonator,” Appl. Opt. 16, 1791–1792 (1977).
    [CrossRef] [PubMed]
  13. V. I. Kravchenko, Y. N. Parkhomenko, and V. A. Sokolov, “Selectivety of dispersive resonator with an inhomogeneous aperture,” Sov. J. Quantum Electron. 18, 54–57 (1988).
    [CrossRef]
  14. Y. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, “Lasers with fast electronic synthesis of spectral line shape,” in Advances in Laser and Optic Research (Nova Science, 2002), Vol.  1, pp. 128–161.
  15. A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).
  16. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
  17. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, pp. 325–386.
    [CrossRef]
  18. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, World Student series (Addison-Wesley, 1989).

2010 (1)

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with bilens mirror,” IEEE J. Quantum Electron. 46, 478–483 (2010).
[CrossRef]

2008 (3)

Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
[CrossRef] [PubMed]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with conical reflectors,” IEEE J. Quantum Electron. 44, 456–461 (2008).
[CrossRef]

2007 (1)

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in diffraction coupled semi-concentric resonators by means of a biprism-like element,” Opt. Eng. 11, 114201-1–114201-7(2007).

2006 (1)

2005 (1)

1999 (1)

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

1992 (1)

1988 (1)

V. I. Kravchenko, Y. N. Parkhomenko, and V. A. Sokolov, “Selectivety of dispersive resonator with an inhomogeneous aperture,” Sov. J. Quantum Electron. 18, 54–57 (1988).
[CrossRef]

1980 (1)

A. P. Kol’chenko, A. G. Nikitenko, and Y. Troitski, “Control of the structure of transverse modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980).
[CrossRef]

1977 (1)

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

1960 (1)

A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

Ananev, Y.

Y. Ananev, Laser Resonators and the Beam Divergence Problem, Adam Hilger Series on Optics and Optoelectronics (Adam Hilger, 1992).

Anisimova, O. V.

Y. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, “Lasers with fast electronic synthesis of spectral line shape,” in Advances in Laser and Optic Research (Nova Science, 2002), Vol.  1, pp. 128–161.

Baykal, Y.

Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Belanger, P. A.

Cai, Y.

Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Chen, Y.

Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Cornier, M.

Danziger, Y.

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

Davidson, N.

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, pp. 325–386.
[CrossRef]

Eyyuboglu, H. T.

Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

Friesem, A. A.

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, pp. 325–386.
[CrossRef]

Galkin, O. N.

Y. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, “Lasers with fast electronic synthesis of spectral line shape,” in Advances in Laser and Optic Research (Nova Science, 2002), Vol.  1, pp. 128–161.

Gerald, C. F.

C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, World Student series (Addison-Wesley, 1989).

Hasman, E.

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, pp. 325–386.
[CrossRef]

Kol’chenko, A. P.

A. P. Kol’chenko, A. G. Nikitenko, and Y. Troitski, “Control of the structure of transverse modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980).
[CrossRef]

Kravchenko, V. I.

V. I. Kravchenko, Y. N. Parkhomenko, and V. A. Sokolov, “Selectivety of dispersive resonator with an inhomogeneous aperture,” Sov. J. Quantum Electron. 18, 54–57 (1988).
[CrossRef]

Lachance, R. L.

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

Nikitenko, A. G.

A. P. Kol’chenko, A. G. Nikitenko, and Y. Troitski, “Control of the structure of transverse modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980).
[CrossRef]

Normatov, A.

Oron, R.

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, pp. 325–386.
[CrossRef]

Pare, C.

Parkhomenko, Y. N.

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with bilens mirror,” IEEE J. Quantum Electron. 46, 478–483 (2010).
[CrossRef]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with conical reflectors,” IEEE J. Quantum Electron. 44, 456–461 (2008).
[CrossRef]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in diffraction coupled semi-concentric resonators by means of a biprism-like element,” Opt. Eng. 11, 114201-1–114201-7(2007).

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Laser mode selection by a combination of biprism-like reflectors with narrow amplitude masks,” Appl. Opt. 45, 2761–2765 (2006).
[CrossRef] [PubMed]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Two regions of mode selection in resonators with biprism-like elements,” Appl. Opt. 44, 2546–2552 (2005).
[CrossRef] [PubMed]

V. I. Kravchenko, Y. N. Parkhomenko, and V. A. Sokolov, “Selectivety of dispersive resonator with an inhomogeneous aperture,” Sov. J. Quantum Electron. 18, 54–57 (1988).
[CrossRef]

Y. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, “Lasers with fast electronic synthesis of spectral line shape,” in Advances in Laser and Optic Research (Nova Science, 2002), Vol.  1, pp. 128–161.

Rioux, M.

Shamir, J.

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with bilens mirror,” IEEE J. Quantum Electron. 46, 478–483 (2010).
[CrossRef]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with conical reflectors,” IEEE J. Quantum Electron. 44, 456–461 (2008).
[CrossRef]

B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
[CrossRef] [PubMed]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in diffraction coupled semi-concentric resonators by means of a biprism-like element,” Opt. Eng. 11, 114201-1–114201-7(2007).

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Laser mode selection by a combination of biprism-like reflectors with narrow amplitude masks,” Appl. Opt. 45, 2761–2765 (2006).
[CrossRef] [PubMed]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Two regions of mode selection in resonators with biprism-like elements,” Appl. Opt. 44, 2546–2552 (2005).
[CrossRef] [PubMed]

Sokolov, V. A.

V. I. Kravchenko, Y. N. Parkhomenko, and V. A. Sokolov, “Selectivety of dispersive resonator with an inhomogeneous aperture,” Sov. J. Quantum Electron. 18, 54–57 (1988).
[CrossRef]

Spektor, B.

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with bilens mirror,” IEEE J. Quantum Electron. 46, 478–483 (2010).
[CrossRef]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with conical reflectors,” IEEE J. Quantum Electron. 44, 456–461 (2008).
[CrossRef]

B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
[CrossRef] [PubMed]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in diffraction coupled semi-concentric resonators by means of a biprism-like element,” Opt. Eng. 11, 114201-1–114201-7(2007).

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Laser mode selection by a combination of biprism-like reflectors with narrow amplitude masks,” Appl. Opt. 45, 2761–2765 (2006).
[CrossRef] [PubMed]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Two regions of mode selection in resonators with biprism-like elements,” Appl. Opt. 44, 2546–2552 (2005).
[CrossRef] [PubMed]

Troitski, Y.

A. P. Kol’chenko, A. G. Nikitenko, and Y. Troitski, “Control of the structure of transverse modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980).
[CrossRef]

Wheatley, P. O.

C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, World Student series (Addison-Wesley, 1989).

Appl. Opt. (4)

Appl. Phys. B (1)

Y. Chen, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Discontinuous phase elements for transverse mode selection in laser resonators,” Appl. Phys. Lett. 74, 1373–1375(1999).
[CrossRef]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

IEEE J. Quantum Electron. (2)

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with conical reflectors,” IEEE J. Quantum Electron. 44, 456–461 (2008).
[CrossRef]

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in resonators with bilens mirror,” IEEE J. Quantum Electron. 46, 478–483 (2010).
[CrossRef]

Opt. Eng. (1)

Y. N. Parkhomenko, B. Spektor, and J. Shamir, “Mode selection in diffraction coupled semi-concentric resonators by means of a biprism-like element,” Opt. Eng. 11, 114201-1–114201-7(2007).

Opt. Lett. (1)

Proc. IRE (1)

A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

Sov. J. Quantum Electron. (2)

V. I. Kravchenko, Y. N. Parkhomenko, and V. A. Sokolov, “Selectivety of dispersive resonator with an inhomogeneous aperture,” Sov. J. Quantum Electron. 18, 54–57 (1988).
[CrossRef]

A. P. Kol’chenko, A. G. Nikitenko, and Y. Troitski, “Control of the structure of transverse modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980).
[CrossRef]

Other (4)

Y. Ananev, Laser Resonators and the Beam Divergence Problem, Adam Hilger Series on Optics and Optoelectronics (Adam Hilger, 1992).

Y. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, “Lasers with fast electronic synthesis of spectral line shape,” in Advances in Laser and Optic Research (Nova Science, 2002), Vol.  1, pp. 128–161.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, pp. 325–386.
[CrossRef]

C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, World Student series (Addison-Wesley, 1989).

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Figures (16)

Fig. 1
Fig. 1

Schematic diagram of the resonator: M 1 , a conical reflector (CR) with head angle β and diameter 2 a 1 ; M 2 , a spherical mirror with curvature radius R and diameter 2 a 2 ; and L, resonator length.

Fig. 2
Fig. 2

Dependence of | Λ j ( n ) ( α ) | 2 for mode subgroups R T j n in resonators with (a)  g = 0.9 , b 1 = 3.1 , b 2 = 4 ; (b)  g = 0.9 , b 1 = 3.6 , b 2 = 5 . Here and in Figs. 3, 4, 5: solid lines, R T j 0 ; dashed lines, R T j 1 ; short dashed line, R T 02 , dashed–dotted line, R T 03 .

Fig. 3
Fig. 3

Same as Fig. 2 for resonators with (a)  g = 0.9 , b 1 = 4.2 , b 2 = 5 , and (b)  g = 1 , b 1 = b 2 = 2.4 . Dashed–double-dotted line, R T 04 .

Fig. 4
Fig. 4

Dependence of | Λ ( α ) | 2 for subgroups R E 00 , R T 01 , and R E 02 in resonators with (a)  g = 0.9 , b 1 = 3.1 , b 2 = 4 , and disk with D 2 = d 2 / a 2 = 0.1 ; and (b)  g = 1 , b 1 = b 2 = 2.4 , D 2 = 0.15 .

Fig. 5
Fig. 5

Dependence of | Λ ( α ) | 2 for subgroups R E 00 , R T 10 , R T 01 , R E 02 , and R E 03 for g = 1 , b 1 = b 2 = 2.67 , with the disk on (a) CR ( D 1 = 0.2 ) and (b)  M 2 ( D 2 = 0.1 ).

Fig. 6
Fig. 6

Dependence of | Λ j ( n ) ( α ) | 2 for modes E ( k , m ) n in resonators with g = 0.9 , b 1 = 3.1 , b 2 = 4 : (a) without filter, and (b) with filter as an absorbing strip mask of w = a 1 / 60 . Here and in Figs. 8, 9, 10, 11, 12, 14, 15, 16: solid line, mode subset E ( 1 , 1 ) n ; dashed line, E ( 1 , 1 ) n ; dashed–dotted line, E ( 1 , 1 ) n ; and dotted line, E ( 1 , 1 ) n ).

Fig. 7
Fig. 7

Configuration of modes: (a)  E ( 1 , 1 ) 0 , and (b)  E ( 1 , 1 ) 1 in resonator without filter.

Fig. 8
Fig. 8

Dependence of | Λ ( α ) | 2 for mode subsets (a)  E ( 1 , 1 ) n , (b)  E ( 1 , 1 ) n , and (c)  E ( 1 , 1 ) n for the resonator with g = 0.9 , b 1 = 3.6 , b 2 = 5 .

Fig. 9
Fig. 9

Dependence of | Λ ( α ) | 2 for the resonator with g = 0.9 , b 1 = 3.6 , b 2 = 5 : (a) without spatial filter, and (b) with filter as the strip mask along the O x axis with w = a 1 / 30 .

Fig. 10
Fig. 10

Same as in Fig. 9 for g = 1 , b 1 = b 2 = 2.4 , w = a 1 / 60 .

Fig. 11
Fig. 11

Same as in Fig. 9a for g = 1 , b 1 = b 2 = 2.67 (without filter).

Fig. 12
Fig. 12

Dependence of | Λ ( α ) | 2 for subsets: (a)  E ( 1 , 1 ) n and (b)  E ( 1 , 1 ) n for g = 1 , b 1 = b 2 = 2.67 with the strip mask along O x .

Fig. 13
Fig. 13

Spatial configuration of the mode E ( 1 , 1 ) 0 for g = 1 , b 1 = b 2 = 2.67 , and with strip mask along O x .

Fig. 14
Fig. 14

Same as in Fig. 11 for the resonator with the crossed filter: two crossed strip masks with w 1 = a 1 / 40 .

Fig. 15
Fig. 15

Selection of the mode E ( 1 , 1 ) 0 in the resonator with different shape diaphragms and the filter as the strip masks along O x ( w 1 = a 1 / 40 ), and (a)  g = 0.9 , b 1 = 3.1 , b 2 = 5 , (b)  g = 1 , b 1 = b 2 = 2.3 .

Fig. 16
Fig. 16

Same as in Fig. 15 for selection of modes E ( 1 , 1 ) 0 and E ( 1 , 1 ) 0 for g = 0.9 , b 1 = 3.6 , with strip masks with w 1 = a 1 / 60 .

Equations (7)

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Λ ˜ ( n ) U ( n ) ( η 1 ) = 0 1 G n ( η 1 , η 2 ) U ( n ) ( η 2 ) η 2 d η 2 ,
G n ( η 1 , η 2 ) = i 2 ( n + 1 ) b 1 2 b 2 2 P 1 ( η 2 ) 0 1 exp [ i 2 g 2 b 2 2 η 3 2 ] J n ( b 1 b 2 η 1 η 3 ) J n ( b 1 b 2 η 2 η 3 ) η 3 d η 3 ,
R T j n ( η 2 , φ , φ 0 ) = U j ( n ) ( η 2 ) cos ( n φ + φ 0 ) ; R E j n ( η 2 , φ ) = U j ( n ) ( η 2 ) exp ( i n φ + φ 0 ) ,
E ( 1 , 1 ) n ( x , y ) , E ( 1 , 1 ) n ( x , y ) , E ( 1 , 1 ) n ( x , y ) , E ( 1 , 1 ) n ( x , y ) ,
Λ ˜ ( k , m ) n E ( k , m ) n ( 1 ) ( x 3 , y 3 ) = S 1 d x 1 d y 1 P 1 ( x 1 , y 1 ) K ˜ ( k , m ) ( x 1 , y 1 ; x 3 , y 3 ) E ( k , m ) n ( 1 ) ( x 1 , y 1 )
K ˜ ( k , m ) ( x 1 , y 1 ; x 3 , y 3 ) = S M 2 d x 2 d y 2 P 2 ( x 2 , y 2 ) K ( k , m ) ( x 1 , y 1 ; x 2 , y 2 ) K ( k , m ) ( x 3 , y 3 ; x 2 , y 2 )
K ( k , m ) ( x 1 , y 1 ; x 2 , y 2 ) = [ G ( y 1 , y 2 ) Y ˜ m + G ( y 1 , y 2 ) ] [ G ( x 1 , x 2 ) X ˜ k + G ( x 1 , x 2 ) ] ,

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