Abstract

Coupled simulation based on intracavity partially coherent light model and 3D CFD model is firstly achieved in this paper. The dynamic equation of partially coherent intracavity field is derived based on partially coherent light theory. A numerical scheme for the coupled simulation as well as a method for computing the intracavity partially coherent field is given. The presented model explains the formation of the sugar scooping phenomenon, and enables studies on the dependence of the spatial mode spectrum on physical parameters of laser cavity and gain medium. Computational results show that as the flow rate of iodine increases, higher order mode components dominate in the partially coherent field. Results obtained by the proposed model are in good agreement with experimental results.

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    [CrossRef]
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    [CrossRef]
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2010 (1)

K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “Comparing modeling and measurements of the output power in chemical oxygen-iodine lasers: a stringent test of I2 dissociation mechanisms,” J. Chem. Phys. 133(8), 084301 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

G. F. Calvo, A. Picon, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100(17), 173902 (2008).
[CrossRef] [PubMed]

2007 (1)

M. Endo, T. Masuda, and T. Uchiyama, “Development of hybrid simulation for supersonic chemical oxygen-iodine laser,” AIAA J. 45(1), 90–97 (2007).
[CrossRef]

2006 (1)

M. Suzuki, H. Matsueda, and W. Masuda, “Numerical simulation of Q-switched supersonic flow chemical oxygen-iodine laser by solving time-dependent paraxial wave equation,” JSME Int. J. Ser. B 49, 1212–1219 (2006).

2005 (1)

M. Guizar-Sicairos and J. C. Gutiérrez-Vera, “Coupled mode competition in unstable resonators using the exact cavity equations of motion with dynamic gain,” J. Opt. B Quantum Semiclassical Opt. 7(9), 253–263 (2005).
[CrossRef]

2004 (1)

E. A. Duff and K. A. Truesdell, “Chemical oxygen iodine laser (COIL) technology and development,” Proc. SPIE 5414, 52–68 (2004).
[CrossRef]

2003 (1)

T. J. Madden, “Aspects of 3D chemical oxygen-iodine laser simulation,” Proc. SPIE 5120, 363–375 (2003).
[CrossRef]

2002 (1)

D. Yu, F. Sang, Y. Jin, and Y. Sun, “Output beam analysis of an unstable resonator with a large Fresnel number for a chemical oxygen iodine laser,” Opt. Eng. 41(10), 2668–2674 (2002).
[CrossRef]

1999 (1)

1998 (2)

1996 (1)

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

1993 (3)

D. A. Copeland and A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29(9), 2525–2539 (1993).
[CrossRef]

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-Cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

1990 (1)

D. A. Copeland, C. Warner, and A. H. Bauer, “Simple model for optical extraction from a flowing oxygen-iodine medium using a Fabry-Perot resonator,” Proc. SPIE 1224, 474–499 (1990).
[CrossRef]

1986 (1)

1983 (1)

1982 (1)

J. Bachar and S. Rosenwaks, “An efficient, small scale chemical oxygen-iodine laser,” Appl. Phys. Lett. 41(1), 16–17 (1982).
[CrossRef]

1980 (1)

T. T. Yang, “Modeling of cw HF chemical lasers with rotational nonequilibrium,” J. Phys. C9, 51–57 (1980).

1975 (1)

1974 (1)

1961 (1)

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

Bachar, J.

J. Bachar and S. Rosenwaks, “An efficient, small scale chemical oxygen-iodine laser,” Appl. Phys. Lett. 41(1), 16–17 (1982).
[CrossRef]

Barmashenko, B.

Barmashenko, B. D.

K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “Comparing modeling and measurements of the output power in chemical oxygen-iodine lasers: a stringent test of I2 dissociation mechanisms,” J. Chem. Phys. 133(8), 084301 (2010).
[CrossRef] [PubMed]

B. D. Barmashenko, “Analysis of lasing in chemical oxygen-iodine lasers with unstable resonators using a geometric-optics model,” Appl. Opt. 48(13), 2542–2550 (2009).
[CrossRef] [PubMed]

Bauer, A. H.

D. A. Copeland and A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29(9), 2525–2539 (1993).
[CrossRef]

D. A. Copeland, C. Warner, and A. H. Bauer, “Simple model for optical extraction from a flowing oxygen-iodine medium using a Fabry-Perot resonator,” Proc. SPIE 1224, 474–499 (1990).
[CrossRef]

Bhowmik, A.

Borghi, R.

Calvo, G. F.

G. F. Calvo, A. Picon, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100(17), 173902 (2008).
[CrossRef] [PubMed]

Copeland, D. A.

D. A. Copeland and A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29(9), 2525–2539 (1993).
[CrossRef]

D. A. Copeland, C. Warner, and A. H. Bauer, “Simple model for optical extraction from a flowing oxygen-iodine medium using a Fabry-Perot resonator,” Proc. SPIE 1224, 474–499 (1990).
[CrossRef]

Crowell, P.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

Duff, E. A.

E. A. Duff and K. A. Truesdell, “Chemical oxygen iodine laser (COIL) technology and development,” Proc. SPIE 5414, 52–68 (2004).
[CrossRef]

Endo, M.

M. Endo, T. Masuda, and T. Uchiyama, “Development of hybrid simulation for supersonic chemical oxygen-iodine laser,” AIAA J. 45(1), 90–97 (2007).
[CrossRef]

M. Endo, M. Kawakami, K. Nanri, S. Takeda, and T. Fujioka, “Two-dimensional simulation of an unstable resonator with a stable core,” Appl. Opt. 38(15), 3298–3307 (1999).
[CrossRef] [PubMed]

Erkkila, J.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

Fox, A. G.

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

Fujioka, T.

Furman, D.

Gori, F.

Gu, C.

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Guattari, G.

Guizar-Sicairos, M.

M. Guizar-Sicairos and J. C. Gutiérrez-Vera, “Coupled mode competition in unstable resonators using the exact cavity equations of motion with dynamic gain,” J. Opt. B Quantum Semiclassical Opt. 7(9), 253–263 (2005).
[CrossRef]

Gutiérrez-Vera, J. C.

M. Guizar-Sicairos and J. C. Gutiérrez-Vera, “Coupled mode competition in unstable resonators using the exact cavity equations of motion with dynamic gain,” J. Opt. B Quantum Semiclassical Opt. 7(9), 253–263 (2005).
[CrossRef]

Hager, G. D.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

Helms, C. A.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

Jin, Y.

D. Yu, F. Sang, Y. Jin, and Y. Sun, “Output beam analysis of an unstable resonator with a large Fresnel number for a chemical oxygen iodine laser,” Opt. Eng. 41(10), 2668–2674 (2002).
[CrossRef]

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Kawakami, M.

Li, F.

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Li, M.

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Li, T.

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

Madden, T. J.

T. J. Madden, “Aspects of 3D chemical oxygen-iodine laser simulation,” Proc. SPIE 5120, 363–375 (2003).
[CrossRef]

Masuda, T.

M. Endo, T. Masuda, and T. Uchiyama, “Development of hybrid simulation for supersonic chemical oxygen-iodine laser,” AIAA J. 45(1), 90–97 (2007).
[CrossRef]

Masuda, W.

M. Suzuki, H. Matsueda, and W. Masuda, “Numerical simulation of Q-switched supersonic flow chemical oxygen-iodine laser by solving time-dependent paraxial wave equation,” JSME Int. J. Ser. B 49, 1212–1219 (2006).

Matsueda, H.

M. Suzuki, H. Matsueda, and W. Masuda, “Numerical simulation of Q-switched supersonic flow chemical oxygen-iodine laser by solving time-dependent paraxial wave equation,” JSME Int. J. Ser. B 49, 1212–1219 (2006).

Nanri, K.

Pang, J.

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Picon, A.

G. F. Calvo, A. Picon, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100(17), 173902 (2008).
[CrossRef] [PubMed]

Plummer, D.

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

Rosenwaks, S.

K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “Comparing modeling and measurements of the output power in chemical oxygen-iodine lasers: a stringent test of I2 dissociation mechanisms,” J. Chem. Phys. 133(8), 084301 (2010).
[CrossRef] [PubMed]

B. Barmashenko, D. Furman, and S. Rosenwaks, “Analysis of lasing in gas-flow lasers with stable resonators,” Appl. Opt. 37(24), 5697–5705 (1998).
[CrossRef] [PubMed]

J. Bachar and S. Rosenwaks, “An efficient, small scale chemical oxygen-iodine laser,” Appl. Phys. Lett. 41(1), 16–17 (1982).
[CrossRef]

Sang, F.

D. Yu, F. Sang, Y. Jin, and Y. Sun, “Output beam analysis of an unstable resonator with a large Fresnel number for a chemical oxygen iodine laser,” Opt. Eng. 41(10), 2668–2674 (2002).
[CrossRef]

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Santarsiero, M.

Siegman, A. E.

Sun, Y.

D. Yu, F. Sang, Y. Jin, and Y. Sun, “Output beam analysis of an unstable resonator with a large Fresnel number for a chemical oxygen iodine laser,” Opt. Eng. 41(10), 2668–2674 (2002).
[CrossRef]

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

Suzuki, M.

M. Suzuki, H. Matsueda, and W. Masuda, “Numerical simulation of Q-switched supersonic flow chemical oxygen-iodine laser by solving time-dependent paraxial wave equation,” JSME Int. J. Ser. B 49, 1212–1219 (2006).

Sziklas, E. A.

Takeda, S.

Townsend, S. W.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-Cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

Truesdell, K. A.

E. A. Duff and K. A. Truesdell, “Chemical oxygen iodine laser (COIL) technology and development,” Proc. SPIE 5414, 52–68 (2004).
[CrossRef]

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

Uchiyama, T.

M. Endo, T. Masuda, and T. Uchiyama, “Development of hybrid simulation for supersonic chemical oxygen-iodine laser,” AIAA J. 45(1), 90–97 (2007).
[CrossRef]

Waichman, K.

K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “Comparing modeling and measurements of the output power in chemical oxygen-iodine lasers: a stringent test of I2 dissociation mechanisms,” J. Chem. Phys. 133(8), 084301 (2010).
[CrossRef] [PubMed]

Warner, C.

D. A. Copeland, C. Warner, and A. H. Bauer, “Simple model for optical extraction from a flowing oxygen-iodine medium using a Fabry-Perot resonator,” Proc. SPIE 1224, 474–499 (1990).
[CrossRef]

Wolf, E.

Yang, T. T.

T. T. Yang, “Modeling of cw HF chemical lasers with rotational nonequilibrium,” J. Phys. C9, 51–57 (1980).

Yu, D.

D. Yu, F. Sang, Y. Jin, and Y. Sun, “Output beam analysis of an unstable resonator with a large Fresnel number for a chemical oxygen iodine laser,” Opt. Eng. 41(10), 2668–2674 (2002).
[CrossRef]

Zambrini, R.

G. F. Calvo, A. Picon, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100(17), 173902 (2008).
[CrossRef] [PubMed]

Zhuang, Q.

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

AIAA J. (1)

M. Endo, T. Masuda, and T. Uchiyama, “Development of hybrid simulation for supersonic chemical oxygen-iodine laser,” AIAA J. 45(1), 90–97 (2007).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

J. Bachar and S. Rosenwaks, “An efficient, small scale chemical oxygen-iodine laser,” Appl. Phys. Lett. 41(1), 16–17 (1982).
[CrossRef]

Bell Syst. Tech. J. (1)

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

High Power Laser Part. Beams (1)

F. Sang, C. Gu, J. Pang, M. Li, F. Li, Y. Sun, Y. Jin, and Q. Zhuang, “Experimental study of a cw chemical oxygen-iodine laser,” High Power Laser Part. Beams 5, 389–393 (1993) (In Chinese).

IEEE J. Quantum Electron. (3)

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-Cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

D. A. Copeland and A. H. Bauer, “Optical saturation and extraction from the chemical oxygen-iodine laser medium,” IEEE J. Quantum Electron. 29(9), 2525–2539 (1993).
[CrossRef]

G. D. Hager, C. A. Helms, K. A. Truesdell, D. Plummer, J. Erkkila, and P. Crowell, “A simplified analytic model for gain saturation and power extraction in the flowing chemical oxygen-iodine laser,” IEEE J. Quantum Electron. 32(9), 1525–1536 (1996).
[CrossRef]

J. Chem. Phys. (1)

K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “Comparing modeling and measurements of the output power in chemical oxygen-iodine lasers: a stringent test of I2 dissociation mechanisms,” J. Chem. Phys. 133(8), 084301 (2010).
[CrossRef] [PubMed]

J. Opt. B Quantum Semiclassical Opt. (1)

M. Guizar-Sicairos and J. C. Gutiérrez-Vera, “Coupled mode competition in unstable resonators using the exact cavity equations of motion with dynamic gain,” J. Opt. B Quantum Semiclassical Opt. 7(9), 253–263 (2005).
[CrossRef]

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

J. Phys. (1)

T. T. Yang, “Modeling of cw HF chemical lasers with rotational nonequilibrium,” J. Phys. C9, 51–57 (1980).

JSME Int. J. Ser. B (1)

M. Suzuki, H. Matsueda, and W. Masuda, “Numerical simulation of Q-switched supersonic flow chemical oxygen-iodine laser by solving time-dependent paraxial wave equation,” JSME Int. J. Ser. B 49, 1212–1219 (2006).

Opt. Eng. (1)

D. Yu, F. Sang, Y. Jin, and Y. Sun, “Output beam analysis of an unstable resonator with a large Fresnel number for a chemical oxygen iodine laser,” Opt. Eng. 41(10), 2668–2674 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

G. F. Calvo, A. Picon, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100(17), 173902 (2008).
[CrossRef] [PubMed]

Proc. SPIE (3)

T. J. Madden, “Aspects of 3D chemical oxygen-iodine laser simulation,” Proc. SPIE 5120, 363–375 (2003).
[CrossRef]

E. A. Duff and K. A. Truesdell, “Chemical oxygen iodine laser (COIL) technology and development,” Proc. SPIE 5414, 52–68 (2004).
[CrossRef]

D. A. Copeland, C. Warner, and A. H. Bauer, “Simple model for optical extraction from a flowing oxygen-iodine medium using a Fabry-Perot resonator,” Proc. SPIE 1224, 474–499 (1990).
[CrossRef]

Other (7)

A. I. Lampson, D. N. Plummer, J. Erkkila, and P. G. Crowell, “Chemical oxygen iodine laser (COIL) beam quality predictions using 3-d Navier-Stokes (MINT) and wave optics (OCELOT) codes,” presented at 29th AIAA Plasmadynamics and Lasers Conference, Albuquerque, NM, 15–18 June, 1998.

R. C. Buggeln, S. Shamroth, A. Lampson, and P. G. Crowell, “Three-dimensional (3-D) Navier-Stokes analysis of the mixing and power extraction in a supersonic chemical oxygen iodine laser (COIL) with transverse I2 injection,” presented at 25th AIAA Plasmadynamics and Lasers Conference, Colorado Springs, CO, 20–23 June, 1994.

J. Paschkewitz, J. Shang, J. Miller, and T. Madden, “An assessment of COIL physical property and chemical kinetic modeling methodologies,” presented at 31st AIAA Plasmadynamics and Lasers Conference, Denver, CO, 19–22 June, 2000.

M. Hishida, N. Azami, K. Iwamoto, W. Masuda, H. Fujii, T. Atsutu, and M. Muro, “Flow and optical fields in a supersonic flow chemical oxygen-iodine laser,” presented at 28th Plasmadynamics and Lasers Conference, Atlanta, GA, 23–25 June, 1997.

Y. Huai, S. Jia, and Y. Jin, “Analysis and optimization of mixing process with large eddy simulation: An application to SCOIL,” presented at 40th AIAA Plasmadynamics and Lasers Conference, San Antonio, TX, 22–25 June, 2009.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

K. Shimizu and S. Yoshida, “High power chemical oxygen-iodine laser of good beam quality,” in Lasers '89; Proceedings of the International Conference (STS Press, 1990), pp. 218—222.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the coupled computation process.

Fig. 2
Fig. 2

Output intensity of the stable resonator.

Fig. 3
Fig. 3

Saturated gain.

Fig. 4
Fig. 4

Small signal gain.

Fig. 5
Fig. 5

Normalized spatial mode spectrum of the output field (mass fraction of molecular Iodine is 30%).

Fig. 6
Fig. 6

Normalized spatial mode spectrum under different mass fraction of molecular Iodine (a) 10% (b) 15% (c) 20% (d) 50%

Fig. 7
Fig. 7

Calculated intensity distribution by (a) the Fabry-Perot model and (b) the roof-top model.

Fig. 8
Fig. 8

Comparison of intensity distribution along x axis. (a) The experimental result. (b) The proposed model. (c) The Fabry-Perot model. (d) The roof-top model.

Tables (1)

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Table 1 Boundary Conditions for the CFD Calculation

Equations (30)

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U( r )= n a n ϕ n ( r ),
a n * a m = | a n | 2 δ mn .
U( ρ,z )= n a n ( z ) ϕ n ( ρ,z ),
ϕ n * ( ρ,z ) ϕ n ( ρ,z )dρ = δ n n .
I( ρ,z )= | U( ρ,z ) | 2 = n λ n ( z ) | ϕ n ( ρ,z ) | 2 .
[ ρ 2 2ik( z 1 2 g( ρ,z ) ) ]U( ρ,z )=0,
[ ρ 2 2ik z ] ϕ n ( ρ,z )=0,
z a n ( z ) 1 2 n G n n (z) a n (z) =0,
G n n (z)= ϕ n * ( ρ,z )g( ρ,z ) ϕ n ( ρ,z )dρ.
z [ a n ( z ) a n * ( z ) ] 1 2 n Re[ G n n (z) a n * ( z ) a n (z) ] =0,
z λ n + ( z ) λ n + ( z )Re( G nn (z) )=0.
z λ n ( z )+ λ n ( z )Re( G nn (z) )=0.
ϕ n ( ρ,z )= ϕ n 1 ( x,z ) ϕ n 2 ( y,z ),
ϕ n 1 ( x,z )= [ Q * ( z ) n 1 Q( z ) ] 1/2 2x ω( z ) ϕ n 1 1 ( x,z ) ( n 1 1 n 1 ) 1/2 Q * ( z ) Q( z ) ϕ n 1 2 ( x,z ),
ϕ 0 ( x,z )= ( 2 π ) 1/4 [ 1 ω 0 Q( z ) ] 1/2 exp[ jk x 2 2q( z ) ],
ϕ 1 ( x,z )= [ Q * ( z ) Q( z ) ] 1/2 [ 2x ω( z ) ] ϕ 0 ( x,z ).
1 q = 1 R( z ) jλ π ω 2 ( z ) ,
q( z )=q( z 0 )+z z 0 ,
Q( z )=q( z )/q( z 0 ),
1 q ( z 2 ) = 1 q( z 2 ) 2 R 2 ,
q ( z )= q ( z 2 )z+ z 2 ,
Q ( z )= q ( z ) q ( z 2 ) Q( z 2 ).
ω 0 2 = λ π [ ( R 1 L )( R 2 L )( R 1 + R 2 L )L ( R 1 + R 2 2L ) 2 ] 1/2
z 0 = z M 1 + ( R 2 L )L R 1 + R 2 2L
q( z 0 )= jπ ω 0 2 λ
λ n ( z 2 )= λ n + ( z 2 ).
λ n + ( z 1 )=( 1α ) λ n ( z 1 ),
λ n out ( z 1 )=α λ n ( z 1 ).
n xmax ω 0 a x .
n ymax ω 0 a y .

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