Abstract

Gaussian beam propagation in a gradient-radial-index medium is presented and discussed. A simple geometrical method for modeling the Gaussian beam inside the gradient-index medium has been developed. The results with this method and those predicted by wave optics have been compared in a number of practical cases. Results show the validity of the ray approach. This is also confirmed by experimental tests on a high-power laser–fiber coupler.

© 1992 Optical Society of America

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References

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  1. S. D. Cusworth, J. M. Senior, “Angular tilt misalignment loss at a GRIN rod lens,” Appl. Opt. 25, 1775–1779 (1986).
    [CrossRef] [PubMed]
  2. T. Sakamoto, “Coupling loss analysis on a multimode fiber directional coupler using GRIN-rod lenses,” Appl. Opt. 25, 2620–2625 (1986).
    [CrossRef] [PubMed]
  3. R. Falciai, A. M. Scheggi, F. Cosi, “Micro-optic two-wavelength demultiplexer,” Int. J. Opt. 3, 509–512 (1988).
  4. M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).
  5. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  6. K. Koizumi, K. Nishizawa, K. Sono, eds., Selfoc Handbook (Nippon Sheet Glass, Tokyo, 1980).

1988

R. Falciai, A. M. Scheggi, F. Cosi, “Micro-optic two-wavelength demultiplexer,” Int. J. Opt. 3, 509–512 (1988).

1986

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Brenci, M.

M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).

Conforti, G.

M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).

Cosi, F.

R. Falciai, A. M. Scheggi, F. Cosi, “Micro-optic two-wavelength demultiplexer,” Int. J. Opt. 3, 509–512 (1988).

Cusworth, S. D.

Falciai, R.

R. Falciai, A. M. Scheggi, F. Cosi, “Micro-optic two-wavelength demultiplexer,” Int. J. Opt. 3, 509–512 (1988).

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Mencaglia, A.

M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).

Mignani, A. G.

M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).

Sakamoto, T.

Scheggi, A. M.

R. Falciai, A. M. Scheggi, F. Cosi, “Micro-optic two-wavelength demultiplexer,” Int. J. Opt. 3, 509–512 (1988).

M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).

Senior, J. M.

Appl. Opt.

Int. J. Opt.

R. Falciai, A. M. Scheggi, F. Cosi, “Micro-optic two-wavelength demultiplexer,” Int. J. Opt. 3, 509–512 (1988).

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other

K. Koizumi, K. Nishizawa, K. Sono, eds., Selfoc Handbook (Nippon Sheet Glass, Tokyo, 1980).

M. Brenci, G. Conforti, A. Mencaglia, A. G. Mignani, A. M. Scheggi, “Fiber to fiber coupling in a multichannel sensor system,” in Optical Interconnections, O. D. D. Soares, G. C. Righini, eds., Proc. Soc. Photo-Opt. Instrum. Eng.862, 88–90 (1987).

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Figures (4)

Fig. 1
Fig. 1

Propagation of a Gaussian beam in a graded medium.

Fig. 2
Fig. 2

Behavior of (a) the spot size w and (b) the divergence dw/dz in a GRIN rod illuminated by single-mode fiber as a function of Ωz. (c) wg − w and (d) dwg/dz − dw/dz show the difference between Gaussian beam and ray optics treatment method.

Fig. 3
Fig. 3

Coupling efficiency of the laser–fiber coupler versus fiber transversal misalignments.

Fig. 4
Fig. 4

Laser–fiber connector before final assembling.

Equations (17)

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2 ψ + k 2 n 2 ψ = 0
ψ = A ( z ) exp [ - i k r 2 Q ( z ) ] , Q ( z ) = 1 R ( z ) - 2 i k w ( z ) 2 ,
w 2 ( z ) = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ] , R ( z ) = z [ 1 + ( π w 0 2 λ z ) 2 ] = w d w d z ,
d Q d z + Q 2 + Ω 2 = 0.
Q ( z ) = Ω ( Q 0 Ω - tan Ω z ) 1 + Q 0 Ω tan Ω z , Q 0 = Q ( 0 ) = 1 n 0 R 1 - 2 i k n 0 w 1 2 ,
w ( z ) = [ - 2 k n 0 1 Im ( Q ) ] 1 / 2 . = [ ( w 1 cos Ω z + d w 1 d z 1 Ω n 0 sin Ω z ) 2 + 4 sin 2 Ω z Ω 2 k 2 n 0 2 w 1 2 ] 1 / 2 ,
d w d z = ( w 1 cos Ω z + d w 1 d z 1 Ω n 0 sin Ω z ) ( 1 n 0 d w 1 d z cos Ω z - Ω w 1 sin Ω z ) + λ 2 sin Ω z cos Ω z π w 1 2 Ω n 0 2 [ ( w 1 cos Ω z + d w 1 d z 1 Ω n 0 sin Ω z ) 2 + 4 sin 2 Ω z Ω 2 k 2 n 0 2 w 1 2 ] 1 / 2 .
tan 2 Ω z = - π w 1 3 d w 1 d z Ω n 0 λ 2 + ( d w 1 d z ) 2 π w 1 2 - Ω 2 w 1 4 π n 0 2 .
r ( z ) = r 1 cos Ω z + d r 1 d z 1 Ω n 0 sin Ω z ,
d r d z = - Ω r 1 sin Ω z + d r 1 1 d z n 0 cos Ω z ,
w g ( z ) = w 1 cos Ω z + Θ Ω n 0 sin Ω z ,
d w g d z = - Ω w 1 sin Ω z + Θ n 0 cos Ω z .
w = ( w 0 2 cos 2 Ω z + λ 2 π 2 w 0 2 Ω 2 n 0 2 sin 2 Ω z ) 1 / 2 ,
d w d z = 1 2 ( - Ω w 0 2 + λ 2 π 2 w 0 2 Ω n 0 2 ) × sin 2 Ω z ( w 0 2 cos 2 Ω z + λ 2 sin 2 Ω z Ω 2 π 2 w 0 2 n 0 2 ) 1 / 2 .
w ( z ) = ( w g 2 + 4 sin 2 Ω z Ω 2 k 2 n 0 2 w 1 2 ) 1 / 2 ,
Ω 2 ¯ = Ω 2 ( 1 - r 1 2 Ω 2 ) cos 2 d r 1 d z ,
w g ( L ) a f ,             | d w g ( L ) d z | NA ,

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