Abstract

From consideration of the physical mechanism of the matching of gaussian modes by means of a single thin lens, a straightforward method is developed for determining the position and the focal length of this lens, when the position and the size of the beam waists are given. The analytic results, although based on a quadratic equation, are unambiguous. An approximate solution is given for the case when the ratio of the beam waists approaches unity. The position of the matching lens may be determined graphically.

© 1971 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  2. S. A. Collins, Appl. Opt. 3, 1263 (1964).
    [CrossRef]
  3. T. Li, Appl. Opt. 3, 1315 (1964).
    [CrossRef]
  4. T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).
  5. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  6. P. Laures, Appl. Opt. 6, 747 (1967).
    [CrossRef] [PubMed]

1967 (1)

1966 (2)

T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

1964 (2)

Chu, T. S.

T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).

Collins, S. A.

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Laures, P.

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

T. Li, Appl. Opt. 3, 1315 (1964).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (2)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

T. S. Chu, Bell Syst. Tech. J. 45, 287 (1966).

Proc. IEEE (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Matching of two coaxially aligned gaussian modes. This figure is a meridian cross section showing the 1/e2 surfaces of the beams, the beam waists 2w01 and 2w02, and the effect of a thin lens on the curvature of the wavefronts.

Fig. 2
Fig. 2

Diagram for determining the normalized distances d1/L and d2/L in function of (w01/w02)2 and the normalized minimum focal length f0/L.

Equations (16)

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f 0 = ( π w 01 w 02 ) / λ .
1 / f = 1 / R 1 + 1 / R 2 ,
w 2 = w 0 2 { 1 + [ ( λ d ) / ( π w 0 2 ) ] 2 } ,
w 01 2 [ 1 + ( λ d 1 π w 01 2 ) 2 ] = w 02 2 [ 1 + ( λ ( L - d 1 ) π w 02 2 ) 2 ] .
( w 02 2 - w 01 2 ) d 1 2 + 2 L w 01 2 d 1 + ( w 01 2 - w 02 2 ) × [ ( π 2 w 01 2 w 02 2 ) / λ 2 ] - w 01 2 L 2 = 0.
d 1 = w 01 2 L ± w 01 w 02 [ L 2 + π 2 ( w 01 2 - w 02 2 ) 2 / λ 2 ] 1 2 w 01 2 - w 02 2 ,
d 2 = w 02 2 L ± w 01 w 02 [ L 2 + π 2 ( w 01 2 - w 02 2 ) 2 / λ 2 ] 1 2 w 02 2 - w 01 2 .
d 1 + d 2 = L .
d 1 = w 01 2 L - w 01 w 02 [ L 2 + π 2 ( w 01 2 - w 02 2 ) 2 / λ 2 ] 1 2 w 01 2 - w 02 2
d 2 = w 02 2 L - w 01 w 02 [ L 2 + π 2 ( w 01 2 - w 02 2 ) 2 / λ 2 ] 1 2 w 02 2 - w 01 2 .
d 1 = L 2 { 1 - w 01 - w 02 w 01 + w 02 [ 4 ( π w 01 w 02 ) λ L ) 2 - 1 ] }
d 2 = L 2 { 1 - w 02 - w 01 w 01 + w 02 [ 4 ( π w 01 w 02 λ L ) 2 - 1 ] } .
d 1 = L 2 { 1 - w 01 - w 02 w 01 + w 02 [ 4 f 0 2 L 2 - 1 ] }
d 2 = L 2 { 1 - w 02 - w 01 w 01 + w 02 [ 4 f 0 2 L 2 - 1 ] } .
1 R i = λ 2 d i π w i 2 w 0 i 2 = d i [ ( π w 0 i 2 ) / λ ] 2 + d i 2             ( i = 1 , 2 )
1 f = d 1 [ ( π w 01 2 ) / λ ] 2 + d 1 2 + d 2 , [ ( π w 02 2 ) / λ ] 2 + d 2 2

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