Abstract

By careful consideration of the precision of available alignment apparatus, mode matching can be accomplished most easily. A function representing the degree of mode matching in a confocal Fabry-Perot interferometer is defined and then evaluated as a function of the angular and translational alignment errors. A method is given for optimizing the beam diameter to take advantage of the available precision of specific instruments. Experimental verification is given for a specific case.

© 1970 Optical Society of America

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References

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  1. R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
    [CrossRef]
  2. M. Hercher, Appl. Opt. 7, 951 (1968).
    [CrossRef] [PubMed]
  3. D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), p. 88ff.

1968 (1)

1964 (1)

Bell, W. E.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), p. 88ff.

Fork, R. L.

Hercher, M.

Herriott, D. R.

Kogelnik, H.

Sinclair, D. C.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), p. 88ff.

Appl. Opt. (2)

Other (1)

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), p. 88ff.

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Figures (6)

Fig. 1
Fig. 1

A schematic diagram of a confocal cavity showing one entering ray and two sets of exiting rays. When the entering ray is exactly aligned with the optical axis of the FPS there will be only one exiting ray path.

Fig. 2
Fig. 2

A family of curves showing the desired beam size for given values of translational and angular adjustment precisions. The values of wd are for λ = 0.6328 μm. For different wavelengths wd = wd (λ′/0.6328)1/2.

Fig. 3
Fig. 3

Experimental setup for mode matching an unexpanded laser beam with FPS spectrum analyzer. Translational controls are incorporated on the beam director attached to the laser. Angular adjustment is provided by the gimbaled analyzer mount.

Fig. 4
Fig. 4

A plot of the degree of mode matching vs translational error for a laser beam of w = 0.43 mm and λ = 0.6328 μm. The solid line represents theoretical values from Eq. (8); experimental values obtained with the setup in Fig. (3) are also shown.

Fig. 5
Fig. 5

A plot of the degree of mode matching vs angular error for a laser beam with w = 0.43 mm and λ = 0.6328 μm. The solid line represents theoretical values from Eq. (11); measured values are plotted showing experimental verification.

Fig. 6
Fig. 6

The output spectra of a He–Ne laser with two axial modes (a) in the nonmode matched and (b) matched conditions. In the mode matched case the transmission and FSR are doubled.

Equations (21)

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D = ( T m - T n ) / ( T m + T n ) ,
A ( x , y ) = A 0 exp [ - ( x 2 + y 2 ) / w 2 ] ,
A 1 ( x , y ) = A 1 exp { - [ ( x + s ) 2 + y 2 ] / w 2 } ,
A 2 ( x , y ) = A 2 exp { - [ ( x - s ) 2 + y 2 ] / w 2 } .
A T = A 1 + A 2 .
I = I ( x , y ) d x d y = ( A 1 A 1 * + A 2 A 2 * + A 1 A 2 * + A 1 * A 2 ) d x d y .
A 1 A 2 * = A 1 * A 2 = A 1 A 2 ;
I m = I 1 + I 2 + 2 ( I 1 I 2 ) 1 2 exp ( - 2 s 2 / w 2 ) .
A 1 A 2 * = A 1 * A 2 = - A 1 A 2 .
I n = I 1 + I 2 - 2 ( I 1 I 2 ) 1 2 exp ( - 2 s 2 / w 2 ) .
T m = I m / I = 1 + exp ( - 2 s 2 / w 2 ) ,
T n = I n / I = 1 - exp ( - 2 s 2 / w 2 ) .
D s = exp ( - 2 s 2 / w 2 ) .
ϕ = ( 2 π / λ ) α x .
A 1 = A 1 exp ( - i ϕ - r 2 / w 2 ) ,
A 2 = A 2 exp ( i ϕ - r 2 / w 2 ) , r 2 = x 2 + y 2 .
I T = I ± [ 1 ± exp ( - 2 π 2 α 2 w 2 / λ 2 ) ] ,
D α = exp ( - 2 π 2 α 2 w 2 / λ 2 ) .
D = D s D α .
D = exp { - 2 [ ( π 2 w 2 Δ α 2 / λ 2 ) + ( Δ s 2 / w 2 ) ] } .
w d = ( λ Δ s / π Δ α ) 1 2 .

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