Abstract

A grating may be employed to fold a laser resonator, thereby enabling the simultaneous extraction of several frequencies from the laser. This limits the range of useful grating constants as compared to the Littrow case. The astigmatism introduced by the grating is examined through the use of σ-circle diagrams. It will be seen that the grating performs an angle-dependent transformation on the σ-circle representative of the plane of diffraction to make the mode astigmatic. Some consequences of this related to the design of similar resonators are discussed.

© 1978 Optical Society of America

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References

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  1. A. Hardy, D. Treves, Appl. Opt. 14, 589 (1975).
    [CrossRef] [PubMed]
  2. T. M. Hard, Appl. Opt. 9, 1825 (1970).
    [CrossRef] [PubMed]
  3. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  4. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  5. M. J. Offerhaus, Philips Res. Rep. 19, 520 (1964).
  6. G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 379.
  7. P. Laures, Appl. Opt. 6, 747 (1967).
    [CrossRef] [PubMed]
  8. D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart and Winston, New York, 1969), Chap. 4.
  9. S. O. Kanstad, A. Bjerkestrand, T. Lund, J. Phys. E: Sci Instr. (in press).

1975 (1)

1970 (1)

1967 (1)

1966 (1)

1965 (1)

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

1964 (1)

M. J. Offerhaus, Philips Res. Rep. 19, 520 (1964).

Bell, W. E.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart and Winston, New York, 1969), Chap. 4.

Bjerkestrand, A.

S. O. Kanstad, A. Bjerkestrand, T. Lund, J. Phys. E: Sci Instr. (in press).

Deschamps, G. A.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 379.

Hard, T. M.

Hardy, A.

Kanstad, S. O.

S. O. Kanstad, A. Bjerkestrand, T. Lund, J. Phys. E: Sci Instr. (in press).

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

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

Laures, P.

Li, T.

Lund, T.

S. O. Kanstad, A. Bjerkestrand, T. Lund, J. Phys. E: Sci Instr. (in press).

Mast, P. E.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 379.

Offerhaus, M. J.

M. J. Offerhaus, Philips Res. Rep. 19, 520 (1964).

Sinclair, D. C.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart and Winston, New York, 1969), Chap. 4.

Treves, D.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

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

Philips Res. Rep. (1)

M. J. Offerhaus, Philips Res. Rep. 19, 520 (1964).

Other (3)

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 379.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart and Winston, New York, 1969), Chap. 4.

S. O. Kanstad, A. Bjerkestrand, T. Lund, J. Phys. E: Sci Instr. (in press).

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Figures (6)

Fig. 1
Fig. 1

Design of a dual-wavelength laser. M1 and M2 are the spherical rear mirrors, M3 is a plane outcoupling mirror, G is the grating, and A the amplifying medium.

Fig. 2
Fig. 2

Transformation of the beam parameters w (linear extent of cross section) and β (angular extent of the mode) in the xz-plane. Primed and double primed variables describe the mode when incident on the grating from the amplifying medium and after diffraction toward a rear mirror, respectively.

Fig. 3
Fig. 3

Transformation by the grating of the wavefronts and σ-circles in the xz-plane (for kα = 0.67). The heavily drawn lines represent the resulting mode as defined by the rear mirror with radius of curvature Rj at z = −l, by the grating transformation matrix Gα, and by the beam waist at z = z′. To the left of the grating, the mode is given by the σj-circle and to the right by the σe-circle. The σj-circle scales into the σe-circle by the factor kα around the grating at z = 0.

Fig. 4
Fig. 4

An ill-designed laser, using similar rear mirrors (i.e., a common σj-circle) in the two branches, with kα = 0.67 and kα = 1.5, respectively. The corresponding beam waist cross sections are shown to the right.

Fig. 5
Fig. 5

A carefully designed laser with positions and radii of curvature of the rear mirrors chosen to make the σj-circle of one branch transform into the extreme σe-circle of the other branch and vice versa. Note that the ellipticity of the cross sections always changes direction through the grating.

Fig. 6
Fig. 6

As in Fig. 5, but with near-optimum circle diagram for a spherical front mirror. The beam waists of the yz-plane are located at z = z1 and z = z2, respectively, for the two branches, while in the xz-plane the beam waists are found somewhere between. It is seen that the corresponding confocal parameters b o 1 y, b o 2 y, b o k x, and may be made to differ very little.

Equations (23)

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m λ j = d ( sin α + sin α j ) ,
sin α j = m λ j / d sin α > 1
sin α j = m λ j / d sin α < 1 .
1 λ j / d < sin α < 2 λ j / d 1 .
0.5 λ j < d < λ j .
w / w = cos α j / cos α = k α 1 / 2 .
β / β = d α j / d α = k α 1 / 2 .
( w β ) into ( w β ) .
G α = ( k α 1 / 2 0 0 k α 1 / 2 ) .
1 q = 1 R i λ π w 2 ,
G = ( A B C D ) ,
q = k α 1 / 2 q / k α 1 / 2 = k α 1 q .
R = k α 1 R .
R z = ( b o 2 + z 2 ) / z ,
b z = ( b o 2 + z 2 ) / b o ,
R = ( b o 2 + z 2 ) / z ,
b = ( b o 2 + z 2 ) / b o .
R = ( b o 2 + z 2 ) / z ,
b = ( b o 2 + z 2 ) / b o .
b o = k α b o ,
z = k α z .
k α R = [ b o 2 + ( z + k α l ) 2 ] / ( z + k α l )
R j > k α 1 L + l j .

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