Abstract

The characteristics of Gaussian beams which are transmitted through complementary Gaussian reflectivity couplers are analyzed. It is shown that the coupled beam properties are strongly dependent on the coupler focal length. The on-axis far-field brightness can be improved by 1.5 and the misalignment sensitivity minimized when the coupler collimates the transmitted beam.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).
  2. L. W. Casperson, “Mode Stability of Lasers and Periodic Optical Systems,” IEEE J. Quantum Electron. QE-10, 629 (1974).
    [CrossRef]
  3. N. McCarthy, P. Lavigne, “Optical Resonators with Gaussian Reflectivity Mirrors: Misalignment Sensitivity,” Appl. Opt. 22, 2704 (1983).
    [CrossRef] [PubMed]
  4. G. Giuliani, Y. K. Park, R. L. Byer, “Radial Birefringent Element and its Application to Laser Resonator Design,” Opt. Lett. 5, 491 (1980).
    [CrossRef] [PubMed]
  5. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971, Chap. 8, pp. 293–345.
  6. A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211 (1975).
    [CrossRef]
  7. A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277 (1965).
    [CrossRef]
  8. P. Lavigne, J.-L. Lachambre, G. Otis, “Analyse Expérimentale et Théorique d’un Laser CO2 TEA Ovec Résonateurs Instables,” Can. J. Phys. 54, 816 (1976).
    [CrossRef]
  9. A. I. Mahan, C. V. Bitterli, S. M. Cannon, “Diffraction Patterns of Complex Apertures,” J. Opt. Soc. Am. 54, 721 (1964).
    [CrossRef]

1983 (1)

1980 (1)

1976 (1)

P. Lavigne, J.-L. Lachambre, G. Otis, “Analyse Expérimentale et Théorique d’un Laser CO2 TEA Ovec Résonateurs Instables,” Can. J. Phys. 54, 816 (1976).
[CrossRef]

1975 (1)

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211 (1975).
[CrossRef]

1974 (1)

L. W. Casperson, “Mode Stability of Lasers and Periodic Optical Systems,” IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

1970 (1)

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).

1965 (1)

A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277 (1965).
[CrossRef]

1964 (1)

Asakura, T.

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211 (1975).
[CrossRef]

Bitterli, C. V.

Byer, R. L.

Cannon, S. M.

Casperson, L. W.

L. W. Casperson, “Mode Stability of Lasers and Periodic Optical Systems,” IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

Giuliani, G.

Lachambre, J.-L.

P. Lavigne, J.-L. Lachambre, G. Otis, “Analyse Expérimentale et Théorique d’un Laser CO2 TEA Ovec Résonateurs Instables,” Can. J. Phys. 54, 816 (1976).
[CrossRef]

Lavigne, P.

N. McCarthy, P. Lavigne, “Optical Resonators with Gaussian Reflectivity Mirrors: Misalignment Sensitivity,” Appl. Opt. 22, 2704 (1983).
[CrossRef] [PubMed]

P. Lavigne, J.-L. Lachambre, G. Otis, “Analyse Expérimentale et Théorique d’un Laser CO2 TEA Ovec Résonateurs Instables,” Can. J. Phys. 54, 816 (1976).
[CrossRef]

Mahan, A. I.

McCarthy, N.

Otis, G.

P. Lavigne, J.-L. Lachambre, G. Otis, “Analyse Expérimentale et Théorique d’un Laser CO2 TEA Ovec Résonateurs Instables,” Can. J. Phys. 54, 816 (1976).
[CrossRef]

Park, Y. K.

Siegman, A. E.

A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277 (1965).
[CrossRef]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971, Chap. 8, pp. 293–345.

Yoshida, A.

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211 (1975).
[CrossRef]

Zucker, H.

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).

Can. J. Phys. (1)

P. Lavigne, J.-L. Lachambre, G. Otis, “Analyse Expérimentale et Théorique d’un Laser CO2 TEA Ovec Résonateurs Instables,” Can. J. Phys. 54, 816 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. W. Casperson, “Mode Stability of Lasers and Periodic Optical Systems,” IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211 (1975).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

A. E. Siegman, “Unstable Optical Resonators for Laser Applications,” Proc. IEEE 53, 277 (1965).
[CrossRef]

Other (1)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971, Chap. 8, pp. 293–345.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Geometry of a misaligned Gaussian beam coupled through a complementary Gaussian reflectivity coupler.

Fig. 2
Fig. 2

Near-field intensity distribution along the x axis and the misalignment y axis for different central reflectivity r 0 2. (A) w i 2 / w c 2 = 1 and (B) w i 2 / w c 2 = 5. It is assumed that the beam center is displaced relative to the mirror center by ηi/wc = 0.1 along the y axis.

Fig. 3
Fig. 3

Far-field intensity distribution and corresponding encircled energy for r 0 2 = 1: (A) V = 0, w i 2 / w c 2 = 5; (B) V = O, w i 2 / w c 2 = 0.1; (C) V = 10, w i 2 / w c 2 = 0.2. The curves labeled r 0 2 = 0 give the unperturbed Gaussian beam characteristics.

Fig. 4
Fig. 4

Strehl ratio of a Gaussian beam coupled through a complementary Gaussian reflectivity coupler (A) vs V for different ratio w i 2 / w c 2 and (B) vs w i 2 / w c 2 for different V. In (B), the solid lines assume r 0 2 = 1, while the dotted lines assume r 0 2 = 0.5. The Gaussian beam is normally incident on the mirror center.

Fig. 5
Fig. 5

Qualitative representation of the perturbing effects of a n-ranked Gaussian hole on the far-field properties of the coupled beam (A) V = 0 and (B) V > 1 + 2 n w i 2 / w c 2.

Fig. 6
Fig. 6

Strehl ratio of the output of a misaligned resonator (ηi/wc = 0.3) as a function of w i 2 / w c 2 when the coupler collimates the output beam. The dotted curve represents the aligned case (ηi/wc = 0). It is assumed that r 0 2 = 1.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

u ( x , y ) = exp ( - i π λ { x 2 [ 1 R - i λ π w 2 ( z ) ] + [ y - ρ ( z ) ] 2 R ( z ) - i λ [ y - η ( z ) ] 2 π w 2 ( z ) } ) .
t ( x , y ) = { 1 - r 0 2 exp [ - 2 ( x 2 + y 2 ) w c 2 ] } 1 / 2 × exp [ i π λ ( x 2 + y 2 ) f c ] ,
u t ( x , y , 0 ) = u t 0 { 1 - r 0 2 exp [ - 2 ( x 2 + y 2 ) w c 2 ] } 1 / 2 × exp [ - i π λ x 2 + ( y - ρ t ) 2 R t - x 2 + ( y - η i ) 2 w i 2 ] ,
u t 0 = exp [ - i π λ ρ i 2 R i ( 1 - R t R i ) ] ,
I n f = u t ( x , y , 0 ) u t * ( x , y , 0 ) = exp [ - 2 x 2 + ( y - η i ) 2 w i 2 ] [ 1 - r 0 2 exp ( - 2 x 2 + y 2 w c 2 ) ] .
u t ( x , y , 0 ) = u t 0 exp [ - i π λ x 2 + ( y - ρ t ) 2 R t ] × ( exp { - x 2 + [ y - η 0 ( 0 ) ] 2 w 0 2 ( 0 ) } - n = 1 c n exp { - x 2 + [ y - η n ( 0 ) ] 2 w n 2 ( 0 ) } ) ,
w n 2 ( 0 ) = w 0 2 ( 0 ) / S n , η n ( 0 ) = η 0 ( 0 ) / S n , S n = 1 + 2 n w 0 2 ( 0 ) / w c 2 , c n = r 0 2 n ( 2 n - 3 ) ! ! 2 n n ! exp { - η 0 2 ( 0 ) w 0 2 ( 0 ) [ 1 - w n 2 ( 0 ) w 0 2 ( 0 ) ] } ,
u t ( x , y , z ) u t 0 = exp { - i P 0 ( z ) - x 2 + [ y - η 0 ( z ) ] 2 w 0 2 ( z ) - i π λ x 2 + [ y - ρ 0 ( z ) ] 2 R 0 ( z ) } - n = 1 c n exp { - i P n ( z ) - x 2 + [ y - η n ( z ) ] 2 w n 2 ( z ) - i π λ x 2 + [ y - ρ n ( z ) ] 2 R n ( z ) }
η n ( z ) = η n ( 0 ) + z β n , ρ n ( z ) = η n ( z ) - β n R n ( z ) , w n 2 ( z ) = w n 2 ( 0 ) { ( 1 + z R t ) 2 + [ z λ π w n 2 ( 0 ) ] 2 } , 1 R n ( z ) = [ 1 R t ( 1 + z R t ) + z λ 2 π 2 w n 4 ( 0 ) ] { ( 1 + z R t ) 2 + [ z λ π w n 2 ( 0 ) ] 2 } - 1 , exp [ - i P n ( z ) ] = w n ( 0 ) w n ( z ) exp { i tan - 1 [ ( z - z 0 n ) λ π w n 2 ( z 0 n ) ] + i tan - 1 [ z 0 n λ π w n 2 ( z 0 n ) ] - i π λ β n 2 [ z - R n ( z ) + R t ] } .
1 w n 2 ( z 0 n ) = [ π w n ( 0 ) λ R t ] 2 + 1 w n 2 ( 0 ) , z 0 n = - R t { 1 + [ λ R t π w n 2 ( 0 ) ] 2 } - 1 .
I f f π 2 w 0 4 ( z 00 ) λ 2 z 2 | 2 ( V 2 + 1 ) 1 / 2 exp [ - i ϕ 0 - θ x 2 + ( θ y - β 0 ) 2 θ 0 2 ] - n = 1 c n ( V 2 + S n 2 ) 1 / 2 exp [ - i ϕ n - ( V 2 + 1 ) S n V 2 + S n 2 θ x 2 + ( θ y - β n ) 2 θ 0 2 ] | 2
( θ x , θ y ) = ( x / z , y / z ) , V = π w 0 2 ( 0 ) / λ R t , ϕ n = V V 2 + S n 2 [ η 0 ( 0 ) - S n ρ t ] 2 w 0 2 ( 0 ) + tan - 1 ( V S n ) .
I f f π 2 w 0 4 ( z 00 ) λ 2 z 2 | exp [ - i tan - 1 ( V ) ] exp ( - θ 2 / θ 0 2 ) ( V 2 + 1 ) 1 / 2 - n = 1 c n ( V 2 + S n 2 ) 1 / 2 exp [ - i tan - 1 ( V / S n ) ] × exp [ - θ 2 θ 0 2 ( V 2 + 1 ) S n V 2 + S n 2 ] | 2 ,
I f f π 2 w i 4 λ 2 z 2 | exp ( - θ 2 / θ 0 2 ) - n = 1 c n S n exp ( - θ 2 / S n θ 0 2 ) | 2 ,

Metrics