Abstract

A method of calculating the angular alignment tolerance of a Porro prism laser oscillator is presented. The reduction in mode volume due to misalignment is found by a ray optical approach. An analytical expression for alignment tolerance, in terms of resonator length L, prism azimuth angle θ, rod location L2, and rod aperture radius a is obtained. An experiment was set up to measure the laser output reduction due to reflector tilting. Results obtained are in good agreement with the theoretical predictions.

© 1989 Optical Society of America

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References

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  1. M. K. Chun, E. A. Teppo, “Laser Resonator: an Electrooptically Q-Switched Porro Prism Device,” Appl. Opt. 15, 1942–1946 (1976).
    [CrossRef] [PubMed]
  2. G. Zhou, L. W. Casperson, “Modes of a Laser Resonator with a Retroreflecting Roof Mirror,” Appl. Opt. 20, 3542–3546 (1981).
    [CrossRef] [PubMed]
  3. G. Gould, S. Jacobs, P. Rabinowitz, T. Shultz, “Crossed Roof Prism Interferometer,” Appl. Opt. 1, 533–534 (1962).
    [CrossRef]
  4. J. F. Lee, C. Y. Leung, “Beam Pointing Direction Changes in a Misaligned Porro Prism Resonator,” Appl. Opt. 27, 2701–2707 (1988).
    [CrossRef] [PubMed]
  5. J. F. Lee, C. Y. Leung, “Lateral Displacement of the Mode Axis in a Misaligned Porro Prism Resonator,” Submitted to Applied Optics for publication.
  6. R. J. Freiberg, A. S. Halsted, “Properties of Low Order Transverse Modes in Argon Ion Lasers,” Appl. Opt. 8, 355–362 (1969).
    [CrossRef] [PubMed]
  7. I. C. Kuo, T. Ko, “Laser Resonators of a Mirror and Corner Cube Reflector: Analysis by Imaging Method,” Appl. Opt. 23, 53–56 (1984).
    [CrossRef] [PubMed]
  8. T. C. Hsieh, K. Y. Hsu, Y. J. Li, “Misaligned Spherical-Mirror Waveguide Resonators,” Jpn. J. Appl. Phys. 25, 6–00 (1986).
    [CrossRef]

1988 (1)

1986 (1)

T. C. Hsieh, K. Y. Hsu, Y. J. Li, “Misaligned Spherical-Mirror Waveguide Resonators,” Jpn. J. Appl. Phys. 25, 6–00 (1986).
[CrossRef]

1984 (1)

1981 (1)

1976 (1)

1969 (1)

1962 (1)

Casperson, L. W.

Chun, M. K.

Freiberg, R. J.

Gould, G.

Halsted, A. S.

Hsieh, T. C.

T. C. Hsieh, K. Y. Hsu, Y. J. Li, “Misaligned Spherical-Mirror Waveguide Resonators,” Jpn. J. Appl. Phys. 25, 6–00 (1986).
[CrossRef]

Hsu, K. Y.

T. C. Hsieh, K. Y. Hsu, Y. J. Li, “Misaligned Spherical-Mirror Waveguide Resonators,” Jpn. J. Appl. Phys. 25, 6–00 (1986).
[CrossRef]

Jacobs, S.

Ko, T.

Kuo, I. C.

Lee, J. F.

J. F. Lee, C. Y. Leung, “Beam Pointing Direction Changes in a Misaligned Porro Prism Resonator,” Appl. Opt. 27, 2701–2707 (1988).
[CrossRef] [PubMed]

J. F. Lee, C. Y. Leung, “Lateral Displacement of the Mode Axis in a Misaligned Porro Prism Resonator,” Submitted to Applied Optics for publication.

Leung, C. Y.

J. F. Lee, C. Y. Leung, “Beam Pointing Direction Changes in a Misaligned Porro Prism Resonator,” Appl. Opt. 27, 2701–2707 (1988).
[CrossRef] [PubMed]

J. F. Lee, C. Y. Leung, “Lateral Displacement of the Mode Axis in a Misaligned Porro Prism Resonator,” Submitted to Applied Optics for publication.

Li, Y. J.

T. C. Hsieh, K. Y. Hsu, Y. J. Li, “Misaligned Spherical-Mirror Waveguide Resonators,” Jpn. J. Appl. Phys. 25, 6–00 (1986).
[CrossRef]

Rabinowitz, P.

Shultz, T.

Teppo, E. A.

Zhou, G.

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Figures (14)

Fig. 1
Fig. 1

Top view of a Porro prism resonator angular misaligned in the horizontal direction (β is positive).

Fig. 2
Fig. 2

Apex orientations of the two Porro prisms.

Fig. 3
Fig. 3

Conceptual picture of the apex planes of a perfectly aligned Porro prism resonator.

Fig. 4
Fig. 4

Location of the tilted mode axis in a misaligned Porro prism resonator (end view).

Fig. 5
Fig. 5

Location of the tilted mode axis in a misaligned Porro prism resonator (top view).

Fig. 6
Fig. 6

Spot diagram of the path positions of an oscillating ray.

Fig. 7
Fig. 7

Porro prism resonator angularly misaligned in the vertical direction.

Fig. 8
Fig. 8

Experimental setup to measure the alignment tolerance in the horizontal direction.

Fig. 9
Fig. 9

Normalized output energy vs horizontal misalignment angle.

Fig. 10
Fig. 10

Horizontal angular alignment tolerance vs the prism azimuth angle.

Fig. 11
Fig. 11

Horizontal angular alignment tolerance vs pumping energy, when θ = 60°.

Fig. 12
Fig. 12

Normalized output energy vs the vertical misalignment angle, when θ = 70°.

Fig. 13
Fig. 13

Vertical angular alignment tolerance vs the prism azimuth angle.

Fig. 14
Fig. 14

Illustration of the apex planes tilting due to horizontal misalignment.

Tables (1)

Tables Icon

Table I Mode Patterns Recorded on a Slip of Thermal Paper Placed 60 cm from the Beam Splitter (θ = 60°)

Equations (13)

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Δ 1 = O Q ¯ L β cot θ csc θ , Δ 2 = R R ¯ β csc θ ( L 2 cot 2 θ + L 1 2 ) 1 / 2 , Δ 3 = S S ¯ β csc θ ( L 2 cot 2 θ + L 2 2 ) 1 / 2 , Δ 4 = P P ¯ L β csc 2 θ ,
E ( β ) nor = E ( β ) E ( 0 ° ) π ( a Δ 3 ) 2 π a 2 = ( 1 Δ 3 a ) 2 .
E ( β ) nor [ 1 β csc θ ( L 2 cot 2 θ + L 2 2 ) 1 / 2 a ] 2 .
1 2 [ 1 β 1 / 2 csc θ ( L 2 cot 2 θ + L 2 2 ) 1 / 2 a ] 2 .
β 1 / 2 ( θ ) 0 . 293 a sin θ ( L 2 cot 2 θ + L 2 2 ) 1 / 2 .
Δ 3 β { [ L d ( 1 1 n ) ] 2 cot 4 θ + 2 [ L d ( 1 1 n ) ] [ L 2 d ( 1 1 n ) ] cot 2 θ + [ L 2 d ( 1 1 n ) ] 2 + ( L L 2 ) 2 cot 2 θ } 1 / 2 ,
E ( β ) nor [ 1 β H ( θ ) a ] 2 ,
β 1 / 2 ( θ ) 0 . 293 a H ( θ ) ,
H ( θ ) { [ L d ( 1 1 n ) ] 2 cot 4 θ + 2 [ L d ( 1 1 n ) ] [ L 2 d ( 1 1 n ) ] cot 2 θ + [ L 2 d ( 1 1 n ) ] 2 + ( L L 2 ) 2 cot 2 θ } 1 / 2 .
Δ 3 2 a > Δ 4 W ,
ψ arctan ( P N L ¯ ) , ϕ arctan ( O M L ¯ ) .
P N ¯ = P P ¯ sin θ ( L β csc 2 θ ) sin θ = L β csc θ , O M ¯ = O Q ¯ sin θ ( L β cot θ csc θ ) sin θ = L β cot θ .
ψ arctan ( β csc θ ) β csc θ , ϕ arctan ( β cot θ ) β cot θ .

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