Abstract

Diffraction power losses for tilted (open) resonators with both circular and strip flat reflectors are computed for several tilt angles by means of a second-order perturbation technique. Results indicate that the power losses increase as the square of the perturbation parameter. The strips exhibit a greater additional power loss per transit for the TEM0 mode than for the TEM1 mode; the opposite is true in the case of the circular disks. Results are applied to the He–Ne 6328-Å line.

© 1980 Optical Society of America

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References

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  1. J. L. Remo, Opt. Lett. 3, 193 (1978).
    [Crossref] [PubMed]
  2. A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
    [Crossref]
  3. L. A. Vainstein, Sov. Phys. JETP 17, 709 (1963).
  4. S. R. Barone, J. Appl. Phys. 34, 831 (1963).
    [Crossref]
  5. H. Risken, Z. Phys. 180, 150 (1964).
    [Crossref]
  6. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
    [Crossref] [PubMed]
  7. See, for example, B. A. Lengyel, Lasers (Wiley-Interscience, New York, 1971), Chap. 3.
  8. H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 20, 598 (1965).
    [Crossref]
  9. W. H. Wells, IEEE J. Quantum Electron. QE-2, 94 (1966).
    [Crossref]
  10. H. Ogura, Y. Yoshida, Jpn J. Appl. Phys 3, 546 (1964).
    [Crossref]
  11. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  12. H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 22, 1434 (1967); L. Ronchi, Appl. Opt. 9, 733 (1970); R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
    [Crossref] [PubMed]
  13. The unperturbed mode functions are valid solutions if (a/L)2 ≪ 1 (parabolic approximation). A further condition is that a2/λL ≪ (L/a)2, which indicates that the only parameter of importance is the Fresnel number N, and is approximately equal to the number of Fresnel zones observed in one reflector from the center of the other reflector. This determines the number of ripples in the field distribution so that the larger the N, the weaker the field intensity at the reflector edge and the smaller the power loss due to spillover. Therefore, as N increases, λ becomes smaller, and the geometric approximation is more accurate as is the walk-out picture of energy loss.
  14. R. J. Pressley, Ed., Handbook for Lasers (Chemical Rubber Co., Cleveland, 1971), Chap. 6.
  15. A. E. Siegmann, IEEE J. Quantum Electron. QE-13, 334 (1977).
    [Crossref]
  16. J. L. Remo, “Phase perturbations and laser resonator beam quality,” in preparation (1979).

1978 (1)

1977 (1)

A. E. Siegmann, IEEE J. Quantum Electron. QE-13, 334 (1977).
[Crossref]

1969 (1)

1967 (1)

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 22, 1434 (1967); L. Ronchi, Appl. Opt. 9, 733 (1970); R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
[Crossref] [PubMed]

1966 (1)

W. H. Wells, IEEE J. Quantum Electron. QE-2, 94 (1966).
[Crossref]

1965 (1)

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 20, 598 (1965).
[Crossref]

1964 (2)

H. Risken, Z. Phys. 180, 150 (1964).
[Crossref]

H. Ogura, Y. Yoshida, Jpn J. Appl. Phys 3, 546 (1964).
[Crossref]

1963 (3)

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

L. A. Vainstein, Sov. Phys. JETP 17, 709 (1963).

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

1961 (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Barone, S. R.

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

Fox, A. G.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Ikenoue, J.-I.

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 22, 1434 (1967); L. Ronchi, Appl. Opt. 9, 733 (1970); R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
[Crossref] [PubMed]

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 20, 598 (1965).
[Crossref]

Lengyel, B. A.

See, for example, B. A. Lengyel, Lasers (Wiley-Interscience, New York, 1971), Chap. 3.

Li, T.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Ogura, H.

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 22, 1434 (1967); L. Ronchi, Appl. Opt. 9, 733 (1970); R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
[Crossref] [PubMed]

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 20, 598 (1965).
[Crossref]

H. Ogura, Y. Yoshida, Jpn J. Appl. Phys 3, 546 (1964).
[Crossref]

Remo, J. L.

J. L. Remo, Opt. Lett. 3, 193 (1978).
[Crossref] [PubMed]

J. L. Remo, “Phase perturbations and laser resonator beam quality,” in preparation (1979).

Risken, H.

H. Risken, Z. Phys. 180, 150 (1964).
[Crossref]

Sanderson, R. L.

Siegmann, A. E.

A. E. Siegmann, IEEE J. Quantum Electron. QE-13, 334 (1977).
[Crossref]

Streifer, W.

Vainstein, L. A.

L. A. Vainstein, Sov. Phys. JETP 17, 709 (1963).

Wells, W. H.

W. H. Wells, IEEE J. Quantum Electron. QE-2, 94 (1966).
[Crossref]

Yoshida, Y.

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 22, 1434 (1967); L. Ronchi, Appl. Opt. 9, 733 (1970); R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
[Crossref] [PubMed]

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 20, 598 (1965).
[Crossref]

H. Ogura, Y. Yoshida, Jpn J. Appl. Phys 3, 546 (1964).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

IEEE J. Quantum Electron. (2)

A. E. Siegmann, IEEE J. Quantum Electron. QE-13, 334 (1977).
[Crossref]

W. H. Wells, IEEE J. Quantum Electron. QE-2, 94 (1966).
[Crossref]

J. Appl. Phys. (1)

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

J. Phys. Soc. Jpn. (2)

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 20, 598 (1965).
[Crossref]

H. Ogura, Y. Yoshida, J.-I. Ikenoue, J. Phys. Soc. Jpn. 22, 1434 (1967); L. Ronchi, Appl. Opt. 9, 733 (1970); R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
[Crossref] [PubMed]

Jpn J. Appl. Phys (1)

H. Ogura, Y. Yoshida, Jpn J. Appl. Phys 3, 546 (1964).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (1)

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

Sov. Phys. JETP (1)

L. A. Vainstein, Sov. Phys. JETP 17, 709 (1963).

Z. Phys. (1)

H. Risken, Z. Phys. 180, 150 (1964).
[Crossref]

Other (4)

See, for example, B. A. Lengyel, Lasers (Wiley-Interscience, New York, 1971), Chap. 3.

The unperturbed mode functions are valid solutions if (a/L)2 ≪ 1 (parabolic approximation). A further condition is that a2/λL ≪ (L/a)2, which indicates that the only parameter of importance is the Fresnel number N, and is approximately equal to the number of Fresnel zones observed in one reflector from the center of the other reflector. This determines the number of ripples in the field distribution so that the larger the N, the weaker the field intensity at the reflector edge and the smaller the power loss due to spillover. Therefore, as N increases, λ becomes smaller, and the geometric approximation is more accurate as is the walk-out picture of energy loss.

R. J. Pressley, Ed., Handbook for Lasers (Chemical Rubber Co., Cleveland, 1971), Chap. 6.

J. L. Remo, “Phase perturbations and laser resonator beam quality,” in preparation (1979).

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Figures (4)

Fig. 1
Fig. 1

Symmetrically perturbed Fabry-Perot flat disk resonator of length L, mirror radius r, and tilt α.

Fig. 2
Fig. 2

Additional power losses per transit vs tilt angle for flat strip mirrors. The corresponding unperturbed values are given at the left.

Fig. 3
Fig. 3

Additional power losses per transit vs tilt angle for flat disk mirrors. The corresponding unperturbed values are given at the left.

Fig. 4
Fig. 4

Total power loss per transit vs Fresnel number for both perturbed (solid line) and unperturbed (dashed line) flat disk resonators with tilt displacement = λ/144. Note the beating between the TEM1, TEM2, and TEM3 modes.

Tables (2)

Tables Icon

Table I Comparison of the Percent Power Loss per Transit as Computed by Ogura et al.18 and This Worka

Tables Icon

Table II Unperturbed P0 and Perturbed P (% Power Loss per Transit) and Relative Power Loss Increase P/P0 at N = 19.8 is Shown for the TEM0, TEM1, TEM2, and TEM3 Modes for Circular Disks Tilted at α = 0.18 Sec of Arca

Equations (21)

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γ m ( 0 ) ψ m ( 0 ) = K ^ ψ m ( 0 ) ,
γ m ( 0 ) ψ m ( 0 ) ( x 1 ) = - a a K ( x , x 1 ) ψ m ( 0 ) ( x ) d x
γ m n ( 0 ) ψ m n ( 0 ) ( r 1 , θ 1 ) = 0 2 π 0 a K ( r , θ ; r 1 , θ 1 ) ψ m n ( 0 ) ( r , θ ) r d r d θ
ψ m ( 0 ) ( x ) = cos m π 2 x a ( 1 + 1 i z - 1 z 2 + ) ,
1 z = [ exp ( i π 2 ) 6 π N ] 1 / 2
ψ m ( 0 ) ( x ) = sin m π 2 x a ( 1 + 1 i z - 1 z 2 + ) ,
ψ m n ( 0 ) ( r θ ) = 2 J m ( K + r ) J m + 1 ( K + ) { cos n θ sin n θ } ,
K + = ν m n ( 1 + 1 / i z - 1 / z 2 + ) ,
- a a ψ m ( 0 ) ( x ) ψ ρ ( 0 ) ( x ) d x = δ m ρ ,
0 a [ J m ( K + r ) ] 2 r d r = a 2 2 [ J m + 1 ( K + ) ] 2 .
ψ m o ( 0 ) ( r θ ) = ψ m ( 0 ) ( r ) .
Φ = α K x = α ( 2 π / λ ) x ,
γ m ψ m ( x 1 ) = exp [ i 2 Φ ( x 1 ) ] - a a K ( x , x 1 ) ψ m ( x ) d x ,
Φ = α K r cos θ .
γ m ( 1 ) = i 2 γ m ( 0 ) m Φ m ,
γ m ( 2 ) = γ m ( 0 ) { - 2 m Φ 2 m + 4 n m p m Φ p 2 / [ 1 - γ m ( 0 ) / γ p ( 0 ) ] } ,
n cos θ n = 0 2 π cos θ { sin 2 n θ cos 2 n θ } d θ = 0 ,
n cos θ q = 0 2 π cos θ { sin n θ sin q θ cos n θ cos q θ } d θ = 0.
n cos 2 θ n 0 ,
m 0 r 2 cos 2 θ m 0 = π m r 2 m .
gain = 3.0 × 10 - 2 L / 2 a % ,

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