Abstract

Waveguide gas lasers (CO2 ones especially) continue to be widely used. We have previously studied simple resonator designs with plane mirrors close to each end of the waveguide. Here we examine theoretical predictions concerning hybrid waveguide/free-space resonators with square-bore guides and curved mirrors. We show how resonator mode losses vary as functions of guide length and width, guide-to-mirror distance, mirror radius of curvature, and mirror tilt. We have tested a 7-W cw rf-excited CO2 laser with unusually good near-TEM00 transverse-mode quality; it is one of many promising resonator geometries not covered by earlier published research. The common case 3 reflector, sometimes viewed as guaranteeing near-TEM00 mode performance, is shown to be alarmingly sensitive to small misalignments for certain guide geometries.

© 1995 Optical Society of America

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References

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  1. J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
    [Crossref]
  2. R. L. Abrams, “Waveguide gas lasers,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), Vol. 3.
  3. D. R. Hall, C. A. Hill, “RF-excited CO2 waveguide lasers,” in Handbook of Molecular Lasers, P. K. Cheo, ed. (Dekker, New York, 1987).
  4. L. A. Newman, R. A. Hart, “Recent R&D advances in sealed-off CO2 lasers,” Laser Focus/Electro-Opt. 23, 80–96 (June1987).
  5. P. E. Jackson, H. J. Baker, D. R. Hall, “CO2 large-area discharge laser using an unstable-waveguide hybrid resonator,” Appl. Phys. Lett. 54, 1950–1952 (1989).
    [Crossref]
  6. R. Gerlach, D. Wei, N. M. Amer, “Coupling efficiency of waveguide laser resonators formed by flat mirrors: analysis and experiment,” IEEE J. Quantum Electron. QE-20, 948–963 (1984).
    [Crossref]
  7. C. A. Hill, “Transverse modes of plane-mirror waveguide resonators,” IEEE J. Quantum Electron. QE-24, 1936–1946 (1988).
    [Crossref]
  8. C. A. Hill, A. D. Colley, “Misalignment effects in a CO2 waveguide laser,” IEEE J. Quantum Electron. QE-26, 323–328 (1990).
    [Crossref]
  9. J. J. Degnan, D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
    [Crossref]
  10. J. Banerji, A. R. Davies, R. W. J. Devereux, C. A. Hill, R. M. Jenkins, “Effects of curved mirror misalignment in a folded waveguide,” Appl. Opt. 29, 777–785 (1990).
    [Crossref] [PubMed]
  11. J. Banerji, A. R. Davies, P. E. Jackson, R. M. Jenkins, “Transmission and coupling losses in a folded waveguide,” Appl. Opt. 28, 4637–4643 (1989).
    [Crossref] [PubMed]
  12. C. A. Hill, J. R. Redding, A. D. Colley, “Multimode treatment of misaligned CO2 waveguide lasers,” J. Mod. Opt. 37, 473–481 (1990).
    [Crossref]
  13. C. A. Hill, P. Monk, D. R. Hall, “Tunable rf-excited CO2 waveguide laser with variable guide width,” IEEE J. Quantum Electron. QE-23, 1968–1973 (1987).
    [Crossref]
  14. R. L. Abrams, A. N. Chester, “Resonator theory for hollow waveguide lasers,” Appl. Opt. 13, 2117–2125 (1974).
    [Crossref] [PubMed]
  15. R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
    [Crossref]
  16. C. A. Hill, D. R. Hall, “Coupling loss theory of single-mode waveguide resonators,” Appl. Opt. 24, 1283–1290 (1985).
    [Crossref] [PubMed]
  17. K. D. Laakmann, W. H. Steier, “Waveguides: characteristic modes of hollow rectangular dielectric waveguides,” Appl. Opt. 15, 1334–1340 (1976).
    [Crossref] [PubMed]
  18. L. Rebuffi, J. P. Crenn, “Radiation patterns of the HE11 mode and Gaussian approximations,” Int. J. Infrared Millimeter Waves 10, 291–311 (1989).
    [Crossref]
  19. C. A. Bennett, D. P. Hutchinson, “Far-field propagation of the truncated EH11 dielectric waveguide mode,” Appl. Opt. 28, 2581–2583 (1989).
    [Crossref] [PubMed]
  20. J.-L. Boulnois, G. P. Agrawal, “Mode discrimination and coupling losses in rectangular waveguide resonators with conventional and phase-conjugate mirrors,” J. Opt. Soc. Am. 72, 853–860 (1982).
    [Crossref]
  21. D. He, D. R. Hall, “A 30-W radio frequency excited waveguide CO2 laser,” Appl. Phys. Lett. 43, 726–728 (1983).
    [Crossref]

1990 (3)

C. A. Hill, A. D. Colley, “Misalignment effects in a CO2 waveguide laser,” IEEE J. Quantum Electron. QE-26, 323–328 (1990).
[Crossref]

J. Banerji, A. R. Davies, R. W. J. Devereux, C. A. Hill, R. M. Jenkins, “Effects of curved mirror misalignment in a folded waveguide,” Appl. Opt. 29, 777–785 (1990).
[Crossref] [PubMed]

C. A. Hill, J. R. Redding, A. D. Colley, “Multimode treatment of misaligned CO2 waveguide lasers,” J. Mod. Opt. 37, 473–481 (1990).
[Crossref]

1989 (4)

L. Rebuffi, J. P. Crenn, “Radiation patterns of the HE11 mode and Gaussian approximations,” Int. J. Infrared Millimeter Waves 10, 291–311 (1989).
[Crossref]

C. A. Bennett, D. P. Hutchinson, “Far-field propagation of the truncated EH11 dielectric waveguide mode,” Appl. Opt. 28, 2581–2583 (1989).
[Crossref] [PubMed]

J. Banerji, A. R. Davies, P. E. Jackson, R. M. Jenkins, “Transmission and coupling losses in a folded waveguide,” Appl. Opt. 28, 4637–4643 (1989).
[Crossref] [PubMed]

P. E. Jackson, H. J. Baker, D. R. Hall, “CO2 large-area discharge laser using an unstable-waveguide hybrid resonator,” Appl. Phys. Lett. 54, 1950–1952 (1989).
[Crossref]

1988 (1)

C. A. Hill, “Transverse modes of plane-mirror waveguide resonators,” IEEE J. Quantum Electron. QE-24, 1936–1946 (1988).
[Crossref]

1987 (2)

L. A. Newman, R. A. Hart, “Recent R&D advances in sealed-off CO2 lasers,” Laser Focus/Electro-Opt. 23, 80–96 (June1987).

C. A. Hill, P. Monk, D. R. Hall, “Tunable rf-excited CO2 waveguide laser with variable guide width,” IEEE J. Quantum Electron. QE-23, 1968–1973 (1987).
[Crossref]

1985 (1)

1984 (1)

R. Gerlach, D. Wei, N. M. Amer, “Coupling efficiency of waveguide laser resonators formed by flat mirrors: analysis and experiment,” IEEE J. Quantum Electron. QE-20, 948–963 (1984).
[Crossref]

1983 (1)

D. He, D. R. Hall, “A 30-W radio frequency excited waveguide CO2 laser,” Appl. Phys. Lett. 43, 726–728 (1983).
[Crossref]

1982 (1)

1976 (2)

1974 (1)

1973 (1)

J. J. Degnan, D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

1972 (1)

R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
[Crossref]

Abrams, R. L.

R. L. Abrams, A. N. Chester, “Resonator theory for hollow waveguide lasers,” Appl. Opt. 13, 2117–2125 (1974).
[Crossref] [PubMed]

R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
[Crossref]

R. L. Abrams, “Waveguide gas lasers,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), Vol. 3.

Agrawal, G. P.

Amer, N. M.

R. Gerlach, D. Wei, N. M. Amer, “Coupling efficiency of waveguide laser resonators formed by flat mirrors: analysis and experiment,” IEEE J. Quantum Electron. QE-20, 948–963 (1984).
[Crossref]

Baker, H. J.

P. E. Jackson, H. J. Baker, D. R. Hall, “CO2 large-area discharge laser using an unstable-waveguide hybrid resonator,” Appl. Phys. Lett. 54, 1950–1952 (1989).
[Crossref]

Banerji, J.

Bennett, C. A.

Boulnois, J.-L.

Chester, A. N.

Colley, A. D.

C. A. Hill, J. R. Redding, A. D. Colley, “Multimode treatment of misaligned CO2 waveguide lasers,” J. Mod. Opt. 37, 473–481 (1990).
[Crossref]

C. A. Hill, A. D. Colley, “Misalignment effects in a CO2 waveguide laser,” IEEE J. Quantum Electron. QE-26, 323–328 (1990).
[Crossref]

Crenn, J. P.

L. Rebuffi, J. P. Crenn, “Radiation patterns of the HE11 mode and Gaussian approximations,” Int. J. Infrared Millimeter Waves 10, 291–311 (1989).
[Crossref]

Davies, A. R.

Degnan, J. J.

J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
[Crossref]

J. J. Degnan, D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

Devereux, R. W. J.

Gerlach, R.

R. Gerlach, D. Wei, N. M. Amer, “Coupling efficiency of waveguide laser resonators formed by flat mirrors: analysis and experiment,” IEEE J. Quantum Electron. QE-20, 948–963 (1984).
[Crossref]

Hall, D. R.

P. E. Jackson, H. J. Baker, D. R. Hall, “CO2 large-area discharge laser using an unstable-waveguide hybrid resonator,” Appl. Phys. Lett. 54, 1950–1952 (1989).
[Crossref]

C. A. Hill, P. Monk, D. R. Hall, “Tunable rf-excited CO2 waveguide laser with variable guide width,” IEEE J. Quantum Electron. QE-23, 1968–1973 (1987).
[Crossref]

C. A. Hill, D. R. Hall, “Coupling loss theory of single-mode waveguide resonators,” Appl. Opt. 24, 1283–1290 (1985).
[Crossref] [PubMed]

D. He, D. R. Hall, “A 30-W radio frequency excited waveguide CO2 laser,” Appl. Phys. Lett. 43, 726–728 (1983).
[Crossref]

J. J. Degnan, D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

D. R. Hall, C. A. Hill, “RF-excited CO2 waveguide lasers,” in Handbook of Molecular Lasers, P. K. Cheo, ed. (Dekker, New York, 1987).

Hart, R. A.

L. A. Newman, R. A. Hart, “Recent R&D advances in sealed-off CO2 lasers,” Laser Focus/Electro-Opt. 23, 80–96 (June1987).

He, D.

D. He, D. R. Hall, “A 30-W radio frequency excited waveguide CO2 laser,” Appl. Phys. Lett. 43, 726–728 (1983).
[Crossref]

Hill, C. A.

C. A. Hill, A. D. Colley, “Misalignment effects in a CO2 waveguide laser,” IEEE J. Quantum Electron. QE-26, 323–328 (1990).
[Crossref]

C. A. Hill, J. R. Redding, A. D. Colley, “Multimode treatment of misaligned CO2 waveguide lasers,” J. Mod. Opt. 37, 473–481 (1990).
[Crossref]

J. Banerji, A. R. Davies, R. W. J. Devereux, C. A. Hill, R. M. Jenkins, “Effects of curved mirror misalignment in a folded waveguide,” Appl. Opt. 29, 777–785 (1990).
[Crossref] [PubMed]

C. A. Hill, “Transverse modes of plane-mirror waveguide resonators,” IEEE J. Quantum Electron. QE-24, 1936–1946 (1988).
[Crossref]

C. A. Hill, P. Monk, D. R. Hall, “Tunable rf-excited CO2 waveguide laser with variable guide width,” IEEE J. Quantum Electron. QE-23, 1968–1973 (1987).
[Crossref]

C. A. Hill, D. R. Hall, “Coupling loss theory of single-mode waveguide resonators,” Appl. Opt. 24, 1283–1290 (1985).
[Crossref] [PubMed]

D. R. Hall, C. A. Hill, “RF-excited CO2 waveguide lasers,” in Handbook of Molecular Lasers, P. K. Cheo, ed. (Dekker, New York, 1987).

Hutchinson, D. P.

Jackson, P. E.

J. Banerji, A. R. Davies, P. E. Jackson, R. M. Jenkins, “Transmission and coupling losses in a folded waveguide,” Appl. Opt. 28, 4637–4643 (1989).
[Crossref] [PubMed]

P. E. Jackson, H. J. Baker, D. R. Hall, “CO2 large-area discharge laser using an unstable-waveguide hybrid resonator,” Appl. Phys. Lett. 54, 1950–1952 (1989).
[Crossref]

Jenkins, R. M.

Laakmann, K. D.

Monk, P.

C. A. Hill, P. Monk, D. R. Hall, “Tunable rf-excited CO2 waveguide laser with variable guide width,” IEEE J. Quantum Electron. QE-23, 1968–1973 (1987).
[Crossref]

Newman, L. A.

L. A. Newman, R. A. Hart, “Recent R&D advances in sealed-off CO2 lasers,” Laser Focus/Electro-Opt. 23, 80–96 (June1987).

Rebuffi, L.

L. Rebuffi, J. P. Crenn, “Radiation patterns of the HE11 mode and Gaussian approximations,” Int. J. Infrared Millimeter Waves 10, 291–311 (1989).
[Crossref]

Redding, J. R.

C. A. Hill, J. R. Redding, A. D. Colley, “Multimode treatment of misaligned CO2 waveguide lasers,” J. Mod. Opt. 37, 473–481 (1990).
[Crossref]

Steier, W. H.

Wei, D.

R. Gerlach, D. Wei, N. M. Amer, “Coupling efficiency of waveguide laser resonators formed by flat mirrors: analysis and experiment,” IEEE J. Quantum Electron. QE-20, 948–963 (1984).
[Crossref]

Appl. Opt. (6)

Appl. Phys. (1)

J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
[Crossref]

Appl. Phys. Lett. (2)

P. E. Jackson, H. J. Baker, D. R. Hall, “CO2 large-area discharge laser using an unstable-waveguide hybrid resonator,” Appl. Phys. Lett. 54, 1950–1952 (1989).
[Crossref]

D. He, D. R. Hall, “A 30-W radio frequency excited waveguide CO2 laser,” Appl. Phys. Lett. 43, 726–728 (1983).
[Crossref]

IEEE J. Quantum Electron. (6)

R. Gerlach, D. Wei, N. M. Amer, “Coupling efficiency of waveguide laser resonators formed by flat mirrors: analysis and experiment,” IEEE J. Quantum Electron. QE-20, 948–963 (1984).
[Crossref]

C. A. Hill, “Transverse modes of plane-mirror waveguide resonators,” IEEE J. Quantum Electron. QE-24, 1936–1946 (1988).
[Crossref]

C. A. Hill, A. D. Colley, “Misalignment effects in a CO2 waveguide laser,” IEEE J. Quantum Electron. QE-26, 323–328 (1990).
[Crossref]

J. J. Degnan, D. R. Hall, “Finite-aperture waveguide laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
[Crossref]

C. A. Hill, P. Monk, D. R. Hall, “Tunable rf-excited CO2 waveguide laser with variable guide width,” IEEE J. Quantum Electron. QE-23, 1968–1973 (1987).
[Crossref]

Int. J. Infrared Millimeter Waves (1)

L. Rebuffi, J. P. Crenn, “Radiation patterns of the HE11 mode and Gaussian approximations,” Int. J. Infrared Millimeter Waves 10, 291–311 (1989).
[Crossref]

J. Mod. Opt. (1)

C. A. Hill, J. R. Redding, A. D. Colley, “Multimode treatment of misaligned CO2 waveguide lasers,” J. Mod. Opt. 37, 473–481 (1990).
[Crossref]

J. Opt. Soc. Am. (1)

Laser Focus/Electro-Opt. (1)

L. A. Newman, R. A. Hart, “Recent R&D advances in sealed-off CO2 lasers,” Laser Focus/Electro-Opt. 23, 80–96 (June1987).

Other (2)

R. L. Abrams, “Waveguide gas lasers,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), Vol. 3.

D. R. Hall, C. A. Hill, “RF-excited CO2 waveguide lasers,” in Handbook of Molecular Lasers, P. K. Cheo, ed. (Dekker, New York, 1987).

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Figures (15)

Fig. 1
Fig. 1

Schematic of a linear waveguide resonator with one plane mirror and one curved mirror.

Fig. 2
Fig. 2

Round-trip dissipative resonator loss versus guide length L for a square-bore waveguide resonator with one case 1 plane mirror (z1 = 0) and one distant plane mirror at z2 = 15 mm (see Fig. 1). Guide half-width a = 1.0 mm, radiation wavelength λ = 10.6 μm, loss factors (as in Ref. 7) of 0.8 for alumina walls and 0.25 for aluminum electrodes. (a) First four resonator modes with all EHmn modes with m, n to four included in the matrix model. (b) First two modes with m, n to eight. (c) First two modes with m, n to 16.

Fig. 3
Fig. 3

As in Fig. 2 but with a curved mirror (radius of curvature R) in place of the distant plane mirror. All EHmn modes with m, n to 16 are included. (a) R = 100 cm, (b) R = 50 cm, (c) R = 25 cm.

Fig. 4
Fig. 4

Round-trip dissipative resonator loss versus the mirror 2 radius of curvature R. L = 20 cm [in the promising window in Fig. 3(c)], z1 = 0, z2 = 15 mm, a = 1.0 mm, λ = 10.6 μm. All EHmn modes with m, n to 16 are included.

Fig. 5
Fig. 5

Round-trip dissipative resonator loss versus the guide-to-curved-mirror distance z2. L = 20 cm, z1 = 0, R = 25 cm, a = 1.0 mm, λ = 10.6 μm. All EHmn modes with m, n to 16 are included. Increasing z2 to ≈5 cm enhances the mode discrimination with little penalty in fundamental mode loss.

Fig. 6
Fig. 6

Round-trip dissipative resonator loss versus mirror 2 radius of curvature R. L = 20 cm, z1 = 0, z2 = 4.7 cm (near the mode-discrimination peak in Fig. 5), a = 1.0 mm, λ = 10.6 μm. All EHmn modes with m, n to 16 are included. This figure and Figs. 4 and 5 explore small regions of parameter space, iterating toward one promising geometry that combines low fundamental-mode loss, high higher-order-mode loss, compactness (L ≈ 4z2), and (see Figs. 9 and 10) good tolerance of mirror misalignment.

Fig. 7
Fig. 7

Round-trip dissipative resonator loss versus waveguide half-width a. L = 20 cm, z1 = 0, z2 = 4.7 cm, R = 25 cm, λ = 10.6 μm. Losses for the first four resonator modes are shown; all EHmn modes with m, n to 16 are included.

Fig. 8
Fig. 8

Round-trip dissipative resonator losses as a function of mirror 2 distance z2 when the mirror surface is phase-matched to the EH11 approximating Gaussian [R = z + b2/z, b = π(0.70a)2/λ]. L = 20 cm, z1 = 0, a = 1.0 mm, λ = 10.6 μm. Solid curves, first two resonator modes when all EHmn with m, n to 16 are included (multimode); dashed curves, with m, n to 2 (pure EH11 and pure EH12).

Fig. 9
Fig. 9

Round-trip dissipative resonator loss versus angular misalignment of curved mirror 2. L = 20 cm, z1 = 0, z2 = 4.7 cm, R = 25 cm, a = 1.0 mm, λ = 10.6 μm. Losses for the first four resonator modes are shown; all EHmn modes with m, n to 16 are included.

Fig. 10
Fig. 10

Predicted mode intensity profiles at a distance d = 300 cm from the plane mirror 1 corresponding with curved mirror 2 tilts of ϕ = 0, 4, and 8 mrad (see Fig. 9). The x and y axes are labeled in units of ka/d. Each profile is normalized to the same peak intensity.

Fig. 11
Fig. 11

Round-trip dissipative resonator loss versus guide length for a case 1/case 3 square-bore waveguide resonator: z1 = 0 (plane mirror), z2 = 14.3 cm, R2 = 28.6 cm, a = 1.0 mm, λ = 10.6 μm. Losses for the first four resonator modes are shown; all EHmn modes with m, n to 16 are included. (a) Mirrors well aligned; (b) mirror 2 tilted, ϕ = 0.2 mrad; (c) mirror 2 tilted, ϕ = 1.0 mrad. All EHmn modes with m, n to 16 are included.

Fig. 12
Fig. 12

Output power versus gas pressure (3 He–1 CO2–1 N2 + 5% Xe) for a waveguide laser with two different outcoupler configurations. Radio-frequency powers (in and as far as possible all other experimental conditions) were the same for both configurations.

Fig. 13
Fig. 13

Laser power signatures, taken from a chart recorder linked to a cw powermeter, for the two designs (gas pressure, 120 Torr; input power, 115 W). (a) Dual-case-1 configuration: z1z2 ≈ 2 mm, a = 1.0 mm, λ = 10.6 μm. (b) Case 1/curved experimental configuration: as in (a) except that mirror 2 has a radius of curvature R = 29 cm and a guide-to-mirror distance z2 = 47 mm.

Fig. 14
Fig. 14

Predicted round-trip dissipative resonator losses for the dual-case-1 experimental configuration: z1z2 ≈ 2 mm, a = 1.0 mm, λ = 10.6 μm, loss factors (again as in Ref. 7, although our actual guide has three alumina walls) of 0.8 for alumina walls and 0.25 for aluminum electrodes. The first four resonator mode losses are plotted with all EHmn modes with m, n to 16 included in the matrix model. (a) Losses versus guide length L. (b) Losses versus angular misalignment of one mirror.

Fig. 15
Fig. 15

As in Fig. 14 but for the case 1/curved experimental configuration: z1 ≈ 2 mm, z2 = 47 mm, R = 29 cm. (a) Losses versus guide length L. (b) Losses versus angular misalignment of the curved mirror.

Equations (1)

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Δ L 16 a 2 / λ m 2 + n 2 m 2 n 2 .

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