Abstract

Apertures have been used to select the low-order transverse modes in resonators. The additional diffraction losses result in a change in the transverse-mode structure, and the presence of apertures inside a resonator generally distorts the mode shape. The optimization of a multiple-aperture resonator demands an approach that differs from the conventional method in which the mode theory is used. We demonstrate an iterative design method to find optimal phase profiles for the reflector surfaces to build a resonator with multiple apertures to produce a lowest-order mode with much smaller diffraction loss and to satisfy the phase-conjugation conditions at the mirrors. The results are compared with conventional stable resonators, and we show that substantial improvement in round-trip loss and beam quality can also be obtained.

© 2004 Optical Society of America

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References

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  1. N. Hodgson, H. Weber, Optical Resonators (Springer-Verlag, London, 1997).
    [CrossRef]
  2. A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).
  3. S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
    [CrossRef]
  4. S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
    [CrossRef]
  5. S. Makki, J. R. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37, 80–86 (2001).
    [CrossRef]
  6. U. D. Zeitner, F. Wyrowski, “Design of unstable laser resonators with user-defined mode shape,” IEEE J. Quantum Electron. 37, 1594–1599 (2001).
    [CrossRef]
  7. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).
  8. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).
  9. K. Nemeto, T. Fujii, M. Nagano, “Laser beam forming by fabricated aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1995).
    [CrossRef]

2001

S. Makki, J. R. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37, 80–86 (2001).
[CrossRef]

U. D. Zeitner, F. Wyrowski, “Design of unstable laser resonators with user-defined mode shape,” IEEE J. Quantum Electron. 37, 1594–1599 (2001).
[CrossRef]

1999

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

1961

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Fujii, T.

K. Nemeto, T. Fujii, M. Nagano, “Laser beam forming by fabricated aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1995).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

Hodgson, N.

N. Hodgson, H. Weber, Optical Resonators (Springer-Verlag, London, 1997).
[CrossRef]

Leger, J. R.

S. Makki, J. R. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37, 80–86 (2001).
[CrossRef]

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Makki, S.

S. Makki, J. R. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37, 80–86 (2001).
[CrossRef]

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

Nagano, M.

K. Nemeto, T. Fujii, M. Nagano, “Laser beam forming by fabricated aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1995).
[CrossRef]

Nemeto, K.

K. Nemeto, T. Fujii, M. Nagano, “Laser beam forming by fabricated aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1995).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

Weber, H.

N. Hodgson, H. Weber, Optical Resonators (Springer-Verlag, London, 1997).
[CrossRef]

Wyrowski, F.

U. D. Zeitner, F. Wyrowski, “Design of unstable laser resonators with user-defined mode shape,” IEEE J. Quantum Electron. 37, 1594–1599 (2001).
[CrossRef]

Zeitner, U. D.

U. D. Zeitner, F. Wyrowski, “Design of unstable laser resonators with user-defined mode shape,” IEEE J. Quantum Electron. 37, 1594–1599 (2001).
[CrossRef]

Bell Syst. Tech. J.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

IEEE J. Quantum Electron.

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

S. Makki, J. R. Leger, “Solid-state laser resonators with diffractive optic thermal aberration correction,” IEEE J. Quantum Electron. 35, 1075–1085 (1999).
[CrossRef]

S. Makki, J. R. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37, 80–86 (2001).
[CrossRef]

U. D. Zeitner, F. Wyrowski, “Design of unstable laser resonators with user-defined mode shape,” IEEE J. Quantum Electron. 37, 1594–1599 (2001).
[CrossRef]

Optik

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

Other

K. Nemeto, T. Fujii, M. Nagano, “Laser beam forming by fabricated aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1995).
[CrossRef]

N. Hodgson, H. Weber, Optical Resonators (Springer-Verlag, London, 1997).
[CrossRef]

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

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Figures (9)

Fig. 1
Fig. 1

Resonator and definition of the fields at the mirrors. The aperture radii at the mirrors are a and b. The U + and U - correspond to the field amplitudes propagating along the +z and -z directions at the output coupler located at z = Z 0. The V + and V - correspond to the field amplitudes propagating along the +z and -z directions at the mirror located at z = Z b . The normalized aperture size R ap is either a/w 1 or b/w 2 where w 1 and w 2 are the mode radii at mirror 1 and mirror 2, respectively.

Fig. 2
Fig. 2

Fox-Li simulation results for (a) semiconfocal and (b) semihemispherical resonators. The length of the resonator is 0.56 m, the wavelength is 1.064 μm, both mirrors of the semiconfocal resonator and mirror 2 of the semihemispherical resonator have radii of 0.6 m. The aperture radius of mirror 2 was chosen as a variable, and the aperture radius of mirror 1 is set at four times the fundamental mode radius at mirror 1, which is 0.436 and 0.225 mm for (a) and (b), respectively. The aperture radius R ap is the ratio of the aperture radius to the mode radius at mirror 2, which is 0.436 and 0.872 mm for (a) and (b), respectively. The ideal L-G results in an ideal Gaussian mode solution assuming a cylindrical symmetry of the resonator.

Fig. 3
Fig. 3

Intensity distribution at mirror 2 of the semiconfocal resonator. It can be seen that the mode tries to adjust itself to reduce the loss by lifting the mode amplitude at the center while reducing the amplitude in the vicinity of the aperture. The self-adjustment is significant for a tighter aperture, as can be seen for R ap = 0.8 compared with larger apertures.

Fig. 4
Fig. 4

Phase error at mirror 1 for the semihemispherical resonator. The phase error is calculated when the mirror phase profile is subtracted from the actual phase mode. Because mirror 1 is flat, the flat profile at the center region conforms better to the mirror profile. For a larger aperture, it can be seen that the phase at the center conforms better to the mirror profile than a smaller-aperture case. Nevertheless there is significant phase error for the largest aperture at the center region, which will result in the loss of beam quality (smaller Strehl ratio).

Fig. 5
Fig. 5

Strehl ratio and round-trip mode loss for the semihemispherical resonator. As the aperture size increases, the round-trip loss reduces and the Strehl ratio increases. Even for the largest aperture (R ap = 1.7), the phase error at the center region results in a Strehl ratio smaller than 1.0.

Fig. 6
Fig. 6

Iterative method to find the mirror phase profiles conforming to the desired mode profile with apertures inside the resonator. Given the desired mode profile at mirror 1, the internal loop updates the phase profile of mirror 1 and the outer loop updates the phase profile of mirror 2. The start guess for the phase profile of mirror 2 can be a random-phase profile. Prop., propagation.

Fig. 7
Fig. 7

DPCM design results for a resonator with the desired mode at mirror 1 as an ideal Gaussian of radius 0.225 mm (same as the semihemispherical resonator). The length of the resonator is 0.56 m and the wavelength is 1.064 μm. The aperture radius of mirror 2 was chosen as the variable and the aperture radius of mirror 1 is set at four times the desired mode radius at mirror 1. The aperture radius R ap is the ratio of the aperture radius to the mode radius at mirror 2, which is 0.872 mm. The ideal L-G results in an ideal Gaussian mode solution assuming a cylindrical symmetry of the resonator.

Fig. 8
Fig. 8

Phase profile of mirror 1 and 2 for the same desired mode as in Fig. 7. It can be seen that (a) the mirror 1 phase profile approaches a flat profile as the aperture size grows and (b) the mirror 2 phase profile approaches to the phase profile of an ideal spherical mirror of radius 60 cm. The phase deviation of mirror 1 from the flat profile at the center is required to form a desired mode with lower loss that satisfies the phase-conjugation condition.

Fig. 9
Fig. 9

Strehl ratio and round-trip mode loss for the same desired mode as in Fig. 7. Compared with the semihemispherical resonator (Fig. 5), the DPCM resonator achieves a much smaller round-trip loss and higher Strehl ratio.

Tables (1)

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Table 1 Comparison of Resonator Performance for the Semihemispherical Resonator and the DPCM Resonator

Equations (3)

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V-u, ν, zb=ik2πLexp-ikLAU-x, y, z0×expikx-u2+y-ν2/2Ldxdy,U+x, y, z0=ik2πLexp-ikLBV+u, ν, zb×expikx-u2+y-ν2/2Ldudv,
U-=U+*,V+=V-*.
SRexp-4π2σ2,

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