Abstract

In lasers with nonorthogonal eigenmodes the excess-noise factor K can be large, especially in unstable-cavity lasers with hard-edged intracavity apertures. To the best of our knowledge, we report the first detailed study of the dependence of K on aperture shape. Calculations and measurements of K for unstable-cavity lasers with variable-size apertures of triangular, square, pentagonal, hexagonal, octagonal, and rhomboid symmetries are summarized. It is shown that both the magnitude of K and its resonant behavior strongly depend on aperture shape and that many aspects of this dependence can be explained in terms of one-dimensional resonance lengths.

© 1999 Optical Society of America

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  1. K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
    [CrossRef]
  2. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989); “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
    [CrossRef] [PubMed]
  3. G. P. Karman, J. P. Woerdman, “Fractal structure of eigenmodes of unstable-cavity lasers,” Opt. Lett. 23, 1909–1911 (1998).
    [CrossRef]
  4. A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
    [CrossRef]
  5. G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
    [CrossRef]
  6. A. E. Siegman, “Lasers without photons—or should it be lasers with too many photons?,” Appl. Phys. B 60, 247–257 (1995).
    [CrossRef]
  7. M. A. Rippin, G. H. C. New, “Excess-noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
    [CrossRef]
  8. Y. J. Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
    [CrossRef] [PubMed]
  9. M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
    [CrossRef] [PubMed]
  10. M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
    [CrossRef]
  11. W. H. Southwell, “Unstable-resonator-mode derivation using virtual-source theory,” J. Opt. Soc. Am. A 3(11) , 1885–1891 (1986).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  13. Å. M. Lindberg, M. A. van Eijkelenborg, J. P. Woerdman, “Measuring the quantum-limited linewidth of a laser using the Zeeman effect,” IEEE J. Quantum Electron. 33, 1767–1773 (1997).
    [CrossRef]
  14. G. P. Karman, Å. M. Lindberg, J. P. Woerdman, “Observed factorization of excess quantum noise that is due to both polarization and spatial mode nonorthogonality,” Opt. Lett. 23, 1698–1700 (1998).
    [CrossRef]
  15. G. S. McDonald, W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
    [CrossRef]
  16. G. S. McDonald, G. H. C. New, J. P. Woerdman, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1998).
    [CrossRef]

1998 (3)

1997 (2)

Å. M. Lindberg, M. A. van Eijkelenborg, J. P. Woerdman, “Measuring the quantum-limited linewidth of a laser using the Zeeman effect,” IEEE J. Quantum Electron. 33, 1767–1773 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
[CrossRef]

1996 (3)

M. A. Rippin, G. H. C. New, “Excess-noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

Y. J. Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

1995 (2)

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

A. E. Siegman, “Lasers without photons—or should it be lasers with too many photons?,” Appl. Phys. B 60, 247–257 (1995).
[CrossRef]

1993 (1)

G. S. McDonald, W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

1989 (1)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989); “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

1986 (1)

W. H. Southwell, “Unstable-resonator-mode derivation using virtual-source theory,” J. Opt. Soc. Am. A 3(11) , 1885–1891 (1986).
[CrossRef]

1979 (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[CrossRef]

1958 (1)

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Cheng, Y. J.

Y. J. Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Fanning, C. G.

Y. J. Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Firth, W. J.

G. S. McDonald, W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

Karman, G. P.

Lindberg, Å. M.

G. P. Karman, Å. M. Lindberg, J. P. Woerdman, “Observed factorization of excess quantum noise that is due to both polarization and spatial mode nonorthogonality,” Opt. Lett. 23, 1698–1700 (1998).
[CrossRef]

Å. M. Lindberg, M. A. van Eijkelenborg, J. P. Woerdman, “Measuring the quantum-limited linewidth of a laser using the Zeeman effect,” IEEE J. Quantum Electron. 33, 1767–1773 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

McDonald, G. S.

G. S. McDonald, G. H. C. New, J. P. Woerdman, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1998).
[CrossRef]

G. S. McDonald, W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

New, G. H. C.

G. S. McDonald, G. H. C. New, J. P. Woerdman, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1998).
[CrossRef]

M. A. Rippin, G. H. C. New, “Excess-noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[CrossRef]

Rippin, M. A.

M. A. Rippin, G. H. C. New, “Excess-noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

Schawlow, A. L.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Siegman, A. E.

Y. J. Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

A. E. Siegman, “Lasers without photons—or should it be lasers with too many photons?,” Appl. Phys. B 60, 247–257 (1995).
[CrossRef]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989); “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

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

Southwell, W. H.

W. H. Southwell, “Unstable-resonator-mode derivation using virtual-source theory,” J. Opt. Soc. Am. A 3(11) , 1885–1891 (1986).
[CrossRef]

Thijssen, M. S.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Townes, C. H.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

van Eijkelenborg, M. A.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
[CrossRef]

Å. M. Lindberg, M. A. van Eijkelenborg, J. P. Woerdman, “Measuring the quantum-limited linewidth of a laser using the Zeeman effect,” IEEE J. Quantum Electron. 33, 1767–1773 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Woerdman, J. P.

G. P. Karman, J. P. Woerdman, “Fractal structure of eigenmodes of unstable-cavity lasers,” Opt. Lett. 23, 1909–1911 (1998).
[CrossRef]

G. P. Karman, Å. M. Lindberg, J. P. Woerdman, “Observed factorization of excess quantum noise that is due to both polarization and spatial mode nonorthogonality,” Opt. Lett. 23, 1698–1700 (1998).
[CrossRef]

G. S. McDonald, G. H. C. New, J. P. Woerdman, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1998).
[CrossRef]

Å. M. Lindberg, M. A. van Eijkelenborg, J. P. Woerdman, “Measuring the quantum-limited linewidth of a laser using the Zeeman effect,” IEEE J. Quantum Electron. 33, 1767–1773 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
[CrossRef]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Appl. Phys. B (1)

A. E. Siegman, “Lasers without photons—or should it be lasers with too many photons?,” Appl. Phys. B 60, 247–257 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

Å. M. Lindberg, M. A. van Eijkelenborg, J. P. Woerdman, “Measuring the quantum-limited linewidth of a laser using the Zeeman effect,” IEEE J. Quantum Electron. 33, 1767–1773 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[CrossRef]

J. Mod. Opt. (3)

M. A. Rippin, G. H. C. New, “Excess-noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

G. S. McDonald, W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

W. H. Southwell, “Unstable-resonator-mode derivation using virtual-source theory,” J. Opt. Soc. Am. A 3(11) , 1885–1891 (1986).
[CrossRef]

Opt. Commun. (2)

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Influence of transverse resonator geometry on excess noise,” Opt. Commun. 137, 303–307 (1997).
[CrossRef]

G. S. McDonald, G. H. C. New, J. P. Woerdman, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989); “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

Phys. Rev. (1)

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Phys. Rev. Lett. (2)

Y. J. Cheng, C. G. Fanning, A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable-cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Other (1)

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

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Figures (5)

Fig. 1
Fig. 1

All the aperture shapes are regular polygons, except for the rhombus. The dashed lines show how some of the shapes can be constructed from equilateral triangles.

Fig. 2
Fig. 2

Experimental setup: M1,2, mirrors; C, capillary; B, beam splitter; D1,2, detectors; P, polarizer; A, aperture (positioned within 1 mm of M2).

Fig. 3
Fig. 3

Measured and calculated K-factor dependencies on equivalent Fresnel numbers for different aperture shapes. Points with solid curves represent experimental results; the dashed curves represent theory.

Fig. 4
Fig. 4

Calculated K factors for different aperture shapes as a function of N eq at M = 1.3: (a) square, (b) octagon, (c) circle, (d) rhombus, (e) hexagon, (f) pentagon, (g) triangle.

Fig. 5
Fig. 5

One-dimensional lengths used in the interpretation of the maxima of K-factor curves.

Equations (2)

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K = 1 -   U s V s d s 2 ,
N eq = M 2 - 1 2 M a 2 λ B ,

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