Abstract

The modes of a stable laser resonator containing, near one mirror, an absorbing mask with two apertures have been calculated on the basis of scalar diffraction theory and experimentally observed in a pulsed Nd:YAG laser. The mode structure has been investigated as a function of the mask geometry, and an interpretation in terms of supermodes is provided.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).
  2. D. G. Youmans, “Phase Locking of Adjacent Channel Leaky Waveguide CO2 Lasers,” Appl. Phys. Lett. 44, 365–367 (1984).
    [CrossRef]
  3. L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
    [CrossRef]
  4. R. A. Hart, L. A. Newman, J. T. Kennedy, “Staggered Hollow-Bore CO2 Waveguide Laser Array,” in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987), pp. 354–356.
  5. D. Cantin, M. Piche, G. A. Heckman, “Phase-Locking of CO2 Laser Array Using Diffraction Coupling,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 82, 83.
  6. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
    [CrossRef]
  7. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Segmented-Mirror Laser Resonators,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 16–17.
  8. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Resonators with Modulated Reflectivity Mirrors,” in Sixteenth International Conference on Quantum Electronics Technical Digest (Japanese Society of Applied Physics, 1988), pp. 98–99.
  9. E. Kapon, J. Katz, A. Yariv, “Supermode Analysis of Phase-Locked Arrays of Semiconductor Lasers,” Opt. Lett. 9, 125–127 (1984).
    [CrossRef] [PubMed]
  10. P. Baues, “Hyugens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37–44 (1969).
    [CrossRef]
  11. A. E. Siegman, Lasers (Oxford U.P., London, 1986), pp. 792–793.
  12. A. E. Siegman, H. Y. Miller, “Unstable Optical Resonator Loss Calculations Using the Prony Method,” Appl. Opt. 9, 2729–2736 (1970).
    [CrossRef] [PubMed]
  13. W. D. Murphy, M. L. Bernabe, “Numerical Procedures for Solving Nonsymmetric Eigenvalue Problems Associated with Optical Resonators,” Appl. Opt. 17, 2358–2365 (1978).
    [CrossRef] [PubMed]
  14. J. Katz, S. Margalit, A. Yariv, “Diffraction Coupled Phase-Locked Semiconductor Laser Arrays,” Appl. Phys. Lett. 42, 554–556 (1983).
    [CrossRef]
  15. J. Katz, J. Cser, W. K. Marshall, “Diffraction-Coupled Real Index-Guided Semiconductor Laser Arrays,” in Conference on Laser and Electro-Optics Technical Digest Series 1986, Vol. 7 (Optical Society of America, Washington, DC, 1986), pp. 78–79.

1987 (1)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
[CrossRef]

1986 (2)

G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

1984 (2)

D. G. Youmans, “Phase Locking of Adjacent Channel Leaky Waveguide CO2 Lasers,” Appl. Phys. Lett. 44, 365–367 (1984).
[CrossRef]

E. Kapon, J. Katz, A. Yariv, “Supermode Analysis of Phase-Locked Arrays of Semiconductor Lasers,” Opt. Lett. 9, 125–127 (1984).
[CrossRef] [PubMed]

1983 (1)

J. Katz, S. Margalit, A. Yariv, “Diffraction Coupled Phase-Locked Semiconductor Laser Arrays,” Appl. Phys. Lett. 42, 554–556 (1983).
[CrossRef]

1978 (1)

1970 (1)

1969 (1)

P. Baues, “Hyugens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37–44 (1969).
[CrossRef]

Baues, P.

P. Baues, “Hyugens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37–44 (1969).
[CrossRef]

Bernabe, M. L.

Cantin, D.

D. Cantin, M. Piche, G. A. Heckman, “Phase-Locking of CO2 Laser Array Using Diffraction Coupling,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 82, 83.

Cantor, A. J.

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

Cross, P.

G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).

Cser, J.

J. Katz, J. Cser, W. K. Marshall, “Diffraction-Coupled Real Index-Guided Semiconductor Laser Arrays,” in Conference on Laser and Electro-Optics Technical Digest Series 1986, Vol. 7 (Optical Society of America, Washington, DC, 1986), pp. 78–79.

De Maria, A. J.

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Segmented-Mirror Laser Resonators,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 16–17.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Resonators with Modulated Reflectivity Mirrors,” in Sixteenth International Conference on Quantum Electronics Technical Digest (Japanese Society of Applied Physics, 1988), pp. 98–99.

Harnagel, G.

G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).

Hart, R. A.

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

R. A. Hart, L. A. Newman, J. T. Kennedy, “Staggered Hollow-Bore CO2 Waveguide Laser Array,” in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987), pp. 354–356.

Heckman, G. A.

D. Cantin, M. Piche, G. A. Heckman, “Phase-Locking of CO2 Laser Array Using Diffraction Coupling,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 82, 83.

Kapon, E.

Katz, J.

E. Kapon, J. Katz, A. Yariv, “Supermode Analysis of Phase-Locked Arrays of Semiconductor Lasers,” Opt. Lett. 9, 125–127 (1984).
[CrossRef] [PubMed]

J. Katz, S. Margalit, A. Yariv, “Diffraction Coupled Phase-Locked Semiconductor Laser Arrays,” Appl. Phys. Lett. 42, 554–556 (1983).
[CrossRef]

J. Katz, J. Cser, W. K. Marshall, “Diffraction-Coupled Real Index-Guided Semiconductor Laser Arrays,” in Conference on Laser and Electro-Optics Technical Digest Series 1986, Vol. 7 (Optical Society of America, Washington, DC, 1986), pp. 78–79.

Kennedy, J. T.

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

R. A. Hart, L. A. Newman, J. T. Kennedy, “Staggered Hollow-Bore CO2 Waveguide Laser Array,” in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987), pp. 354–356.

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Resonators with Modulated Reflectivity Mirrors,” in Sixteenth International Conference on Quantum Electronics Technical Digest (Japanese Society of Applied Physics, 1988), pp. 98–99.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Segmented-Mirror Laser Resonators,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 16–17.

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Segmented-Mirror Laser Resonators,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 16–17.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Resonators with Modulated Reflectivity Mirrors,” in Sixteenth International Conference on Quantum Electronics Technical Digest (Japanese Society of Applied Physics, 1988), pp. 98–99.

Margalit, S.

J. Katz, S. Margalit, A. Yariv, “Diffraction Coupled Phase-Locked Semiconductor Laser Arrays,” Appl. Phys. Lett. 42, 554–556 (1983).
[CrossRef]

Marshall, W. K.

J. Katz, J. Cser, W. K. Marshall, “Diffraction-Coupled Real Index-Guided Semiconductor Laser Arrays,” in Conference on Laser and Electro-Optics Technical Digest Series 1986, Vol. 7 (Optical Society of America, Washington, DC, 1986), pp. 78–79.

Miller, H. Y.

Murphy, W. D.

Newman, L. A.

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

R. A. Hart, L. A. Newman, J. T. Kennedy, “Staggered Hollow-Bore CO2 Waveguide Laser Array,” in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987), pp. 354–356.

Piche, M.

D. Cantin, M. Piche, G. A. Heckman, “Phase-Locking of CO2 Laser Array Using Diffraction Coupling,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 82, 83.

Scifres, D.

G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).

Siegman, A. E.

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Resonators with Modulated Reflectivity Mirrors,” in Sixteenth International Conference on Quantum Electronics Technical Digest (Japanese Society of Applied Physics, 1988), pp. 98–99.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Segmented-Mirror Laser Resonators,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 16–17.

Welch, D.

G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).

Yariv, A.

E. Kapon, J. Katz, A. Yariv, “Supermode Analysis of Phase-Locked Arrays of Semiconductor Lasers,” Opt. Lett. 9, 125–127 (1984).
[CrossRef] [PubMed]

J. Katz, S. Margalit, A. Yariv, “Diffraction Coupled Phase-Locked Semiconductor Laser Arrays,” Appl. Phys. Lett. 42, 554–556 (1983).
[CrossRef]

Youmans, D. G.

D. G. Youmans, “Phase Locking of Adjacent Channel Leaky Waveguide CO2 Lasers,” Appl. Phys. Lett. 44, 365–367 (1984).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

J. Katz, S. Margalit, A. Yariv, “Diffraction Coupled Phase-Locked Semiconductor Laser Arrays,” Appl. Phys. Lett. 42, 554–556 (1983).
[CrossRef]

D. G. Youmans, “Phase Locking of Adjacent Channel Leaky Waveguide CO2 Lasers,” Appl. Phys. Lett. 44, 365–367 (1984).
[CrossRef]

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. De Maria, “High Power Coupled CO2 Waveguide Laser Array,” Appl. Phys. Lett. 48, 1701–1703 (1986).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Segmented-Mirror Phased-Array Lasers,” Appl. Phys. Lett., 51, 1771–1773 (1987).
[CrossRef]

Laser & Appl. (1)

G. Harnagel, D. Welch, P. Cross, D. Scifres, “High Power Laser Arrays: A Progress Report,” Laser & Appl., 5, 135–138 (June1986);sections on semiconductor laser arrays in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987).

Opt. Lett. (1)

Opto-electronics (1)

P. Baues, “Hyugens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” Opto-electronics 1, 37–44 (1969).
[CrossRef]

Other (6)

A. E. Siegman, Lasers (Oxford U.P., London, 1986), pp. 792–793.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Segmented-Mirror Laser Resonators,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 16–17.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Modes of Resonators with Modulated Reflectivity Mirrors,” in Sixteenth International Conference on Quantum Electronics Technical Digest (Japanese Society of Applied Physics, 1988), pp. 98–99.

R. A. Hart, L. A. Newman, J. T. Kennedy, “Staggered Hollow-Bore CO2 Waveguide Laser Array,” in Conference on Laser and Electro-Optics Technical Digest Series 1987, Vol. 14 (Optical Society of America, Washington, DC, 1987), pp. 354–356.

D. Cantin, M. Piche, G. A. Heckman, “Phase-Locking of CO2 Laser Array Using Diffraction Coupling,” in Conference on Laser and Electro-Optics Technical Digest Series 1988, Vol. 7 (Optical Society of America, Washington, DC, 1988), pp. 82, 83.

J. Katz, J. Cser, W. K. Marshall, “Diffraction-Coupled Real Index-Guided Semiconductor Laser Arrays,” in Conference on Laser and Electro-Optics Technical Digest Series 1986, Vol. 7 (Optical Society of America, Washington, DC, 1986), pp. 78–79.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Resonator containing an absorbing mask with two apertures (top). Sketch of the absorbing mask (bottom).

Fig. 2
Fig. 2

Round-trip losses of the first even-parity and odd-parity modes as a function of the aperture distance b for a mask with aperture width a = 1.1 mm. The numbers near the curves refer to subsequent figures that show the mode profiles.

Fig. 3
Fig. 3

Mode intensity profiles generated by the mask with apertures of width a = 1.1 mm separated by b = 0.4 mm (point 3 in Fig. 2). (a) Computed near field, (b) computed far field; (c) and (d) corresponding experimental profiles. The shaded area in (a) represents the part of the beam intercepted by the absorbing mask. The horizontal scale in (c) is 0.44 mm/div, and 0.54 mrad/div in (d).

Fig. 4
Fig. 4

Same as Fig. 3 with a = 1.1 mm and b = 0.88 mm.

Fig. 5
Fig. 5

Same as Fig. 3 with a = 1.1 mm and b = 1.55 mm.

Fig. 6
Fig. 6

Same as Fig. 3 with a = 1.1 mm and b = 1.8 mm.

Fig. 7
Fig. 7

Mode intensity profile (wave incident on the mask) and phase in the resonator containing one off-axis aperture. The mask is the same pertaining to Fig. 4 with one of the apertures blocked. The reference surface for the phase is the surface of the mirror near the mask. The shaded area represents that part of the beam intercepted by the absorbing mask; the corresponding losses are 56.9%.

Fig. 8
Fig. 8

Supermodes in the resonator containing a mask with a = 1.1 mm and b = 0.88 mm (same as that of Fig. 4). The intensity profiles are calculated by superimposition of the off-axis mode of Fig. 8 on the mirror image respect to x = 0. The even supermode (a) and the odd supermode (b) are obtained, respectively, by subtraction and addition.

Fig. 9
Fig. 9

Square of the center-to-center spacing between the two apertures (dm) corresponding to the minima of the loss curve in Fig. 2 as a function of the integer n that numbers the sequence of the minima (dots). The solid line is an interpolating straight line. The value of d m 2 is divided by the wavelength λ and the resonator length L.

Fig. 10
Fig. 10

Round-trip losses of different modes as a function of the aperture width a for aperture separation b = 0.7 mm. The number near a curve refers to Fig. 11 which shows the mode profile.

Fig. 11
Fig. 11

Mode intensity profiles generated by the mask with apertures of width a = 1.65 mm separated by b = 0.7 mm (point 11 in Fig. 10). (a) Computed near field, (b) computed far field; (c) and (d) corresponding experimental profiles. The shaded area in (a) represents that part of the beam intercepted by the absorbing mask. The horizontal scale in (c) is 0.44 mm/div, and 0.54 mrad/div in (d).

Fig. 12
Fig. 12

Round-trip losses of the first even-parity and odd-parity modes as a function of the aperture distance b for a mask with the overall dimension 2a + b = 2.8 mm. The number near one of the curves refers to Fig. 13 which shows the mode profile.

Fig. 13
Fig. 13

Mode intensity profiles generated by the mask with apertures of width a = 0.77 mm separated by b = 1.26 mm (point 13 in Fig. 12). (a) Computed near field, (b) computed far field; (c) and (d) corresponding experimental profiles. The shaded area in (a) represents the part of the beam intercepted by the absorbing mask. The horizontal scale in (c) is 0.44 mm/div, and 0.54 mrad/div in (d).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

σ u ( x ) = τ 2 ( x 1 ) K ( x , x 1 ) u ( x 1 ) d x 1 ,
K ( x , x 1 ) = j / λ B exp [ j ( π / λ B ) ( A x 1 2 2 x x 1 + D x 2 ) ]
A = 1 4 L / R 1 2 L / R 2 + 4 L 2 / R 1 R 2 , B = 2 L ( 1 L / R 2 ) D = 1 2 L / R 2 .
d m 2 λ L = 2 β n ,
d m 2 λ L = 3 . 0 n ,

Metrics