Abstract

Hermite–sinusoidal-Gaussian solutions to the wave equation have recently been obtained. In the limit of large Hermite–Gaussian beam size, the sinusoidal factors are dominant and reduce to the conventional modes of a rectangular waveguide. In the opposite limit the beams reduce to the familiar Hermite–Gaussian form. The propagation of these beams is examined in detail, and resonators are designed that will produce them. As an example, a special resonator is designed to produce hyperbolic-sine-Gaussian beams. This ring resonator contains a hyperbolic-cosine-Gaussian apodized aperture. The beam mode has finite energy and is perturbation stable.

© 1998 Optical Society of America

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References

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  1. R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).
  2. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
    [CrossRef]
  3. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  4. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. 67, 1274–1276 (1977).
    [CrossRef]
  5. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Subsec. 20.5.
  6. Z. Jiang, “Truncation of a two-dimensional nondiffracting cos beam,” J. Opt. Soc. Am. A 14, 1478–1481 (1997).
    [CrossRef]
  7. L. W. Casperson, A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997).
    [CrossRef]
  8. L. W. Casperson, A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  9. C. Pare, L. Gagnon, P. A. Belanger, “Aspheric laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
    [CrossRef]
  10. R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
    [CrossRef]
  11. G. Angelow, F. Laeri, T. Tschudi, “Designing resonators with large mode volume and high mode discrimination,” Opt. Lett. 21, 1324–1326 (1996).
    [CrossRef] [PubMed]
  12. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef]
  13. D. Marcuse, “The effect of the ∇n2 term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
    [CrossRef]
  14. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [CrossRef]
  15. See A. A. Tovar, L. W. Casperson, “Generalized beam matrices. IV. Optical system design,” J. Opt. Soc. Am. A 14, 882–893 (1997), and references therein.
    [CrossRef]
  16. L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
    [CrossRef] [PubMed]

1998 (1)

1997 (3)

1996 (1)

1995 (1)

1994 (1)

R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
[CrossRef]

1992 (1)

C. Pare, L. Gagnon, P. A. Belanger, “Aspheric laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef]

1981 (1)

1977 (1)

1976 (1)

1974 (1)

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

1973 (1)

D. Marcuse, “The effect of the ∇n2 term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
[CrossRef]

1961 (1)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Angelow, G.

Belanger, P. A.

R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
[CrossRef]

C. Pare, L. Gagnon, P. A. Belanger, “Aspheric laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Casperson, L. W.

Dicke, R. H.

R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).

Gagnon, L.

C. Pare, L. Gagnon, P. A. Belanger, “Aspheric laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Jiang, Z.

Lachance, R. L.

R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
[CrossRef]

Laeri, F.

Marcuse, D.

D. Marcuse, “The effect of the ∇n2 term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
[CrossRef]

Pare, C.

R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
[CrossRef]

C. Pare, L. Gagnon, P. A. Belanger, “Aspheric laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef]

Pratesi, R.

Ronchi, L.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Subsec. 20.5.

Tovar, A. A.

Tschudi, T.

Van Neste, R.

R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. Van Neste, C. Pare, R. L. Lachance, P. A. Belanger, “Graded-phase mirror resonators with a super-Gaussian output in a cw CO2 laser,” IEEE J. Quantum Electron. 30, 3663–3669 (1994).
[CrossRef]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

D. Marcuse, “The effect of the ∇n2 term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Phys. Rev. A (1)

C. Pare, L. Gagnon, P. A. Belanger, “Aspheric laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef]

Other (2)

R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Subsec. 20.5.

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Figures (3)

Fig. 1
Fig. 1

(a) Field amplitude before the light beam enters the optical system, (b) example optical system, (c) field amplitude after the beam propagates through the optical system.

Fig. 2
Fig. 2

Resonator design that produces a sinh-Gaussian beam.

Fig. 3
Fig. 3

Curve (a) Gaussian, curve (b) cosh-Gaussian apodized aperture transmission profiles.

Equations (60)

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2E(x, y, z)+k2(x, y, z)E(x, y, z)=0,
E(x, y, z, t)=ReE(x, y, z)exp(iωt)ixiy.
k(x, y, z)2πλn(x, y, z)+iα(x, y, z)
=k0(z)-k1x(z)x/2-k1y(z)y/2-k2x(z)x2/2-k2y(z)y2/2
E(x, y, z)=E0(z)exp-iQx(z)2x2+Qy(z)2y2+Sx(z)x+Sy(z)y+P(z)×Hm2Wx(z)[x-δx(z)]×Hn2Wy(z)[y-δy(z)]×cosh[Ωx(z)x+Φx(z)]sinh[Ωx(z)x+Φx(z)]cos[Ωx(z)x+Φx(z)]sin[Ωx(z)x+Φx(z)]×cosh[Ωy(z)y+Φy(z)]sinh[Ωy(z)y+Φy(z)]cos[Ωy(z)y+Φy(z)]sin[Ωy(z)y+Φy(z)],
E2(x, y)=E02 exp-ik022qx2x2+k022qy2y2+Sx2x+Sy2y+P2×Hm2Wx2(x-δx2)Hn2Wy2(y-δy2)×cosh(Ωx2x+Φx2)sinh(Ωx2x+Φx2)cos(Ωx2x+Φx2)sin(Ωx2x+Φx2)×cosh(Ωy2y+Φy2)sinh(Ωy2y+Φy2)cos(Ωy2y+Φy2)sin(Ωy2y+Φy2),
1qx=1Rx-iλmπwx2
1qy=1Ry-iλmπwy2,
Sxβ0=-1qxdxa+dxa,
Syβ0=-1qydya+dya,
uxux/qxSxux2=AxBx0CxDx0GxHx1uxux/qxSxux1,
uyuy/qySyuy2=AyBy0CyDy0GyHy1uyuy/qySyuy1,
(E0)2=AJ(E0)1.
Mx=1d0010001,
My=1d0010001,
AJ=exp(-iβ0d).
1qx2=Cx+Dx/qx1Ax+Bx/qx1,
1qy2=Cy+Dy/qy1Ay+By/qy1,
Sx2=Sx1+Gx+Hx/qx1Ax+Bx/qx1,
Sy2=Sy1+Gy+Hy/qy1Ay+By/qy1.
Wx22=Wx12(Ax+Bx/qx1)2+4iBx(Ax+Bx/qx1)/k01,
Wy22=Wy12(Ay+By/qy1)2+4iBy(Ay+By/qy1)/k01,
δx2=δx1(Ax+Bx/qx1)+Sx1Bx/k01+(BxGx-AxHx)/k01,
δy2=δy1(Ay+By/qy1)+Sy1Bx/k01+(BxGy-AyHx)/k01,
Ωx2=Ωx1Ax+Bx/qx1,
Ωy2=Ωy1Ay+By/qy1,
Φx2=Φx1-Ωx1Bxk01Sx1+Gx+Hx/qx1Ax+Bx/qx1+Ωx1Hxk01,
Φy2=Φy1-Ωy1Byk01Sy1+Gy+Hy/qy1Ay+By/qy1+Ωy1Hyk01,
P2-P1=i2ln(AxDx-BxCx)-i2ln(Ax+Bx/qx1)-i2ln(Ay+By/qy1)+im2ln1+4ik01Wx12BxAx+Bx/qx1+in2ln1+4ik01Wy12ByAy+By/qy1iΩx122k01BxAx+Bx/qx1iΩy122k01ByAy+By/qy1-Bx2k01(Sx1+Gx+Hx/qx1)2Ax+Bx/qx1
-By2k01(Sy1+Gy+Hy/qy1)2Ay+By/qy1+Hx2k01(2Sx1+Gx+Hx/qx1)+Hy2k01(2Sy1+Gy+Hy/qy1)
+12k010zGxdHxdz-HxdGxdzdz+12k010zGydHydz-HydGydzdz.
ABCD=1f0110-1/f1=0f-1/f1.
1q2=-f-1+q1-1f/q1.
Ω2=Ω1f/q1.
|Ein|=exp(-x2/w12)cosh(Ω1x),
1q1=-iλπw12.
|Eout|=exp[-x2/(λf/πw1)2]cosh(iΩ1πw12λ-1f-1x),
f=πw12λ,
|Eout|=exp(-x2/w12)cosh(iΩ1x),
E1=E1,0 exp-iβ02qx1x2+P1sinh(Ωx1x),
t(x)=taper cosh(Ωaperx).
E2=12taperE1,0 exp-iβ02qx1x2+P1sinh(2Ωx1x),
E3=12rmirrortaper exp(g0l/2)E1,0×exp-iβ02Cx+Dx/qx1Ax+Bx/qx1x2+P1-i2ln(Ax+Bx/qx1)+2β0Ωx12BxAx+Bx/qx1×sinh2Ωx1Ax+Bx/qx1x,
Mx=20Cx1/2.
Cx=-32L-i2λπwaper2.
E3=2-3/2rmirrortaper exp(g0l/2)E1,0 exp-iβ04-32L-i2λπwaper2+12qx1x2+P1sinh(Ωx1x).
1qx1=-1L-i4λ3πwaper2,
rmirrortaper exp(g0l/2)=2-3/2,
Mx=10-iλ/(πwaper2)110-6/L11L/301×10-1/2/L11L/30110-15/(2L)11L/301.
ttotal(x)=taper cosh(Ωaperx)exp(-x2/waper2).
t(x)=tapercapersinh(caperΩaperx)sinh(Ωaperx).
Fs=|A+B/q|>1.
E˜1=E1,0 exp-iβ02qx1x2+P1sinh[(Ωx1+δ)x],
E˜2=taperE1,0 exp-iβ02qx1x2+P1×sinh[(Ωx1+δ)x]cosh(Ωx1x)
=taper2E1,0 exp-iβ02qx1x2+P1×{sinh[(2Ωx1+δ)x]+sinh(δx)}.
E˜3=rmirrortaper exp(g0l/2)2E1,0×exp-iβ02qx3x2+P3×{sinh[(Ωx1+δ/2)x]+sinh(δx/2)}.
E˜3=E1,0 exp-iβ02qx1x2+P1{sinh[(Ωx1+δ)x]}.
sinh[(Ωx1+δ)x]=sinh[(Ωx1+δ/2)x]+sinh(Ωx1x/2).
δδ=1+cosh(Ωx1x)2 cosh(Ωx1x).
δδ=1+cos(Ωx1x)2 cos(Ωx1x),

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