Abstract

The x-polarized and y-polarized beam modes of simple confining circularly symmetric spherical-mirror laser resonators may be represented with Laguerre–Gaussian functions. The corresponding r- and ϕ-polarized beam-mode profiles are found to be represented by a different Laguerre–Gaussian mode set. Though similar in name, these cylindrically polarized modes are fundamentally different from their rectangularly polarized counterparts. For example, the fundamental x-polarized mode is Gaussian, which has a maximum at the center of the beam, while the fundamental ϕ-polarized mode has a null at the beam center. The complete propagation characteristics of several types of azimuthally polarized Laguerre–Gaussian beams through optical systems representable by complex ABCD matrices are obtained. Cylindrically polarized Laguerre–Gaussian beams may be produced by inserting the appropriate Brewster window into a simple confining circularly symmetric laser resonator.

© 1998 Optical Society of America

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References

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  1. See, for example, J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), pp. 164–176.
  2. E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
    [CrossRef]
  3. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  4. A. A. Tovar, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997).
    [CrossRef]
  5. D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
    [CrossRef]
  6. P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
    [CrossRef]
  7. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.
  9. H. W. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  10. 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]
  11. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Subsec. 20.5.
  12. See, for example, A. L. Bloom, Gas Lasers (Wiley, New York, 1968), pp. 82–85.
  13. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  14. A. A. Tovar, “Phase compensation of azimuthally polarized J1 Bessel–Gaussian laser beams,” Appl. Opt. 37, 540–545 (1998).
    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1973 (1)

1972 (1)

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965 (1)

H. W. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

1964 (1)

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Casperson, L. W.

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Greene, P. L.

Hall, D. G.

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Kogelnik, H. W.

H. W. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Kong, J. A.

See, for example, J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), pp. 164–176.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

Pohl, D.

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

Siegman, A. E.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.

Tovar, A. A.

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

H. W. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

J. Opt. Soc. Am. A (3)

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (4)

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

See, for example, A. L. Bloom, Gas Lasers (Wiley, New York, 1968), pp. 82–85.

See, for example, J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), pp. 164–176.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.

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Figures (2)

Fig. 1
Fig. 1

If a Brewster window is defined as an optical element that completely transmits one polarization while providing some loss to the corresponding orthogonal polarization, then it is possible to define r- and ϕ-Brewster windows.

Fig. 2
Fig. 2

The beam profile of the fundamental mode of a laser with an x- or y-Brewster window is Gaussian, but when an r- or ϕ-Brewster window is used, the mode has an axial null.

Equations (72)

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2E(r, ϕ, z)+k2(r, ϕ, z)E(r, ϕ, z)
=-2k(r, ϕ, z)k(r, ϕ, z)·E(r, ϕ, z).
E(r, ϕ, z)=Er(r, ϕ, z)ir+Eϕ(r, ϕ, z)iϕ.
2Er+k2(r, ϕ, z)Er=1r2Er+2 Eϕϕ,
2Eϕ+k2(r, ϕ, z)Eϕ=1r2Eϕ-2 Erϕ.
Er(r, ϕ, z)=T±(r, z)sin(nϕ),
Eϕ(r, ϕ, z)=T±(r, z)cos(nϕ),
2T±r2+1rT±r+2T±z2
-(n±1)2r2T±+k2(r, z)T±=0.
Er(r, ϕ, z)=T±(r, z)cos(nϕ),
Eϕ(r, ϕ, z)=±T±(r, z)sin(nϕ),
T±(r, z)=U±(r, z)exp-i0zk0(z)dz.
2U±r2+1rU±r-U±r2-2ik0(z) U±z-(n±1)2r2U±
+k2(r, z)-k02(z)-i dk0dzU±=0,
k(r, z)=k0(z)-k2(z)r2/2,
k2(r, z)k0(z)[k0(z)-k2(z)r2].
2U±r2+1rU±r-U±r2
-2ik0(z) U±z-(n±1)2r2U±
-k0(z)k2(z)r2+i dk0dzU±=0.
U±(r, z)=L(r, z)exp{-i[Q(z)r2/2+P(z)]}.
2Lr2+1r-2iQ(z)r Lr-2ik0(z) Lz-(n±1)2r2L
-Q2(z)+k0(z) Qz+k0(z)k2(z)r2L
-2k0(z) Pz+2iQ(z)+i dk0dzL=0.
Q2(z)+k0(z) dQdz+k0(z)k2(z)=0,
2k0(z) dPdz=-2iQ(z)-i dk0dz-8p+4(n±1)W2(z),
2Lr2+1r-2iQ(z)r Lr-2ik0(z) Lz-(n±1)2r2L
+[8p+4(n±1)] LW2(z)=0.
ρ(r, z)=21/2rW(z),
ζ(r, z)=z,
2Lρ2+1ρ-iρW2(z)Q(z)-k0(z)W(z)dWdz Lρ
-ik0(z)W2(z) Lζ
+[4p+2(n±1)]-(n±1)2ρ2L=0.
iW2(z)Q(z)-k0(z)W(z)dWdz=2,
d2Ldρ2+1ρ-2ρ dLdρ+[4p+2(n±1)]
-(n±1)2ρ2L=0.
τ=ρ2,
4τ d2Ldτ2+(4-4τ) dLdτ+[4p+2(n±1)]
-(n±1)2τL=0.
L(τ)=τ(n±1)/2Lpn±1(τ)
τ d2Lpn±1dτ2+[1+(n±1)-τ]dLpn±1dτ+pLpn±1(τ)=0.
E(r, ϕ, z)=E0 exp-i0zk0(z)dz×exp-ik0q(z)r22+P(z)×2r2W2(z)(n±1)/2Lpn±12r2W2(z)×cos(nϕ)iϕsin(nϕ)ir±sin(nϕ)iϕ+cos(nϕ)ir,
Q2(z)+k0(z) dQdz+k0(z)k2(z)=0.
1q(z)Q(z)k0(z)=1u(z)dudz,
ddzk0(z) dudz+k2(z)u(z)=0.
u(z)=A(z)u(0)+B(z)u(0)/q(0).
u(z)/q(z)=C(z)u(0)+D(z)u(0)/q(0),
u(z)u(z)/q(z)=A(z)B(z)C(z)D(z)u(0)u(0)/q(0).
1q(z)=C(z)+D(z)/q(0)A(z)+B(z)/q(0).
iW2(z)Q(z)-k0(z)W(z)dWdz=2.
i1u(z)dudz-1W(z)dWdz=2k0(z)W2(z).
W2(z)=W(0)2[A(z)+B(z)/q(0)]2+4ik0(0)B(z)[A(z)+B(z)/q(0)],
A(z)D(z)-B(z)C(z)=k0(0)/k0(z)
2 dPdz=-2iu(z)dudz-ik0(z)dk0dz-i[4p+2(n±1)]×1u(z)dudz-1W(z)dWdz.
P(z)-P(0)
=-i ln[A(z)+B(z)/q(0)]+i2ln[A(z)D(z)-B(z)C(z)]+i[4p+2(n±1)]4
×ln1+4ik0(0)W2(0)B(z)A(z)+B(z)/q(0).
u2u2/q2=A1B1C1D1u1u1/q1,
u3u3/q3=A2B2C2D2u2u2/q2.
u3u3/q3=A2B2C2D2A1B1C1D1u1u1/q1.
Msyst=A2B2C2D2A1B1C1D1=M2M1.
Msyst=MnMn-1M3M2M1.
1q2=C+D/q1A+B/q1.
1q=1R-i λn0πw2.
E0,2=AJE0,1,
AJ=exp-iz1z2k0(z)dz.
W22=W12(A+B/q1)2+4ik01B(A+B/q1).
P2-P1=-i ln(A+B/q1)
+i2ln(AD-BC)+i[4p+2(n±1)]4×ln1+4ik01W12BA+B/q1.
E2=E0,2cos(nϕ)iϕsin(nϕ)ir±sin(nϕ)iϕ+cos(nϕ)ir×exp-ik02q2r22+P22r2W22(n±1)/2Lpn±12r2W22,
E2=E0,1AJ exp-ik02q2r22+P2×2r2W221/2Lp12r2W22iϕir.
E2=E0,2 exp-ik0,2q2r22+P22r2w221/2Lp12r2w22iϕir.
E2=E0,221/2rw2exp-ik0,2q2r22+P2iϕir.

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