Abstract

We examine the free-space propagation characteristics of an azimuthally polarized, circularly symmetric beam, such as that emitted by a concentric-circle-grating surface-emitting laser. We begin with the appropriate scalar wave equation and then find the azimuthal Bessel–Gauss beam solution, using both a Cartesian decomposition method and an angular spectrum representation. We find a general azimuthal diffraction integral for circularly symmetric disturbances and examine two special cases, a thin lens and a circular aperture; the azimuthally polarized beam remains well behaved in both cases. Plots of radial field profiles and longitudinal beam-waist evolution are presented.

© 1996 Optical Society of America

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References

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  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).
  2. D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
    [Crossref]
  3. Y. Mushiake, K. Matsumura, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE Lett. 60, 1107–1109 (1972).
    [Crossref]
  4. J. Wynne, “Generation of the rotationally symmetric TE01and TM01modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
    [Crossref]
  5. M. E. Marhic, E. Garmire, “Low-order TE0qoperation of a CO2laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
    [Crossref]
  6. 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]
  7. T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [Crossref]
  8. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [Crossref] [PubMed]
  9. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  10. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  11. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [Crossref]
  12. S. De Nicola, “Irradiance from an aperture with a truncated J0Bessel beam,” Opt. Commun. 80, 299–302 (1991).
    [Crossref]
  13. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  14. S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
    [Crossref]
  15. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 12, pp. 248–259.
  16. R. H. Jordan, D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,” J. Opt. Soc. Am. A 12, 84–94 (1995).
    [Crossref]
  17. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), Chap. 11, p. 585.
  18. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, New York, 1966), Sec. 3.8.3, p. 93.
  19. Ref. 18, Sec. 3.6.1, p. 79.
  20. R. G. Schell, G. Tyras, “Irradiance from an aperture with a truncated-Gaussian field distribution,” J. Opt. Soc. Am. 61, 31–35 (1971).
    [Crossref]

1995 (1)

1994 (2)

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]

1991 (2)

1990 (1)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

1987 (3)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

1981 (1)

M. E. Marhic, E. Garmire, “Low-order TE0qoperation of a CO2laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

1974 (1)

J. Wynne, “Generation of the rotationally symmetric TE01and TM01modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
[Crossref]

1972 (2)

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

Y. Mushiake, K. Matsumura, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE Lett. 60, 1107–1109 (1972).
[Crossref]

1971 (1)

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).

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]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), Chap. 11, p. 585.

De Nicola, S.

S. De Nicola, “Irradiance from an aperture with a truncated J0Bessel beam,” Opt. Commun. 80, 299–302 (1991).
[Crossref]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

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]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

Garmire, E.

M. E. Marhic, E. Garmire, “Low-order TE0qoperation of a CO2laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Hall, D. G.

R. H. Jordan, D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,” J. Opt. Soc. Am. A 12, 84–94 (1995).
[Crossref]

R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[Crossref] [PubMed]

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]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

Jordan, R. H.

Kenney, C. S.

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]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 12, pp. 248–259.

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, New York, 1966), Sec. 3.8.3, p. 93.

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).

Marhic, M. E.

M. E. Marhic, E. Garmire, “Low-order TE0qoperation of a CO2laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

Matsumura, K.

Y. Mushiake, K. Matsumura, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE Lett. 60, 1107–1109 (1972).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Mushiake, Y.

Y. Mushiake, K. Matsumura, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE Lett. 60, 1107–1109 (1972).
[Crossref]

Nakajima, N.

Y. Mushiake, K. Matsumura, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE Lett. 60, 1107–1109 (1972).
[Crossref]

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, New York, 1966), Sec. 3.8.3, p. 93.

Overfelt, P. L.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[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]

Ruschin, S.

Schell, R. G.

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).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 12, pp. 248–259.

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, New York, 1966), Sec. 3.8.3, p. 93.

Tyras, G.

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]

Wynne, J.

J. Wynne, “Generation of the rotationally symmetric TE01and TM01modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
[Crossref]

Appl. Phys. Lett. (3)

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

M. E. Marhic, E. Garmire, “Low-order TE0qoperation of a CO2laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[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. (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).

IEEE J. Quantum Electron. (1)

J. Wynne, “Generation of the rotationally symmetric TE01and TM01modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
[Crossref]

J. Appl. Phys. (1)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

S. De Nicola, “Irradiance from an aperture with a truncated J0Bessel beam,” Opt. Commun. 80, 299–302 (1991).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Proc. IEEE Lett. (1)

Y. Mushiake, K. Matsumura, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE Lett. 60, 1107–1109 (1972).
[Crossref]

Other (4)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), Chap. 11, p. 585.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, New York, 1966), Sec. 3.8.3, p. 93.

Ref. 18, Sec. 3.6.1, p. 79.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 12, pp. 248–259.

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Figures (3)

Fig. 1
Fig. 1

Intensity profiles for an ABG beam (solid curve) and a Gaussian beam (dashed curve). The intensity of each beam has been normalized to 1 at its peak value. The parameters for the ABG beam are λ = 632.8 nm, w0 = 1 mm, and θ = 1 mrad; for the Gaussian beam, w0 = 1.5 mm. The 1/e2 width of each beam is 1.5 mm.

Fig. 2
Fig. 2

(a) Intensity profiles for the ABG (solid curve) and Gaussian (dashed curve) beams shown in Fig. 1 at the focus of a thin, nonaperturing f/1 lens (z = f = 3.0 mm). The beams have been normalized to reach peak values of 1. The 1/e2 width of the focused ABG beam is 0.9 μm, and that of the Gaussian is 0.4 μm. (b) Beam width, in micrometers, as a function of distance from the lens.

Fig. 3
Fig. 3

(a) Representative intensity profiles for the beams shown in Fig. 1 taken 1 cm after a 25-μm-diameter pinhole aperture. The ABG beam profile is shown as a solid curve; the Gaussian, a dashed curve. The intensity of each beam has been normalized to 1 at its peak. (b) Beam width, measured in millimeters, as a function of distance from the pinhole. The ABG beam diverges at an angle of 1.98° to the axis, and the Gaussian diverges at 1.19°.

Equations (24)

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E ( r , z ) = ϕ ^ E ϕ ( r , z ) = ϕ ^ f ( r , z ) exp ( i k z ) ,
1 r r ( r E ϕ r ) + 2 f z 2 exp ( i k z ) + 2 i k f z exp ( i k z ) - E ϕ r 2 = 0.
1 r r ( r f r ) - f r 2 + 2 i k f z = 0.
E ( x , y , z ) = - i exp ( i k z ) λ z - E ( x , y , 0 ) × exp { i π λ z [ ( x 2 - x 2 ) + ( y 2 - y 2 ) ] } d x d y ,
0 J v ( α t ) J v ( β t ) exp ( - γ t 2 ) t d t = ( 1 / 2 γ ) exp [ - ( α 2 + β 2 ) / 4 γ ] I v ( α β / 2 γ )
E ( ρ , z ) = ϕ ^ G ( ρ , z ) exp [ i ( k z - ω t ) ] ,
G ( ρ , z ) = A J 1 ( β ρ 1 + i z / L ) g ( ρ , z ) Q ( z ) , g ( ρ , z ) = w 0 w ( z ) exp [ - i Φ ( z ) ] exp ( - ρ 2 / w 0 2 1 + i z / L ) , Q ( z ) = exp [ i β 2 z / ( 2 k ) 1 + i z / L ] .
E ϕ ( r , z ) = 2 π i 0 F ϕ ( α ) J 1 ( α r ) exp ( i ω c z ) α d α ,
F ϕ ( α ) = 1 2 π i 0 E ϕ ( r , 0 ) J 1 ( α r ) r d r .
E ϕ ( r , z ) 2 π i exp ( i k z ) 0 F ϕ ( α ) J 1 ( α r ) × exp [ - i ( α 2 / 2 k ) z ] α d α ,
E ϕ ( r , 0 ) = A J 1 ( β r ) exp ( - r 2 / w 0 2 ) ,
F ϕ ( α ) - A 2 π w 0 2 2 exp [ - ( α 2 + β 2 ) w 0 2 4 ] J 1 ( i α β w 0 2 2 ) .
E ( r , z ) = - ϕ ^ k z exp [ i k ( z + r 2 2 z ) ] 0 P ( r ) E ϕ ( r , 0 ) × J 1 ( k r r z ) exp ( i k r 2 2 z ) r d r .
J 0 ( a ) = 1 π 0 π exp ( i a cos θ ) d θ
E ( r , z ) = - i k z exp [ i k ( z + r 2 2 z ) ] 0 P ( r ) E ( r , 0 ) × J 0 ( k r r z ) exp ( i k r 2 2 z ) r d r .
E ϕ ( r , 0 ) = A J 1 ( β r ) exp ( - r 2 / w 0 2 ) ,
P ( r ) = exp ( - i k r 2 / 2 f ) ,
E ϕ ( r , z ) = - A k 2 F z exp [ i k ( z + r 2 2 z ) ] × exp [ 1 4 F ( β 2 + k 2 r 2 z 2 ) ] I 1 ( β k r 2 F z ) ,
F - 1 w 0 2 + i k 2 ( 1 z - 1 f ) .
0 J v ( α t ) exp ( - γ t 2 ) t v + 1 d t = α v ( 2 γ ) - v - 1 exp ( - α 2 / 4 γ ) ,
E ( r , z ) = A i k 2 F z exp [ i k ( z + k r 2 2 z ) ] exp ( k 2 r 2 4 F z 2 ) ,
P ( r ) = circ ( r / a ) = { 1 0 r a 0 r > a .
E ϕ ( r , z ) = - A k z exp [ i k ( z + r 2 2 z ) ] 0 a J 1 ( β r ) J 1 ( k r r z ) × exp [ ( i k 2 z - 1 w 0 2 ) r 2 ] r d r ,
E ( r , z ) = - A i k z exp [ i k ( z + r 2 2 z ) ] 0 a J 0 ( k r r z ) × exp [ ( i k 2 z - 1 w 0 2 ) r 2 ] r d r .

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