Abstract

As other semiconductor lasers, concentric-circle-grating, surface-emitting lasers are compact, light, and efficient. However, unlike other semiconductor lasers, they emit high-power, low-divergence azimuthally polarized J 1 Bessel–Gaussian beams. Because of their azimuthal polarization, they have a null at the center of the beam that makes them undesirable for certain applications. Binary phase compensation, a lossless technique previously used to improve the far-field profile of linearly polarized Hermite–Gaussian beams, is adapted to these azimuthally polarized beams to rid them of their axial nulls and improve their beam profile.

© 1998 Optical Society of America

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References

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  1. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  2. N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
    [CrossRef]
  3. D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
    [CrossRef]
  4. N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
    [CrossRef]
  5. S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
    [CrossRef]
  6. T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [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. R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
    [CrossRef]
  9. L. W. Casperson, “Phase compensation of laser beam modes,” Opt. Quantum Electron. 8, 537–544 (1976).
    [CrossRef]
  10. L. W. Casperson, N. K. Kinchloe, O. M. Stafsudd, “Phase plates for laser beam compensation,” Opt. Commun. 21, 1–4 (1977).
    [CrossRef]
  11. L. W. Casperson, “How phase plates transform and control laser beams,” Laser Focus World 30 (5), 223–228 (1994).
  12. 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.
  13. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), pp. 355–494.

1997

1994

L. W. Casperson, “How phase plates transform and control laser beams,” Laser Focus World 30 (5), 223–228 (1994).

1992

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]

1990

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

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

1989

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

1980

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

1977

N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
[CrossRef]

L. W. Casperson, N. K. Kinchloe, O. M. Stafsudd, “Phase plates for laser beam compensation,” Opt. Commun. 21, 1–4 (1977).
[CrossRef]

1976

L. W. Casperson, “Phase compensation of laser beam modes,” Opt. Quantum Electron. 8, 537–544 (1976).
[CrossRef]

1972

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Alexopoulos, N. G.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
[CrossRef]

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]

Bour, D. P.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Carlson, N. W.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Casperson, L. W.

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.

L. W. Casperson, “How phase plates transform and control laser beams,” Laser Focus World 30 (5), 223–228 (1994).

L. W. Casperson, N. K. Kinchloe, O. M. Stafsudd, “Phase plates for laser beam compensation,” Opt. Commun. 21, 1–4 (1977).
[CrossRef]

L. W. Casperson, “Phase compensation of laser beam modes,” Opt. Quantum Electron. 8, 537–544 (1976).
[CrossRef]

Cordero-Iannarella, R. F.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

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]

Evans, G. A.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Hall, D. G.

R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
[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]

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

Hardy, A.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Jordan, R. H.

Kerner, S. R.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
[CrossRef]

Kinchloe, N. K.

L. W. Casperson, N. K. Kinchloe, O. M. Stafsudd, “Phase plates for laser beam compensation,” Opt. Commun. 21, 1–4 (1977).
[CrossRef]

King, O.

R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
[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]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Liew, S. K.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Parke, R.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Rishton, S.

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]

Scifres, D. R.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Stafsudd, O. M.

L. W. Casperson, N. K. Kinchloe, O. M. Stafsudd, “Phase plates for laser beam compensation,” Opt. Commun. 21, 1–4 (1977).
[CrossRef]

Streifer, W.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Tovar, A. A.

Welch, D. F.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Wicks, 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]

Appl. Phys. Lett.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[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]

IEEE Trans. Microwave Theory Tech.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

J. Appl. Phys.

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

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Laser Focus World

L. W. Casperson, “How phase plates transform and control laser beams,” Laser Focus World 30 (5), 223–228 (1994).

Opt. Commun.

L. W. Casperson, N. K. Kinchloe, O. M. Stafsudd, “Phase plates for laser beam compensation,” Opt. Commun. 21, 1–4 (1977).
[CrossRef]

Opt. Quantum Electron.

L. W. Casperson, “Phase compensation of laser beam modes,” Opt. Quantum Electron. 8, 537–544 (1976).
[CrossRef]

Other

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), pp. 355–494.

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Figures (3)

Fig. 1
Fig. 1

Polarization vector points azimuthally at each point over the transverse cross section of the beam. The polarization vector direction at the center of the beam is undefined, implying an axial null.

Fig. 2
Fig. 2

Far-field intensity profiles of several different J1 Bessel–Gaussian beams with and without phase compensation: w, width of the Gaussian portion of the beam; W, width of the Bessel portion of the beam.

Fig. 3
Fig. 3

Sidelobe of the Bessel–Gaussian beam represents an additional phase discontinuity that must also be compensated in beams with significant power in their sidelobes.

Equations (38)

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E ¯ out x ,   y = E out , x x ,   y i ¯ x + E out , y x ,   y i ¯ y ,
E out , x x ,   y = exp - i β 0 z - -   T x 0 ,   y 0 × K x 0 ,   y 0 ,   x ,   y E in , x x 0 ,   y 0 d x 0 d y 0 ,
E out , y x ,   y = exp - i β 0 z - -   T x 0 ,   y 0 × K x 0 ,   y 0 ,   x ,   y E in , y x 0 ,   y 0 d x 0 d y 0 .
K x 0 ,   y 0 ,   x ,   y = i λ m B x 1 / 2 B y 1 / 2 × exp - i   π λ m D x x 2 - 2 x 0 x + A x x 0 2 B x + D y y 2 - 2 y 0 y + A y y 0 2 B y .
I out x ,   y = c ε 0 n 0 2 E out , x * x ,   y E out , x x ,   y + E out , y * x ,   y E out , y x ,   y ,
E ¯ out r ,   ϕ = E out , r r ,   ϕ i ¯ r + E out , ϕ r ,   ϕ i ¯ ϕ ,
E out , r r ,   ϕ = exp - i β 0 z 0 0 2 π   T r 0 ,   ϕ 0 K r 0 ,   ϕ 0 ,   r ,   ϕ × E in , r r 0 ,   ϕ 0 cos ϕ 0 - ϕ - E in , ϕ r 0 ,   ϕ 0 × sin ϕ 0 - ϕ r 0 d ϕ 0 d r 0 ,
E out , ϕ r ,   ϕ = exp - i β 0 z 0 0 2 π   T r 0 ,   ϕ 0 K r 0 ,   ϕ 0 ,   r ,   ϕ × E in , r r 0 ,   ϕ 0 sin ϕ 0 - ϕ + E in , ϕ r 0 ,   ϕ 0 × cos ϕ 0 - ϕ r 0 d ϕ 0 d r 0 .
K r 0 ,   ϕ 0 ,   r ,   ϕ = i λ m B   exp - i   π λ m × Dr 2 - 2 r 0 r   cos ϕ 0 - ϕ + Ar 0 2 B .
I out r ,   ϕ = c ε 0 n 0 2 E out , r * r ,   ϕ E out , r r ,   ϕ + E out , ϕ * r ,   ϕ E out , ϕ r ,   ϕ .
A B C D = 1 z 0 1 1 0 - 1 / f 1 ,
= 1 - z / f z - 1 / f 1 .
T r 0 ,   ϕ 0 = exp - i ϕ 0 .
E in , ϕ r 0 ,   ϕ 0 = E 0 exp - r 0 2 / w 2 J 1 r 0 / W .
E out , r r ,   ϕ = - i λ m f   E 0 exp - i β 0 f 0 0 2 π exp - i ϕ 0 × exp - i   π λ m r 2 - 2 r 0 r   cos ϕ 0 - ϕ f × exp - r 0 2 / w 2 J 1 r 0 / W sin ϕ 0 - ϕ × r 0 d ϕ 0 d r 0 ,
E out , ϕ r ,   ϕ = i λ m f   E 0 exp - i β 0 f 0 0 2 π exp - i ϕ 0 × exp - i   π λ m r 2 - 2 r 0 r   cos ϕ 0 - ϕ f × exp - r 0 2 / w 2 J 1 r 0 / W cos ϕ 0 - ϕ × r 0 d ϕ 0 d r 0 .
θ ϕ 0 - ϕ ,
E out , r r ,   ϕ = - i λ m f   E 0 exp - i β 0 f exp - i   π r 2 λ m f × exp - i ϕ 0   r 0 exp - r 0 w 2 J 1 r 0 W × 0 2 π exp - i θ exp i   π λ m 2 r 0 r   cos θ f × sin θ d θ d r 0 ,
E out , ϕ r ,   ϕ = i λ m f   E 0 exp - i β 0 f exp - i   π r 2 λ m f × exp - i ϕ 0   r 0 exp - r 0 w 2 J 1 r 0 W × 0 2 π exp - i θ exp i   π λ m 2 r 0 r   cos θ f × cos θ d θ d r 0 .
0 2 π exp ix   cos θ d θ = 2 π J 0 x ,
0 2 π sin 2 θ exp ix   cos θ d θ = 0 ,
0 2 π cos 2 θ exp ix   cos θ d θ = - 2 π J 2 x ,
2 nx - 1 J n x = J n - 1 x + J n + 1 x ,
2 J n x = J n - 1 x - J n + 1 x ,
E out , r r ,   ϕ = i   E 0 r exp - i β 0 f exp - i   π r 2 λ m f exp - i ϕ × 0 exp - r 0 w 2 J 1 r 0 W J 1 2 π r λ m f   r 0 d r 0 ,
E out , ϕ r ,   ϕ = i   E 0 r exp - i β 0 f exp - i   π r 2 λ m f exp - i ϕ × 0   r 0 exp - r 0 w 2 J 1 r 0 W × d J 1 2 π r λ m f   r 0 d r 0 d r 0 .
I out r = I 0 r 2 0 exp - r 0 w 2 J 1 r 0 W J 1 2 π r λ m f   r 0 d r 0 2 + 0   r 0 exp - r 0 w 2 J 1 r 0 W × d J 1 2 π r λ m f   r 0 d r 0 d r 0 2 ,
r = 2 π rw λ m - 1 f - 1 ,
r 0 = r 0 / w ,
I 0 = 4 I 0 w 4 π 2 λ m - 2 f - 2 .
I out r I 0 = 1 r 2 0 exp - r 0 2 J 1 ar 0 J 1 r r 0 d r 0 2 + 0   r 0   exp - r 0 2 J 1 ar 0 d J 1 r r 0 d r r 0 d r 0 2 ,
a w / W .
I out 0 I 0 = π a 2 / 128 exp - a 2 / 4 I 0 a 2 / 8 - I 1 a 2 / 8 2 .
T r 0 ,   ϕ 0 = exp - i ϕ 0 2 u 3.83171 × W - 1 .
E 2 r ,   ϕ = exp - i   2 π λ nz + L - z E in r ,   ϕ .
z = ϕ 2 π L ,
T ϕ = exp - i   2 π λ exp - i   n - 1 L λ   ϕ .
L = λ n - 1 .

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