Abstract

Previous analytical approximations for the far-field radiation patterns of a double-heterostructure laser have been restricted by TE-mode propagation in slab waveguides. A model that is applicable to both TE- and TM-mode propagation either in a slab waveguide or in a multiple-quantum-well structure is developed. Results of computations agree with the measured irradiance profiles of visible diode lasers of wavelengths from 635 to 670 nm. A process for deriving far-field expressions in terms of the diode-laser parameters listed in data books is suggested. Astigmatic aberration present in the wave front is determined, and the final result is expressed in a form convenient for diffraction analysis of truncated focused (or collimated) diode-laser beams.

© 1996 Optical Society of America

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  1. D. Botez, “Near and far-field analytical approximations for the fundamental mode in symmetric waveguide DH lasers,” RCA Rev. 39, 577–603 (1978).
  2. B. Mroziewicz, M. Bugajski, W. Nakwaski, Physics of Semiconductor Lasers (Elsevier, New York, 1991), Chap. 4.
  3. A. Naqwi, F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
  4. Y. Li, “Focusing of diode laser beams: a simple mathematical model: comments,” Appl. Opt. 31, 3392–3393 (1992).
  5. A. Naqwi, “Focusing of diode laser beams: a simple mathematical model: reply to comment,” Appl. Opt. 31, 3394 (1992).
  6. X. Zeng, A. Naqwi, “Far-field distribution of double-heterostructure diode laser beams,” Appl. Opt. 32, 4491–4494 (1993).
  7. S. Nemoto, “Experimental evaluation of a new expression for the far field of a diode laser beam,” Appl. Opt. 33, 6387–6392 (1994).
  8. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active region,” IEEE J. Quantum Electron. QE-11, 400–402 (1975).
  9. K. Tatsuno, A. Arimoto, “Measurement and analysis of diode laser wave front,” Appl. Opt. 20, 3520–3525 (1981).
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 9.
  11. W. Streifer, D. R. Scifres, D. Burnham, “Optical analysis of multiple-quantum-well lasers,” Appl. Opt. 18, 3547–3548 (1979).
  12. A. Harada, K. Kishino, “674 nm wavelength planar-buried-heterostructure GaInAsP/AlGaAs visible laser diode grown on GaAs by LPE,” Trans. Inst. Electron. Inf. Commun. Eng. 70, 17–19 (1987).
  13. K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).
  14. Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).
  15. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 13.
  16. D. D. Cook, F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1672 (1975).
  17. J. F. Lotspeich, “Explicit general eigenvalue solutions for dielectric slab waveguides,” Appl. Opt. 14, 327–335 (1975);Y. Li, “Method of successive approximations for calculating the eigenvalues of optical thin film waveguides,” Appl. Opt. 20, 2595–2596 (1981).
  18. P. A. Kirkby, G. H. B. Thompson, “The effect of double heterojunction waveguide parameters on the far field emission patterns of lasers,” Optoelectronics 4, 323–334 (1972).
  19. H. C. Casey, M. B. Panish, J. L Merz, “Beam divergence of the emission from double-heterostructure injection lasers,” J. Appl. Phys. 44, 5470–5475 (1973).
  20. M. B. Panish, “Heterostructure injunction lasers,” Proc. IEEE 64, 1512–1540 (1976).
  21. Toshiba Optoelectronic Semiconductor Laser Diode Product Guide (Toshiba America Electronic Components, Inc., Irvine, Calif., 1994), pp. 10–13.
  22. Sanyo Red Laser Diode DL-3038-033 data sheet (Sanyo Electric Co., Ltd., Semiconductor business Headquarters, Mountain View, Calif., 1995), pp. 12–13.
  23. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
  24. R. Simon, “Anisotropic Gaussian beams,” Opt. Commun. 46, 265–268 (1983).

1994

1993

1992

1990

A. Naqwi, F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).

Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).

1987

A. Harada, K. Kishino, “674 nm wavelength planar-buried-heterostructure GaInAsP/AlGaAs visible laser diode grown on GaAs by LPE,” Trans. Inst. Electron. Inf. Commun. Eng. 70, 17–19 (1987).

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

1983

R. Simon, “Anisotropic Gaussian beams,” Opt. Commun. 46, 265–268 (1983).

1981

1979

1978

D. Botez, “Near and far-field analytical approximations for the fundamental mode in symmetric waveguide DH lasers,” RCA Rev. 39, 577–603 (1978).

1976

M. B. Panish, “Heterostructure injunction lasers,” Proc. IEEE 64, 1512–1540 (1976).

1975

D. D. Cook, F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1672 (1975).

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active region,” IEEE J. Quantum Electron. QE-11, 400–402 (1975).

J. F. Lotspeich, “Explicit general eigenvalue solutions for dielectric slab waveguides,” Appl. Opt. 14, 327–335 (1975);Y. Li, “Method of successive approximations for calculating the eigenvalues of optical thin film waveguides,” Appl. Opt. 20, 2595–2596 (1981).

1973

H. C. Casey, M. B. Panish, J. L Merz, “Beam divergence of the emission from double-heterostructure injection lasers,” J. Appl. Phys. 44, 5470–5475 (1973).

1972

P. A. Kirkby, G. H. B. Thompson, “The effect of double heterojunction waveguide parameters on the far field emission patterns of lasers,” Optoelectronics 4, 323–334 (1972).

W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).

Arimoto, A.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 9.

Botez, D.

D. Botez, “Near and far-field analytical approximations for the fundamental mode in symmetric waveguide DH lasers,” RCA Rev. 39, 577–603 (1978).

Bugajski, M.

B. Mroziewicz, M. Bugajski, W. Nakwaski, Physics of Semiconductor Lasers (Elsevier, New York, 1991), Chap. 4.

Burnham, D.

Carter, W. H.

Casey, H. C.

H. C. Casey, M. B. Panish, J. L Merz, “Beam divergence of the emission from double-heterostructure injection lasers,” J. Appl. Phys. 44, 5470–5475 (1973).

Cook, D. D.

D. D. Cook, F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1672 (1975).

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active region,” IEEE J. Quantum Electron. QE-11, 400–402 (1975).

Durst, F.

Feit, Z.

Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).

Gomyo, A.

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

Harada, A.

A. Harada, K. Kishino, “674 nm wavelength planar-buried-heterostructure GaInAsP/AlGaAs visible laser diode grown on GaAs by LPE,” Trans. Inst. Electron. Inf. Commun. Eng. 70, 17–19 (1987).

Hino, I.

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

Kawata, S.

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

Kirkby, P. A.

P. A. Kirkby, G. H. B. Thompson, “The effect of double heterojunction waveguide parameters on the far field emission patterns of lasers,” Optoelectronics 4, 323–334 (1972).

Kishino, K.

A. Harada, K. Kishino, “674 nm wavelength planar-buried-heterostructure GaInAsP/AlGaAs visible laser diode grown on GaAs by LPE,” Trans. Inst. Electron. Inf. Commun. Eng. 70, 17–19 (1987).

Kobayashi, K.

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

Kostyk, D.

Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).

Li, Y.

Lotspeich, J. F.

Mak, P.

Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).

Merz, J. L

H. C. Casey, M. B. Panish, J. L Merz, “Beam divergence of the emission from double-heterostructure injection lasers,” J. Appl. Phys. 44, 5470–5475 (1973).

Mroziewicz, B.

B. Mroziewicz, M. Bugajski, W. Nakwaski, Physics of Semiconductor Lasers (Elsevier, New York, 1991), Chap. 4.

Nakwaski, W.

B. Mroziewicz, M. Bugajski, W. Nakwaski, Physics of Semiconductor Lasers (Elsevier, New York, 1991), Chap. 4.

Naqwi, A.

Nash, F. R.

D. D. Cook, F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1672 (1975).

Nemoto, S.

Panish, M. B.

M. B. Panish, “Heterostructure injunction lasers,” Proc. IEEE 64, 1512–1540 (1976).

H. C. Casey, M. B. Panish, J. L Merz, “Beam divergence of the emission from double-heterostructure injection lasers,” J. Appl. Phys. 44, 5470–5475 (1973).

Scifres, D. R.

Simon, R.

R. Simon, “Anisotropic Gaussian beams,” Opt. Commun. 46, 265–268 (1983).

Streifer, W.

Suzuki, T.

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

Tatsuno, K.

Thompson, G. H. B.

P. A. Kirkby, G. H. B. Thompson, “The effect of double heterojunction waveguide parameters on the far field emission patterns of lasers,” Optoelectronics 4, 323–334 (1972).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 9.

Woods, R. J.

Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 13.

Zeng, X.

Appl. Opt.

IEEE J. Quantum Electron.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active region,” IEEE J. Quantum Electron. QE-11, 400–402 (1975).

K. Kobayashi, I. Hino, A. Gomyo, S. Kawata, T. Suzuki, “AlGaInP double heterostructure visible-light laser diodes with a GaInP active layer grown by metal-organic vapor phase epitaxy,” IEEE J. Quantum Electron. QE-23, 704–711 (1987).

IEEE Photon. Technol. Lett.

Z. Feit, D. Kostyk, R. J. Woods, P. Mak, “Molecular beam epitaxy-grown PbSnTe-PbEuSeTe buried heterostructure diode lasers,” IEEE Photon. Technol. Lett. 2, 860–868 (1990).

J. Appl. Phys.

D. D. Cook, F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1672 (1975).

H. C. Casey, M. B. Panish, J. L Merz, “Beam divergence of the emission from double-heterostructure injection lasers,” J. Appl. Phys. 44, 5470–5475 (1973).

J. Opt. Soc. Am.

Opt. Commun.

R. Simon, “Anisotropic Gaussian beams,” Opt. Commun. 46, 265–268 (1983).

Optoelectronics

P. A. Kirkby, G. H. B. Thompson, “The effect of double heterojunction waveguide parameters on the far field emission patterns of lasers,” Optoelectronics 4, 323–334 (1972).

Proc. IEEE

M. B. Panish, “Heterostructure injunction lasers,” Proc. IEEE 64, 1512–1540 (1976).

RCA Rev.

D. Botez, “Near and far-field analytical approximations for the fundamental mode in symmetric waveguide DH lasers,” RCA Rev. 39, 577–603 (1978).

Trans. Inst. Electron. Inf. Commun. Eng.

A. Harada, K. Kishino, “674 nm wavelength planar-buried-heterostructure GaInAsP/AlGaAs visible laser diode grown on GaAs by LPE,” Trans. Inst. Electron. Inf. Commun. Eng. 70, 17–19 (1987).

Other

B. Mroziewicz, M. Bugajski, W. Nakwaski, Physics of Semiconductor Lasers (Elsevier, New York, 1991), Chap. 4.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 13.

Toshiba Optoelectronic Semiconductor Laser Diode Product Guide (Toshiba America Electronic Components, Inc., Irvine, Calif., 1994), pp. 10–13.

Sanyo Red Laser Diode DL-3038-033 data sheet (Sanyo Electric Co., Ltd., Semiconductor business Headquarters, Mountain View, Calif., 1995), pp. 12–13.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 9.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram showing a diode laser, the far-field emission pattern, and the cylindrical wave front propagating in the active layer.

Fig. 2
Fig. 2

Cross-sectional view of the fields and waves in the vicinity of the active region. (a) side view, (b) top view.

Fig. 3
Fig. 3

(a) Schematic representation of the source-field distributions on the η axis and (b) the corresponding Gaussian approximation.

Fig. 4
Fig. 4

Diagrams showing the intensity distributions (a) in the plane normal to the junction at beam-divergence angles θ = 30° and 50°, and (b) in the plane parallel to the junction at angles θ|| = 8° and 10°.

Fig. 5
Fig. 5

Comparison of the beam-divergence angle θ|| obtained from Eq. (29) and the prediction of paraxial Gaussian beam theory.

Fig. 6
Fig. 6

Comparison of the calculated (solid curves) and the measured irradiance profiles. The open circles represent experimental data on the irradiance profiles of a standard-type TE-mode IG diode laser (Toshiba TOLD9211) of wavelength λ = 670 nm. The plane of observation is 4 mm in front of the flat glass cap of the laser. (a) Isophotes (contours of the irradiance) of far-field distribution, (b) profile normal to the junction, (c) profiles parallel with the junction.

Fig. 7
Fig. 7

Comparison of the calculated and the measured irradiance profiles. The open circles represent experimental data of the irradiance profiles of a standard-type TE-mode GG diode laser (Toshiba TOLD9200) of wavelength λ = 670 nm. The plane of observation is 4 mm in front of the flat glass cap of the laser. (a) Profile normal to the junction, (b) profile a parallel with the junction.

Fig. 8
Fig. 8

Comparison of the calculated (solid curves) and the measured irradiance profiles. The open circles represent experimental data of the irradiance of a MQW TM-mode diode laser (Sanyo DL-3038-033) of wavelength λ = 635 nm. The plane of observation is 4 mm in front of the flat glass cap of the laser. (a) Profile normal to the junction, (b) profile parallel with the junction.

Fig. 9
Fig. 9

Schematic representation of the isophotes of actual devices. (a) Effect of beam truncation by the aperture of the capsule, (b) expansion of outer rings in the isophotes caused by high-order modes in the source field.

Tables (1)

Tables Icon

Table 1 Active-Layer Thickness of DH Diode Lasers of Different Wavelengths

Equations (45)

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U 0 ( Q ) = A 0 e η ( ξ ) e ξ ( η ) exp ( i k η 2 2 ρ A ) ,
k = 2 π / λ
e η ( ξ ) = cos ( q ξ ) for   | ξ | d / 2 , e η ( ξ ) = cos ( q ξ ) exp { p [ | ξ | ( d / 2 ) ] } for   |  ξ  | > d / 2.
e ξ ( η ) = exp ( η 2 w 1 2 ) ,
w 1 w 0 [ 1 + ( λΔ Z / π w 0 2 ) 2 ] 1 / 2 ,
e ξ ( η ) = cos  ( q   η ) for   | η | B / 2 , e ξ ( η ) = cos  ( q   η ) exp { p [ | η | ( B / 2 ) ] } for   | η | > B / 2.
κ p = q   tan ( q d / 2 ) .
( p d ) 2 + ( q d ) 2 = ( 2 π d λ ) 2 ( n 1 2 n 2 2 ) D 2 ,
p = k ( 2 π d λ ) n 1 2 n 2 2 2 κ ,    q = k ( n 1 2 n 2 2 ) 1 / 2 ,
U ( P ) = i λ S U 0 ( Q ) exp ( i k s ) s K ( χ ) d S .
r = O P ¯ = ( x 2 + y 2 + z 0 2 ) 1 / 2 ,
r e ^ ρ = r ( ξ  cos  α + η  cos  β ) ,
K ( χ ) =   cos  γ z 0 / r ,
U ( P ) = i A 0 λ ( z 0 r ) exp ( i k r ) r S U 0 ( Q )    ×   exp [ i k ( ξ  cos  α + η  cos  β ) ] dξdη .
U ( P ) = A ( z 0 r )   exp ( i k r i Φ ) r [ K T ( 1 + x 2 h 1 2 ) 1 ] exp ( y 2 w y 2 ) ,
Φ = k Δ Z 2 r y 2 , r = ( x 2 + y 2 + z 0 2 ) 1 / 2 ,
h 1 = ( p / k ) r ,
K T = { [ 1 + ( q p ) p 2 + ( k   cos  α ) 2 q 2 ( k   cos  α ) 2   tan   ( q d / 2 ) ]    ×   cos   [ ( k   cos  α ) d / 2 ] k   cos   α p p 2 + q 2 q 2 ( k   cos  α ) 2    × sin [ ( k   cos  α ) d / 2 ] } q q + p   tan ( q d / 2 ) ,
w y = ( λ / π w 0 ) r .
I ( P ) = U ( P ) U * ( P ) .
I ( P ) = I 0 { [ 1 + ( x 2 + y 2 ) ] / z 0 2 } 2 [ K T ( 1 + x 2 h 1 2 ) 1 ] 2   exp ( 2 y 2 w y 2 ) ,
I 0 = ( | A | / z 0 ) 2
1 [ 1 + ( x 2 + y 2 ) / z 0 2 ] 2 , [ K T ( 1 + x 2 h 1 2 ) 1 ] 2 ,  exp ( 2 y 2 w y 2 ) ,
I ( x , 0 ) = I 0 [ K T ( 1 + x 2 h x 2 ) 1 ] 2 ,
h x 2 = z 0 2 p 2 p 2 + k 2 .
K T = 1 ( π  sin  θ ) 2 6 [ 3 2 κ ( n 1 2 n 2 2 ) 1 / 2 ] ( d λ ) 2 + ,
K T 1.
δ = 33.3 [ 3 2 κ ( n 1 2 n 2 2 ) 1 / 2 ] ( d λ ) 2 ( % ) .
d λ 0.2.
I ( x , 0 ) = I 0 ( 1 + x 2 h x 2 ) 2 ,
tan ( θ / 2 ) = [ ( 2 1 ) p 2 p 2 + k 2 ] 1 / 2 .
θ = 2   arctan { D 2 κ [ 2 1 1 + ( D / 2 κ ) 2 ] 1 / 2 } .
θ = 2   arctan 0.644 D [ D 2 + 4 / ( n 1 2 n 2 2 ) ] 1 / 2 .
I ( 0 , y ) = I 0 ( 1 + y 2 / z 0 2 ) 2   exp ( 2 y 2 w y 2 ) .
  exp [ ( π w 0 / λ ) 2 sin 2 ( θ / 2 ) ] = 2   cos 2 ( θ / 2 ) .
θ = 2   arctan  ( 0.1874 λ / w 0 ) .
r = ( x 2 + y 2 + z 0 2 ) 1 / 2 z 0 + x 2 + y 2 2 z 0 .
exp ( i k r i Φ ) exp ( i k R x ) exp [ i k ( x 2 2 R x + y 2 2 R y ) ] ,
R x z 0 ,    R y z 0 + Δ Z ,
U ( P ) = A U 1 ( x ) U 2 ( y ) ,
U 1 ( x ) = ( 1 + x 2 h 2 2 ) 1   exp ( i k x 2 2 R x ) .
U 2 ( y ) =   exp ( y 2 w y 2 + i k y 2 2 R y ) ,
h 1 = r sin ( θ / 2 ) [ 2   cos 2 ( θ / 2 ) 1 ] 1 / 2 , h x = 1.55 z 0 tan ( θ ) .
w y = 1.7 r    tan ( θ / 2 ) .
2 cos 2 ( θ / 2 ) 1 > 0 , θ < 2   arccos  ( 1 / 2 4 ) = 65.5 ° ,

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