Abstract

A simplified mathematical model for the far field of a monomode diode laser is employed for easy but fairly accurate computations of the optical field in the focal region. The present treatment is concerned with laser junctions significantly narrower than the wavelength. The field distribution in the plane perpendicular to the diode junction is considered in detail. The results of computations are shown to agree well with the measurements. Hence, the computational code is valuable for the designing of optical devices, such as diode–fiber couplings and laser Doppler anemometers. The present work is not concerned with design calculations for specific applications. Instead, it is intended to illustrate the general features of the proposed mathematical model of monomode diode laser beams.

© 1990 Optical Society of America

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References

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  1. H. C. Casey, M. B. Panish, Heterostructure Lasers (Academic, New York, 1978).
  2. M. Saruwatari, K. Nawata, “Semiconductor Laser to Single-Mode Fiber Coupler,” Appl. Opt. 18, 1847–1856 (1979).
    [CrossRef] [PubMed]
  3. M. Sumida, K. Takemoto, “Lens Coupling of Laser Diodes to Single-Mode Fibers,” IEEE IOSA J. Lightwave Tech. LT-2, 305–311 (1984).
    [CrossRef]
  4. W. B. Joyce, B. C. DeLoach, “Alignment of Gaussian Beams,” Appl. Opt. 23, 4187–4196 (1984).
    [CrossRef] [PubMed]
  5. S. N. Sarkar, B. P. Pal, K. Thyagrajan, “Lens Coupling of Laser Diodes to Monomode Elliptic Core Fibers,” J. Opt. Commun. 7, 92–96 (1986).
    [CrossRef]
  6. H. Karstensen, “Laser Diode to Single-Mode Fiber Coupling with Ball Lenses,” J. Opt. Commun. 9, 42–49 (1988).
    [CrossRef]
  7. D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660–1672 (1975).
    [CrossRef]
  8. W. P. Dumke, “The Angular Beam Divergence in Double-Heterojunction Lasers with Very Thin Active Regions,” IEEE J. Quantum Electron. QE-11 7, 400–402 (1975).
    [CrossRef]
  9. S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.
  10. T. L. Paoli, “Waveguiding in a Stripe-Geometry Junction Laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
    [CrossRef]
  11. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).
  13. R. Muller, A. Naqwi, Optimization of a Laser Diode Anemometry System (U. Erlangen, 1988, Report LSTM-239).
  14. H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  15. S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658–662 (1983).
    [CrossRef] [PubMed]
  16. E. Wolf, Y. Li, “Conditions for the Validity of the Debye Integral Representation of Focused Fields,” Opt. Commun. 39, 205–210 (Oct.1981).
    [CrossRef]
  17. Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211–215 (Oct.1981).
    [CrossRef]
  18. J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” Opt. Commun. 40, 81–85 (Dec.1981).
    [CrossRef]
  19. M. P. Givens, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 41, 145 (Apr.1981).
    [CrossRef]
  20. Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151–156 (Dec.1982).
    [CrossRef]

1988 (1)

H. Karstensen, “Laser Diode to Single-Mode Fiber Coupling with Ball Lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

1986 (1)

S. N. Sarkar, B. P. Pal, K. Thyagrajan, “Lens Coupling of Laser Diodes to Monomode Elliptic Core Fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

1984 (2)

M. Sumida, K. Takemoto, “Lens Coupling of Laser Diodes to Single-Mode Fibers,” IEEE IOSA J. Lightwave Tech. LT-2, 305–311 (1984).
[CrossRef]

W. B. Joyce, B. C. DeLoach, “Alignment of Gaussian Beams,” Appl. Opt. 23, 4187–4196 (1984).
[CrossRef] [PubMed]

1983 (1)

1982 (1)

Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151–156 (Dec.1982).
[CrossRef]

1981 (4)

E. Wolf, Y. Li, “Conditions for the Validity of the Debye Integral Representation of Focused Fields,” Opt. Commun. 39, 205–210 (Oct.1981).
[CrossRef]

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211–215 (Oct.1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” Opt. Commun. 40, 81–85 (Dec.1981).
[CrossRef]

M. P. Givens, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 41, 145 (Apr.1981).
[CrossRef]

1979 (1)

1977 (1)

T. L. Paoli, “Waveguiding in a Stripe-Geometry Junction Laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
[CrossRef]

1975 (2)

D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660–1672 (1975).
[CrossRef]

W. P. Dumke, “The Angular Beam Divergence in Double-Heterojunction Lasers with Very Thin Active Regions,” IEEE J. Quantum Electron. QE-11 7, 400–402 (1975).
[CrossRef]

1965 (1)

Bopp, S.

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

Casey, H. C.

H. C. Casey, M. B. Panish, Heterostructure Lasers (Academic, New York, 1978).

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).
[CrossRef]

DeLoach, B. C.

Dumke, W. P.

W. P. Dumke, “The Angular Beam Divergence in Double-Heterojunction Lasers with Very Thin Active Regions,” IEEE J. Quantum Electron. QE-11 7, 400–402 (1975).
[CrossRef]

Durst, F.

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

Givens, M. P.

M. P. Givens, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 41, 145 (Apr.1981).
[CrossRef]

Joyce, W. B.

Karstensen, H.

H. Karstensen, “Laser Diode to Single-Mode Fiber Coupling with Ball Lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

Kogelnik, H.

Li, Y.

Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151–156 (Dec.1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the Validity of the Debye Integral Representation of Focused Fields,” Opt. Commun. 39, 205–210 (Oct.1981).
[CrossRef]

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211–215 (Oct.1981).
[CrossRef]

Muller, R.

R. Muller, A. Naqwi, Optimization of a Laser Diode Anemometry System (U. Erlangen, 1988, Report LSTM-239).

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

Naqwi, A.

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

R. Muller, A. Naqwi, Optimization of a Laser Diode Anemometry System (U. Erlangen, 1988, Report LSTM-239).

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).
[CrossRef]

Nawata, K.

Pal, B. P.

S. N. Sarkar, B. P. Pal, K. Thyagrajan, “Lens Coupling of Laser Diodes to Monomode Elliptic Core Fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

Panish, M. B.

H. C. Casey, M. B. Panish, Heterostructure Lasers (Academic, New York, 1978).

Paoli, T. L.

T. L. Paoli, “Waveguiding in a Stripe-Geometry Junction Laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
[CrossRef]

Sarkar, S. N.

S. N. Sarkar, B. P. Pal, K. Thyagrajan, “Lens Coupling of Laser Diodes to Monomode Elliptic Core Fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

Saruwatari, M.

Self, S. A.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” Opt. Commun. 40, 81–85 (Dec.1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” Opt. Commun. 40, 81–85 (Dec.1981).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

Sumida, M.

M. Sumida, K. Takemoto, “Lens Coupling of Laser Diodes to Single-Mode Fibers,” IEEE IOSA J. Lightwave Tech. LT-2, 305–311 (1984).
[CrossRef]

Takemoto, K.

M. Sumida, K. Takemoto, “Lens Coupling of Laser Diodes to Single-Mode Fibers,” IEEE IOSA J. Lightwave Tech. LT-2, 305–311 (1984).
[CrossRef]

Thyagrajan, K.

S. N. Sarkar, B. P. Pal, K. Thyagrajan, “Lens Coupling of Laser Diodes to Monomode Elliptic Core Fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

Tropea, C.

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

Weber, H.

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

Wolf, E.

Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151–156 (Dec.1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the Validity of the Debye Integral Representation of Focused Fields,” Opt. Commun. 39, 205–210 (Oct.1981).
[CrossRef]

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211–215 (Oct.1981).
[CrossRef]

Appl. Opt. (4)

IEEE IOSA J. Lightwave Tech. (1)

M. Sumida, K. Takemoto, “Lens Coupling of Laser Diodes to Single-Mode Fibers,” IEEE IOSA J. Lightwave Tech. LT-2, 305–311 (1984).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. P. Dumke, “The Angular Beam Divergence in Double-Heterojunction Lasers with Very Thin Active Regions,” IEEE J. Quantum Electron. QE-11 7, 400–402 (1975).
[CrossRef]

T. L. Paoli, “Waveguiding in a Stripe-Geometry Junction Laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
[CrossRef]

J. Appl. Phys. (1)

D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660–1672 (1975).
[CrossRef]

J. Opt. Commun. (2)

S. N. Sarkar, B. P. Pal, K. Thyagrajan, “Lens Coupling of Laser Diodes to Monomode Elliptic Core Fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

H. Karstensen, “Laser Diode to Single-Mode Fiber Coupling with Ball Lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

Opt. Commun. (5)

E. Wolf, Y. Li, “Conditions for the Validity of the Debye Integral Representation of Focused Fields,” Opt. Commun. 39, 205–210 (Oct.1981).
[CrossRef]

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211–215 (Oct.1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” Opt. Commun. 40, 81–85 (Dec.1981).
[CrossRef]

M. P. Givens, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 41, 145 (Apr.1981).
[CrossRef]

Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151–156 (Dec.1982).
[CrossRef]

Other (5)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

R. Muller, A. Naqwi, Optimization of a Laser Diode Anemometry System (U. Erlangen, 1988, Report LSTM-239).

H. C. Casey, M. B. Panish, Heterostructure Lasers (Academic, New York, 1978).

S. Bopp, F. Durst, R. Muller, A. Naqwi, C. Tropea, H. Weber, “Small Laser-Doppler Anemometers Using Semi-Conductor Laser and Avalanche-Photo Diodes,” in Proceedings, Fourth International Symposium on Application of Laser Anemometry to Fluid Mechanics, 6.4.

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Figures (5)

Fig. 1
Fig. 1

Typical laser diode junction and the diverging laser beam.

Fig. 2
Fig. 2

Optical arrangements for the focusing of a diode laser beam.

Fig. 3
Fig. 3

Intensity profile of a collimated beam: comparison between measurement and the truncated Lorentzian model. (Laser diode: Sharp LT024; Collimator: Melles Griot 06GLCO02, 8-mm focal length, 0.5 numerical aperture.)

Fig. 4
Fig. 4

Intensity distribution normal to the junction for a focusing beam of laser diode: comparison between measurement and computation.

Fig. 5
Fig. 5

Wavefront distortions: comparison between a truncated Lorentzian beam a Gaussian beam, and a top-hat beam. (Laser diode: Hitachi HL8314; First lens: 7-mm focal length, 0.35 numerical aperture; Second lens: 100-mm focal length.)

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

u 0 ( x 0 , y 0 ) = u 0 x ( x 0 ) u 0 y ( y 0 ) ,
u 0 ( y 0 ) = exp [ - ( y 0 w 0 ) 2 - i k 2 y 0 2 R 0 y ] ,
u 0 x ( x 0 ) = { γ 2 γ 2 + x 0 2 exp ( - i k 2 x 0 2 R 0 x ) if x 0 l 0 otherwise .
exp [ - i k 2 f ( x 2 + y 2 ) ] .
i ( k x 0 2 2 R 0 x ) 2 ,
u ( x , y , z ) = exp ( - i k z ) u x ( x , z ) u y ( y , z ) .
u y ( y , z ) = i λ z - u 0 y ( y 0 ) exp [ - i k 2 z ( y - y 0 ) 2 ] d y 0 .
u y ( y , z ) = i λ z - exp { - ( y 0 w 0 ) 2 - i k 2 [ y 0 2 R 0 y + ( y - y 0 ) 2 z ] } d y 0 .
u y ( y , z ) = w 0 w exp { - ( y w ) 2 + i [ ϕ A y - k 2 ( y 2 R y ) ] } ,
w = ( 2 R 0 y z ) 2 + [ k w 0 2 ( R 0 y + z ) ] 2 k w 0 R 0 y ,
1 R y = 1 w 2 [ ( w 0 2 R 0 y 2 + 4 ( k w 0 ) 2 ) z + w 0 2 R 0 y ] ,
ϕ A y = 1 / 2 ϕ A y ,
sin ϕ A y ~ 2 R 0 y z , cos ϕ A y ~ k w 0 2 ( R 0 y + z ) ,
u x ( x , z ) = i λ z - l + l γ 2 γ 2 + x 0 2 exp { - i k 2 [ x 0 2 R 0 x + ( x - x 0 ) 2 z ] } d x 0 .
u x ( x , z ) = ( 1 + i ) 2 α λ z exp [ - i k 2 ( x 2 R 0 x + z ) ] × { G if z < - R 0 x , G * if z > - R 0 x ,
G ( δ 1 , δ 2 ) = δ 1 δ 2 γ 2 γ 2 + ( x ^ α + β ) 2 exp ( - i x ^ 2 2 ) d x ^ ,
α = k | z + R 0 x R 0 x z | ,
β = | R 0 x x R 0 x + z | .
G ( δ , ; β ) = α γ 2 { [ P ( δ ) sin ( δ 2 2 ) + Q ( δ ) cos ( δ 2 2 ) ] + i [ P ( δ ) cos ( δ 2 2 ) - Q ( δ ) sin ( δ 2 2 ) ] } ,
P ( δ ) = - a 1 ( δ ) + a 3 ( δ ) - , Q ( δ ) = a 2 ( δ ) - a 4 ( δ ) + .
a 1 = A x ^ | x ^ = δ , a 2 = | 1 x ^ d d x ^ ( A x ^ ) | x ^ = δ , etc . ,
A ( x ^ ) = 1 α γ 2 + ( x ^ + α β ) 2 .

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