Abstract

We describe a computation which models the coupling among five widely separated laser emitters in a single monolithic array due to a GRIN rod lens. Individual emitter modes are characterized as to nearfield rectangular beam waist sizes by using an effective index waveguide calculation. The differential equations describing ray propagation in a medium with nonuniform index of refraction are solved using a Runge-Kutta algorithm. A ray bundle originating at each emitter is traced through free space, the GRIN rod, and back to the array. The fraction of (1/e**2) power entering an emitter from the front then defines the coupling from the source diode to the target diode. Provisions for describing array distances, offsets, and tilts are included. The model was tested for an array with 6-μm stripes on 250-μm centers. An array position and orientation was found which coupled all five laser elements together. The computed coupling coefficients were commensurate with values required to fully couple the array elements. Previous experimental results are now interpreted as a result of injection locking facilitated by the rod lens as opposed to an external cavity configuration.

© 1989 Optical Society of America

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References

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  1. K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
    [CrossRef]
  2. K.-Y. Liou, F. Bosch, C. A. Burrus, “Graded-Index-Rod External-Coupled-Cavity Laser with Backface-Output-Monitor Stabilized Single-Frequency Operation,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington DC, 1985), paper TUP4.
  3. C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
    [CrossRef]
  4. D. L. Begley, D. Martin, B. Vivian, R. R. Rice, “Compact Linear Array External Cavity Laser,” Appl. Opt. 25, 3835–3837 (1986) andin Technical Digest, Annual Meeting Optical Society of America (Optical Society of America, Washington, DC, 1985), paper TUl1.
    [CrossRef] [PubMed]
  5. J. Buus, “The Effective Index Method and its Application to Semiconductor Lasers,” IEEE J. Quantum Electron QE-18, 1083–1089 (1982).
    [CrossRef]
  6. The waveguide mode program was written by J. L. Levy.
  7. T. Okoshi, Optical Fibers (Academic, New York, 1982).
  8. E. G. Rawson, D. R. Herriott, J. McKenna, “Analysis of Refractive Index Distributions in Cylindrical, Graded Index Glass Rods (GRIN Rods) Used as Image Relays,” Appl. Opt. 9, 753–759 (1970).
    [CrossRef] [PubMed]
  9. The ray-tracing program was written by R. Juhala.
  10. F. Mogensen, H. Olsen, G. Jacobsen, “Locking Conditions and Stability Properties for a Semiconductor with External Light Injection,” IEEE J. Quantum Electron. QE-21, 784–793 (1985).
    [CrossRef]

1986 (2)

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

D. L. Begley, D. Martin, B. Vivian, R. R. Rice, “Compact Linear Array External Cavity Laser,” Appl. Opt. 25, 3835–3837 (1986) andin Technical Digest, Annual Meeting Optical Society of America (Optical Society of America, Washington, DC, 1985), paper TUl1.
[CrossRef] [PubMed]

1985 (1)

F. Mogensen, H. Olsen, G. Jacobsen, “Locking Conditions and Stability Properties for a Semiconductor with External Light Injection,” IEEE J. Quantum Electron. QE-21, 784–793 (1985).
[CrossRef]

1984 (1)

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

1982 (1)

J. Buus, “The Effective Index Method and its Application to Semiconductor Lasers,” IEEE J. Quantum Electron QE-18, 1083–1089 (1982).
[CrossRef]

1970 (1)

Begley, D. L.

Besomi, P.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

Bosch, F.

K.-Y. Liou, F. Bosch, C. A. Burrus, “Graded-Index-Rod External-Coupled-Cavity Laser with Backface-Output-Monitor Stabilized Single-Frequency Operation,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington DC, 1985), paper TUP4.

Burnham, R. D.

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

Burrus, C. A.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

K.-Y. Liou, F. Bosch, C. A. Burrus, “Graded-Index-Rod External-Coupled-Cavity Laser with Backface-Output-Monitor Stabilized Single-Frequency Operation,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington DC, 1985), paper TUP4.

Buus, J.

J. Buus, “The Effective Index Method and its Application to Semiconductor Lasers,” IEEE J. Quantum Electron QE-18, 1083–1089 (1982).
[CrossRef]

Chang-Hasnain, C.

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

Dienes, A.

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

Granland, S. W.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

Herriott, D. R.

Jacobsen, G.

F. Mogensen, H. Olsen, G. Jacobsen, “Locking Conditions and Stability Properties for a Semiconductor with External Light Injection,” IEEE J. Quantum Electron. QE-21, 784–793 (1985).
[CrossRef]

Kaminow, I. P.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

Linke, R. A.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

Liou, K.-Y.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

K.-Y. Liou, F. Bosch, C. A. Burrus, “Graded-Index-Rod External-Coupled-Cavity Laser with Backface-Output-Monitor Stabilized Single-Frequency Operation,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington DC, 1985), paper TUP4.

Martin, D.

McKenna, J.

Mogensen, F.

F. Mogensen, H. Olsen, G. Jacobsen, “Locking Conditions and Stability Properties for a Semiconductor with External Light Injection,” IEEE J. Quantum Electron. QE-21, 784–793 (1985).
[CrossRef]

Okoshi, T.

T. Okoshi, Optical Fibers (Academic, New York, 1982).

Olsen, H.

F. Mogensen, H. Olsen, G. Jacobsen, “Locking Conditions and Stability Properties for a Semiconductor with External Light Injection,” IEEE J. Quantum Electron. QE-21, 784–793 (1985).
[CrossRef]

Rawson, E. G.

Rice, R. R.

Scifres, D. R.

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

Swan, C. B.

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

Vivian, B.

Welch, D. F.

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

Whinnery, J. R.

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

K.-Y. Liou, C. A. Burrus, R. A. Linke, I. P. Kaminow, S. W. Granland, C. B. Swan, P. Besomi, “Single-Longitudinal-Mode Stabilized Graded-Index-Rod External Coupled-Cavity Laser,” Appl. Phys. Lett. 45, 729–731 (1984).
[CrossRef]

C. Chang-Hasnain, D. F. Welch, D. R. Scifres, J. R. Whinnery, A. Dienes, R. D. Burnham, “Diffraction-limited Emission from a Diode Laser Array in an Apertured Graded-index Lens External Cavity,” Appl. Phys. Lett. 49, 614–616 (1986).
[CrossRef]

IEEE J. Quantum Electron (1)

J. Buus, “The Effective Index Method and its Application to Semiconductor Lasers,” IEEE J. Quantum Electron QE-18, 1083–1089 (1982).
[CrossRef]

IEEE J. Quantum Electron. (1)

F. Mogensen, H. Olsen, G. Jacobsen, “Locking Conditions and Stability Properties for a Semiconductor with External Light Injection,” IEEE J. Quantum Electron. QE-21, 784–793 (1985).
[CrossRef]

Other (4)

The waveguide mode program was written by J. L. Levy.

T. Okoshi, Optical Fibers (Academic, New York, 1982).

K.-Y. Liou, F. Bosch, C. A. Burrus, “Graded-Index-Rod External-Coupled-Cavity Laser with Backface-Output-Monitor Stabilized Single-Frequency Operation,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington DC, 1985), paper TUP4.

The ray-tracing program was written by R. Juhala.

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Figures (4)

Fig. 1
Fig. 1

Array-GRIN rod system configuration. Linear offset is parallel to the y-axis; left is in the -Y direction.

Fig. 2
Fig. 2

Mode profiles and electric field phases for individual diode emitters: (A) transverse (x-direction for untilted array); (B) lateral (y-direction for untilted array).

Fig. 3
Fig. 3

Case 2 array images as returned to the array emitting facet after traversal of the GRIN rod. Emitter sizes are 1 × 4.4 μm, not drawn to scale. Four diodes are entirely within a light spot.

Fig. 4
Fig. 4

Coupling scheme for Case 2.

Tables (1)

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Table I Coupling Coefficients for Case 2

Equations (6)

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n ( r ) = 1.602 ( 1 A 2 r 2 / 2 ) .
d d s [ n ( r ) d r d s ] = n ( r )
d 2 r d z 2 [ r d θ 2 d z ] = 1 2 sec Γ 2 n 2 ( r 0 ) d d r [ n 2 ( r ) ]
r 2 d θ d z = sec ( Γ 0 ) [ x cos ( β 0 ) y cos ( α 0 ) ] ,
P = 2 π A
σ = L P ,

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