Abstract

A new effective index optical model is presented for the analysis of lateral waveguiding effects in vertical-cavity surface-emitting lasers. In addition to providing a concise formalism for reducing the dimensionality of the Maxwell equations describing the lasing mode, this model also provides new insights into waveguiding phenomena in vertical-cavity lasers. In particular, it is shown that the effective index responsible for waveguiding is dependent only on lateral changes in the Fabry–Perot resonance frequency. This concept leads naturally to new design methods for these lasers that are expected to result in more efficient devices with superior modal characteristics.

© 1995 Optical Society of America

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  1. J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
    [CrossRef]
  2. D. L. Huffaker, J. Shin, D. G. Deppe, Electron Lett. 30, 1946 (1994).
    [CrossRef]
  3. K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
    [CrossRef]
  4. C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
    [CrossRef]
  5. K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
    [CrossRef]
  6. K. L. Lear, Sandia National Laboratories, Albuquerque, N.M. (personal communication, 1995).

1995 (2)

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
[CrossRef]

1994 (1)

D. L. Huffaker, J. Shin, D. G. Deppe, Electron Lett. 30, 1946 (1994).
[CrossRef]

1993 (2)

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

Choquette, K. D.

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
[CrossRef]

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

Deppe, D. G.

D. L. Huffaker, J. Shin, D. G. Deppe, Electron Lett. 30, 1946 (1994).
[CrossRef]

Figiel, J. J.

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

Geib, K. M.

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

Hasnain, G.

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

Huffaker, D. L.

D. L. Huffaker, J. Shin, D. G. Deppe, Electron Lett. 30, 1946 (1994).
[CrossRef]

Kilcoyne, S. P.

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

Lear, K. L.

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
[CrossRef]

K. L. Lear, Sandia National Laboratories, Albuquerque, N.M. (personal communication, 1995).

Li, G. S.

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

Lott, J. A.

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

Schneider, R. P.

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
[CrossRef]

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

Shin, J.

D. L. Huffaker, J. Shin, D. G. Deppe, Electron Lett. 30, 1946 (1994).
[CrossRef]

Wu, Y. A.

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

Appl. Phys. Lett. (2)

C. J. Chang-Hasnain, Y. A. Wu, G. S. Li, G. Hasnain, K. D. Choquette, Appl. Phys. Lett. 63, 1307 (1993).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1995).
[CrossRef]

Electron Lett. (1)

D. L. Huffaker, J. Shin, D. G. Deppe, Electron Lett. 30, 1946 (1994).
[CrossRef]

Electron. Lett. (2)

K. L. Lear, K. D. Choquette, R. P. Schneider, S. P. Kilcoyne, K. M. Geib, Electron. Lett. 31, 208 (1995).
[CrossRef]

J. A. Lott, R. P. Schneider, K. D. Choquette, S. P. Kilcoyne, J. J. Figiel, Electron. Lett. 29, 1693 (1993).
[CrossRef]

Other (1)

K. L. Lear, Sandia National Laboratories, Albuquerque, N.M. (personal communication, 1995).

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Figures (3)

Fig. 1
Fig. 1

Schematic illustrating the use of the effective index method to describe lateral waveguiding in a VCSEL. One-dimensional eigenvalue computations are performed in each structurally distinct region as shown, resulting in a radially dependent effective index profile.

Fig. 2
Fig. 2

Ray diagram illustrating the equivalency between changes in optical cavity length and effective index. Longitudinal propagation vectors must tilt as energy is diffracted into the extended cavity region in order to maintain the Fabry–Perot condition. The amount of tilt corresponds with the angle of total internal reflection if the effective index in extended region is given by relation (11).

Fig. 3
Fig. 3

Schematic of the structure of an index-guided device fabricated using selective oxidation of a single quarter-wave mirror layer. The oxide functions both as a means of current confinement and as an index guide. DBR’s, distributed Bragg reflectors.

Equations (11)

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2 E - ɛ c 2 2 E t 2 = 0 ,
E ( r , z , ϕ , t ) φ i ( z ) E ( r , ϕ , t ) exp ( - i ω 0 t ) .
φ i E + φ i t 2 E + ɛ k 0 2 φ i E + 2 i ɛ k 0 φ i E τ = 0 ,
φ i + k 0 2 ( 1 - ξ i ) ɛ i ( z ) φ i = 0
E τ = i 2 k 0 ɛ i ( t 2 + k 0 2 Δ ɛ eff ) E ,
ɛ i φ * ɛ i ( z ) φ d z
Δ ɛ eff = Δ ( n eff 2 ) = ξ i ɛ i .
ξ i ɛ i = Δ ( n eff 2 ) = ɛ i - 1 k 0 2 φ i 2 d z ,
1 k 0 2 φ i 2 d z = 1 k 0 2 ( k 0 ) 2 ɛ φ i 2 d z = k 0 2 k 0 2 ɛ i ,
Δ ( n eff 2 ) ɛ i [ 1 - ( k 0 ) 2 k 0 2 ] ,
Δ n eff n eff Δ λ 0 λ 0 .

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