Abstract

Two-dimensional reflectivity of transverse-magnetic-like modes (Hx = 0) is considered. The modes are derived by means of the effective-index approximation. Mathematical expressions for the reflectivity and conversion coefficients are similar in form to those of transverse-electric-like modes, although the various functions and parameters in the equations are defined somewhat differently.

© 1984 Optical Society of America

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References

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  1. A. Hardy, “Formulation of two-dimensional reflectivity calculations based on the effective-index method,” J. Opt. Soc. Am. A 1, 550–555 (1984).
    [CrossRef]
  2. G. B. Hocker, W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977), and references therein.
    [CrossRef] [PubMed]
  3. J. Buus, “The effective index method and its application to semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 1083–1089 (1982), and references therein.
    [CrossRef]
  4. W. Streifer, E. Kapon, “Application of the equivalent-index method to DH diode lasers,” Appl. Opt. 18, 3724–3725 (1979).
    [PubMed]
  5. W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of gain-induced waveguiding in stripe geometry diode lasers,” IEEE J. Quantum Electron. QE-14, 418–427 (1978).
    [CrossRef]
  6. E. Kapon, A. Hardy, A. Zussman, “Application of perturbation theory to the analysis of diode lasers with lateral optical confinement,” IEEE J. Quantum Electron. (to be published).
  7. T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
    [CrossRef]
  8. L. Lewin, “A method for the calculation of the radiation pattern and mode-conversion properties of a solid-state heterojunction laser,” IEEE Trans. Microwave Theory Tech. MIT-23, 576–585 (1975).
    [CrossRef]
  9. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1975), Chap. 2.
    [CrossRef]
  10. A. Hardy, “Orthogonality relations of modes obtained by the effective index method,” J. Opt. Soc. Am. A 1, 547–549 (1984).
    [CrossRef]
  11. R. E. Collin, “Radiation from apertures,” in Antenna Theory, R. E. Collin, F. J. Zucker (McGraw-Hill, New York, 1969), Part I, Chap. 3.
  12. L. M. Magid, Electromagnetic Fields, Energy, and Waves (Wiley, New York, 1972).

1984

1982

J. Buus, “The effective index method and its application to semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 1083–1089 (1982), and references therein.
[CrossRef]

1979

1978

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of gain-induced waveguiding in stripe geometry diode lasers,” IEEE J. Quantum Electron. QE-14, 418–427 (1978).
[CrossRef]

1977

1975

L. Lewin, “A method for the calculation of the radiation pattern and mode-conversion properties of a solid-state heterojunction laser,” IEEE Trans. Microwave Theory Tech. MIT-23, 576–585 (1975).
[CrossRef]

1972

T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of gain-induced waveguiding in stripe geometry diode lasers,” IEEE J. Quantum Electron. QE-14, 418–427 (1978).
[CrossRef]

Burns, W. K.

Buus, J.

J. Buus, “The effective index method and its application to semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 1083–1089 (1982), and references therein.
[CrossRef]

Collin, R. E.

R. E. Collin, “Radiation from apertures,” in Antenna Theory, R. E. Collin, F. J. Zucker (McGraw-Hill, New York, 1969), Part I, Chap. 3.

Hardy, A.

Hocker, G. B.

Ikegami, T.

T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

Kapon, E.

W. Streifer, E. Kapon, “Application of the equivalent-index method to DH diode lasers,” Appl. Opt. 18, 3724–3725 (1979).
[PubMed]

E. Kapon, A. Hardy, A. Zussman, “Application of perturbation theory to the analysis of diode lasers with lateral optical confinement,” IEEE J. Quantum Electron. (to be published).

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1975), Chap. 2.
[CrossRef]

Lewin, L.

L. Lewin, “A method for the calculation of the radiation pattern and mode-conversion properties of a solid-state heterojunction laser,” IEEE Trans. Microwave Theory Tech. MIT-23, 576–585 (1975).
[CrossRef]

Magid, L. M.

L. M. Magid, Electromagnetic Fields, Energy, and Waves (Wiley, New York, 1972).

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of gain-induced waveguiding in stripe geometry diode lasers,” IEEE J. Quantum Electron. QE-14, 418–427 (1978).
[CrossRef]

Streifer, W.

W. Streifer, E. Kapon, “Application of the equivalent-index method to DH diode lasers,” Appl. Opt. 18, 3724–3725 (1979).
[PubMed]

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of gain-induced waveguiding in stripe geometry diode lasers,” IEEE J. Quantum Electron. QE-14, 418–427 (1978).
[CrossRef]

Zussman, A.

E. Kapon, A. Hardy, A. Zussman, “Application of perturbation theory to the analysis of diode lasers with lateral optical confinement,” IEEE J. Quantum Electron. (to be published).

Appl. Opt.

IEEE J. Quantum Electron.

T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

J. Buus, “The effective index method and its application to semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 1083–1089 (1982), and references therein.
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of gain-induced waveguiding in stripe geometry diode lasers,” IEEE J. Quantum Electron. QE-14, 418–427 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

L. Lewin, “A method for the calculation of the radiation pattern and mode-conversion properties of a solid-state heterojunction laser,” IEEE Trans. Microwave Theory Tech. MIT-23, 576–585 (1975).
[CrossRef]

J. Opt. Soc. Am. A

Other

E. Kapon, A. Hardy, A. Zussman, “Application of perturbation theory to the analysis of diode lasers with lateral optical confinement,” IEEE J. Quantum Electron. (to be published).

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, New York, 1975), Chap. 2.
[CrossRef]

R. E. Collin, “Radiation from apertures,” in Antenna Theory, R. E. Collin, F. J. Zucker (McGraw-Hill, New York, 1969), Part I, Chap. 3.

L. M. Magid, Electromagnetic Fields, Energy, and Waves (Wiley, New York, 1972).

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Figures (2)

Fig. 1
Fig. 1

Cross section of the laser perpendicular to the guidance axis z.

Fig. 2
Fig. 2

The incident, reflected, and transmitted rays in the plane of incidence xz, which is tilted by an angle ϕ with respect to the x axis.

Equations (61)

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H ( r , t ) = Re { H ^ ( x , y ) exp [ j ( β z - ω t ) ] } .
( x , y ) x [ 1 ( x , y ) H ^ y x ] + 2 H ^ y y 2 + [ k 0 2 n 2 ( x , y ) - β 2 ] H ^ y ( x , y ) = 0 ,
H ^ y ( u , v ) = F u ( x ) G v ( u ) ( y ) ,
( x , y ) x ( 1 F u x ) + [ k 0 2 n 2 ( x , y ) - β u 2 ( y ) ] F u ( x ) = 0 ,
d 2 G v ( u ) ( y ) d y 2 + [ k 0 2 n eff , u 2 ( y ) - β u v 2 ] G v ( u ) ( y ) = 0.
E ^ x ( u , v ) ( x , y ) = k 0 2 ω β u v 0 F ¯ u ( x ) G ¯ v ( u ) ( y ) ,
F ¯ u ( x ) = [ 0 / ( x , 0 ) ] F u ( x ) = [ 1 / n 2 ( x , 0 ) ] F u ( x ) ,
G ¯ v ( u ) ( y ) = ( x 0 , 0 ) [ n eff , u 2 ( y ) / ( x 0 , y ) ] G v ( u ) ( y ) ,
G ˜ v ( u ) ( y ) = [ ( x 0 , 0 ) n eff , u 2 ( y ) / ( x 0 , y ) ] - 1 G v ( u ) ( y ) ,
- F u ( x ) F ¯ l ( x ) d x = δ u , l ,
- G ˜ v ( u ) ( y ) G ¯ m ( l ) ( y ) d y = - G v ( u ) ( y ) G m ( l ) ( y ) d y = δ v , m             for u = l .
H ^ y ( x , y ) = F u ( x ) G v ( u ) ( y ) + l , m R l , m ( u , v ) F l ( x ) G m ( l ) ( y ) ,
E ^ x ( x , y ) = k 0 2 ω 0 [ 1 β u v F ¯ u ( x ) G ¯ v ( u ) ( y ) - l m 1 β l m R l , m ( u , v ) F ¯ l ( x ) G ¯ m ( l ) ( y ) ] ,
H ^ ( r ) = ( 1 2 π ) 2 - A ( k x , k y ) exp ( j kr ) d k x d k y ,
E ^ ( r ) = - 1 ω 0 ( 1 2 π ) 2 - k × A ( k x , k y ) exp ( j kr ) d k x d k y .
k x = ( 2 π / λ ) α x = 2 π ν ,
k y = ( 2 π / λ ) α y = 2 π ξ ,
H ^ y ( x , y , 0 ) = - A y ( ν , ξ ) exp [ j 2 π ( ν x + ξ y ) ] d ν d ξ
E ^ x ( x , y , 0 ) = k 0 2 ω 0 - 1 - ( λ ν ) 2 [ 1 - ( λ ν ) 2 - ( λ ξ ) 2 ] 1 / 2 A y ( ν , ξ ) × exp [ j 2 π ( ν x + ξ y ) ] d ν d ξ ,
A ( ν , ξ ) = f u ( ν ) g v ( u ) ( ξ ) + l , m R l , m ( u , v ) f l ( ν ) g m ( l ) ( ξ ) ,
B ( ν , ξ ) = ( β u v ) - 1 f ¯ u ( ν ) g ¯ v ( u ) ( ξ ) - l , m ( β l m ) - 1 R l , m ( u , v ) f ¯ l ( ν ) g ¯ m ( l ) ( ξ ) ,
B ( ν , ξ ) = A ( ν , ξ ) / C ( ν , ξ ) ,
C ( ν , ξ ) = k 0 [ 1 - ( λ ν ) 2 ] - 1 [ 1 - ( λ ν ) 2 - ( λ ξ ) 2 ] 1 / 2 .
F l ( x ) = - f l ( ν ) exp ( j 2 π ν x ) d ν
f l ( ν ) = - F l ( x ) exp ( - j 2 π ν x ) d x ,
- f u ( - ν ) f ¯ l ( ν ) d ν = δ u , l
- g ˜ v ( u ) ( - ξ ) g ¯ m ( l ) ( ξ ) d ξ = - g v ( u ) ( - ξ ) g m ( l ) ( ξ ) d ξ = δ v , m             for u = l .
δ u , l δ v , m + R l , m ( u , v ) = - C ( ν , ξ ) B ( ν , ξ ) f ¯ l ( - ν ) g m ( l ) ( - ξ ) d ν d ξ
( β u v ) - 1 δ u , l δ v , m - ( β l m ) - 1 R l , m ( u , v ) = - B ( ν , ξ ) f l ( - ν ) g ˜ m ( l ) ( - ξ ) d ν d ξ .
M ( ν , ξ ) = - ( ω 0 / k 0 2 ) e x ( r ) ( ν , ξ ) / h y ( r ) ( ν , ξ ) ,
h y ( r ) ( ν , ξ ) = i = N 1 + 1 j = N 2 + 1 R i j ( u , v ) f i ( ν ) g j ( i ) ( ξ )
e x ( r ) ( ν , ξ ) = - ( k 0 2 / ω 0 ) i = N 1 + 1 j = N 2 + 1 R i j ( u , v ) ( β i j ) - 1 f ¯ i ( ν ) g ¯ j ( i ) ( ξ )
M ( ν , ξ ) = 1 k 0 n 2 U ( ν , ξ ) ( λ ν ) 2 U 2 ( ν , ξ ) Γ + ( λ ξ ) 2 n 2 Γ ( λ ν ) 2 Γ + ( λ ξ ) 2 Γ ,
Γ ( ν , ξ ) = n 2 V ( ν , ξ ) - U ( ν , ξ ) n 2 V ( ν , ξ ) + U ( ν , ξ ) ,
Γ ( ν , ξ ) = U ( ν , ξ ) - V ( ν , ξ ) U ( ν , ξ ) + V ( ν , ξ ) ,
U ( ν , ξ ) = [ n 2 - ( λ ν ) 2 - ( λ ξ ) 2 ] 1 / 2 ,
V ( ν , ξ ) = [ 1 - ( λ ν ) 2 - ( λ ξ ) 2 ] 1 / 2 ,
B ( ν , ξ ) = [ 1 + C ( ν , ξ ) M ( ν , ξ ) ] - 1 [ M ( ν , ξ ) f u ( ν ) g v ( u ) ( ξ ) + ( β u v ) - 1 f ¯ u ( ν ) g ¯ v ( u ) ( ξ ) + i = 0 N 1 j = 0 N 2 R i j ( u , v ) × [ f i ( ν ) g j ( i ) ( ξ ) M ( ν , ξ ) - ( β i j ) - 1 f ¯ i ( ν ) g ¯ j ( i ) ( ξ ) ] } .
i = 0 N 1 j = 0 N 2 R i j ( u , v ) ( q i j l m - δ i l δ j m ) = δ u , l δ v , m - P u v l m ,
i = 0 N 1 j = 0 N 2 [ q ¯ i j l m + ( 1 / β l m ) δ i l δ j m ] = ( 1 / β u v ) δ u l δ v m - P ¯ u v l m ,
q i j k l = - [ C ( ν , ξ ) 1 + C ( ν , ξ ) M ( ν , ξ ) ] × [ M ( ν , ξ ) f i ( ν ) g j ( i ) ( ξ ) - ( 1 / β i j ) f ¯ i ( ν ) g ¯ j ( i ) ( ξ ) ] × f ¯ k ( - ν ) g l ( k ) ( - ξ ) d ν d ξ ,
P i j k l = - [ C ( ν , ξ ) 1 + C ( ν , ξ ) M ( ν , ξ ) ] × [ M ( ν , ξ ) f i ( ν ) g j ( i ) ( ξ ) + ( 1 / β i j ) f ¯ i ( ν ) g ¯ j ( i ) ( ξ ) ] × f ¯ k ( - ν ) g l ( k ) ( - ξ ) d ν d ξ ,
q ¯ i j k l = - [ 1 + C ( ν , ξ ) M ( ν , ξ ) ] - 1 [ M ( ν , ξ ) f i ( ν ) g j ( i ) ( ξ ) - ( 1 / β i j ) f ¯ i ( ν ) g ¯ j ( i ) ( ξ ) ] × f k ( - ν ) g ˜ l ( k ) ( - ξ ) d ν d ξ ,
P ¯ i j k l = - [ 1 + C ( ν , ξ ) M ( ν , ξ ) ] - 1 [ M ( ν , ξ ) f i ( ν ) g j ( i ) ( ξ ) + ( 1 / β i j ) f ¯ i ( ν ) g ¯ j ( i ) ( ξ ) ] × f k ( - ν ) g ˜ l ( k ) ( - ξ ) d ν d ξ ,
S ( θ , ϕ ) ~ ( sin 2 θ sin 2 ϕ + cos 2 θ ) A ( ν , ξ ) 2 ,
e ( i ) = a ^ x e x ( i ) + a ^ z e z ( i ) ,
e ( i ) = a ^ y e y ( i ) ,
e y ( r ) = Γ e y ( i ) = - Γ ( η / cos θ i ) h x ( i ) ,
h x ( r ) = ( 1 / η ) ( cos θ i ) Γ e y ( i ) = - Γ h x ( i )
e x ( r ) = Γ e x ( i ) = Γ η ( cos θ i ) h y ( i ) ,
h y ( r ) = - ( η cos θ i ) - 1 Γ e x ( i ) = - Γ h y ( i )
Γ = η 0 / cos θ t - η / cos θ i η 0 / cos θ t + η / cos θ i
Γ = η 0 cos θ t - η cos θ i η 0 cos θ t + η cos θ i ,
η = ( μ 0 / 0 ) 1 / 2 1 / n = η 0 / n
h x ( i ) = h y ( i ) sin ϕ ,
h y ( i ) = h y ( i ) cos ϕ ,
e x ( r ) = e x ( r ) cos ϕ - e y ( r ) sin ϕ ,
h y ( r ) = h x ( r ) sin ϕ + h y ( r ) cos ϕ ,
M ( ν , ξ ) = - ω 0 k 0 e x ( r ) h y ( r ) = 1 k 0 n Γ cos θ i cos 2 ϕ + Γ ( 1 / cos θ i ) sin 2 ϕ Γ cos 2 ϕ + Γ sin 2 ϕ .
λ ν = k x / k 0 = sin θ t cos ϕ t ,
λ ξ = k y / k 0 sin θ t sin ϕ t ,

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