Abstract

The multiple-quantum-well structure exhibits both a fast and a slow saturation. The theory of mode locking with such a saturable absorber is investigated, and it is found that the fast component contributes substantially to the formation of short pulses. The presence of the fast-absorber component removes the requirement for gain depletion for the stabilization of the pulse against growth of noise following the pulse for pulses shorter than 1.5 psec. However, the buildup of short pulses from longer ones still calls for gain depletion. We also investigate the conditions for self-starting and stability against relaxation oscillations to arrive at criteria for the construction of short resonator structures.

© 1985 Optical Society of America

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References

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  1. Y. Silberberg, P. W. Smith, D. J. Eilenberger, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, “Passive mode locking of a semiconductor diode laser,” Opt. Lett. 9, 507–509 (1984).See also Ref. 2.
    [Crossref] [PubMed]
  2. P. W. Smith, Y. Silberberg, and D. A. B. Miller, “Mode locking of semiconductor diode lasers using saturable excitonic nonlinearities,” J. Opt. Soc. Am. B 2, 1228–1236 (1985).
    [Crossref]
  3. W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
    [Crossref] [PubMed]
  4. D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
    [Crossref]
  5. Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).
  6. D. S. Chemla and D. A. B. Miller, “Room-temperature excitonic nonlinear-optical effects in semiconductor quantum-well structures,” J. Opt. Soc. Am. B 2, 1155–1173 (1985).
    [Crossref]
  7. H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
    [Crossref]
  8. H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
    [Crossref]
  9. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156–158 (1984).
    [Crossref] [PubMed]

1985 (3)

1984 (3)

1975 (2)

H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[Crossref]

Chemla, D. S.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

D. S. Chemla and D. A. B. Miller, “Room-temperature excitonic nonlinear-optical effects in semiconductor quantum-well structures,” J. Opt. Soc. Am. B 2, 1155–1173 (1985).
[Crossref]

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
[Crossref]

Downer, M. C.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

Eilenberger, D. J.

Fork, R. L.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156–158 (1984).
[Crossref] [PubMed]

Gordon, J. P.

Gossard, A. C.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
[Crossref]

Y. Silberberg, P. W. Smith, D. J. Eilenberger, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, “Passive mode locking of a semiconductor diode laser,” Opt. Lett. 9, 507–509 (1984).See also Ref. 2.
[Crossref] [PubMed]

Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).

Haus, H. A.

H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[Crossref]

Knox, W. H.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

Martinez, O. E.

Miller, D. A. B.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

D. S. Chemla and D. A. B. Miller, “Room-temperature excitonic nonlinear-optical effects in semiconductor quantum-well structures,” J. Opt. Soc. Am. B 2, 1155–1173 (1985).
[Crossref]

P. W. Smith, Y. Silberberg, and D. A. B. Miller, “Mode locking of semiconductor diode lasers using saturable excitonic nonlinearities,” J. Opt. Soc. Am. B 2, 1228–1236 (1985).
[Crossref]

Y. Silberberg, P. W. Smith, D. J. Eilenberger, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, “Passive mode locking of a semiconductor diode laser,” Opt. Lett. 9, 507–509 (1984).See also Ref. 2.
[Crossref] [PubMed]

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
[Crossref]

Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).

Shank, C. V.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

Silberberg, Y.

Smith, P. W.

P. W. Smith, Y. Silberberg, and D. A. B. Miller, “Mode locking of semiconductor diode lasers using saturable excitonic nonlinearities,” J. Opt. Soc. Am. B 2, 1228–1236 (1985).
[Crossref]

Y. Silberberg, P. W. Smith, D. J. Eilenberger, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, “Passive mode locking of a semiconductor diode laser,” Opt. Lett. 9, 507–509 (1984).See also Ref. 2.
[Crossref] [PubMed]

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
[Crossref]

Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).

Tell, B.

Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).

Wiegmann, W.

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
[Crossref]

Y. Silberberg, P. W. Smith, D. J. Eilenberger, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, “Passive mode locking of a semiconductor diode laser,” Opt. Lett. 9, 507–509 (1984).See also Ref. 2.
[Crossref] [PubMed]

Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).

IEEE J. Quantum Electron. (2)

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, “Room temperature excitonic nonlinear absorption and refraction in GaAs-AlGaAs multiple quantum well structures,” IEEE J. Quantum Electron. QE-20, 265–275 (1984).
[Crossref]

H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[Crossref]

J. Appl. Phys. (1)

H. A. Haus, “Theory of mode-locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

W. H. Knox, R. L. Fork, M. C. Downer, D. A. B. Miller, D. S. Chemla, C. V. Shank, A. C. Gossard, and W. Wiegmann, “Femtosecond dynamics of resonantly excited excitons in room temperature GaAs quantum wells,” Phys. Rev. Lett. 54, 1306 (1985).
[Crossref] [PubMed]

Other (1)

Y. Silberberg, P. W. Smith, D. A. B. Miller, B. Tell, A. C. Gossard, and W. Wiegmann, “Fast nonlinear optical response from proton-bombarded multiple quantum well structures,” Appl. Phys. Lett. (to be published).

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Figures (4)

Fig. 1
Fig. 1

Schematic of (a) semiconductor laser diode mode locked by MQWS and (b) the components of the theoretical modeling.

Fig. 2
Fig. 2

Pulse buildup under action of combined slow and fast absorber. The parameters used are g0 = 0.8, l0 = 0.5, α0LQW = 0.2, τg = τs = TR, and Ωgτf = 6. The pulse intensity is displayed at intervals of four round trips in the cavity, and the curves are shifted up progressively for clarity. The dotted and solid lines signify regions of net instantaneous loss and gain, respectively. Net loss in front of and behind the pulse is necessary for stability.

Fig. 3
Fig. 3

Pulse buildup under action of fast absorber only. All parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Pulse buildup under action of slow absorber only. All parameters are the same as in Fig. 2.

Equations (63)

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α = α 0 β x N x β e ( N e + N h ) ,
α α 0 = 1 4 A x N x 2 A x N e .
A x d N x d t = τ s I 2 I s A x N x τ f ,
A x d N e d t = A x N x τ f A x N e τ s .
x 4 A x N x ,
y 2 A x N e .
= 1 x y ,
d x d t = 2 τ s I I s x τ f ,
d y d t = 1 2 x τ f y τ s .
x = τ f τ s 2 I I s 1 y 1 + 2 τ t τ s I I s .
d y d t = 1 τ s I I s 1 y 1 + 2 τ f τ s I I s .
y = 1 exp ( d t τ s I I s 1 + 2 τ f τ s I I s ) .
α ( I ) = α 0 { 1 2 τ f τ s I I s 1 1 + 2 τ f τ s I I s exp ( d t τ s I / I s 1 + 2 τ f τ s I I s ) [ 1 exp ( d t τ s I / I s 1 + 2 τ f τ s I I s ) ] }
α ( I ) = α 0 [ 1 2 τ f τ s I I s d t τ s I I s + 1 2 ( d t τ s I I s ) 2 ] .
W s = I s τ s .
W f = 1 2 τ f τ s τ f I s = 1 2 W s .
ν n + 1 = { 1 + [ g ( n ) ω g 2 d 2 d t 2 + g ( n ) exp ( t | ν n | 2 d t / W g ) ] l s ( n ) exp ( t | ν n | 2 d t / W s ) l f ( 1 | ν n | 2 P f ) l 0 } ν n ,
ν n + 1 = ν n + Δ T ν n t .
0 = Δ T d ν d t + g ω g 2 d 2 ν d t 2 + g ν exp t | ν | 2 d t / W g l s ν exp t | ν | 2 d t / W s l f ν [ 1 | ν | 2 P f ] l 0 ν .
l s exp | ν | 2 d t / W s l s [ 1 | ν | 2 d t / W s + 1 2 ( | ν | 2 d t / W s ) 2 ]
ν ( t ) = V 0 sech ( t τ p ) .
d ν d t = V 0 τ p tanh ( t τ p ) sech ( t τ p ) ,
d 2 ν d t 2 = V 0 τ p 2 [ 1 2 sech 2 ( t τ p ) ] sech ( t τ p ) ,
t | ν | 2 d t = V 0 2 τ p [ 1 + tanh ( t τ p ) ] , = W 0 2 [ 1 + tanh ( t τ p ) ] ,
tanh 2 ( t τ p ) = 1 sech 2 ( t τ p ) .
g ( 1 W 0 2 W g + W 0 2 4 W g 2 ) + g ω g 2 τ p 2 l s [ 1 W 0 2 W s + W 0 2 4 W s 2 ] l f l 0 = 0 .
Δ T τ p g ( W 0 2 W g W 0 2 4 W g 2 ) + l s ( W 0 2 W s W 0 2 4 W s 2 ) = 0 .
2 g ω g 2 τ p 2 + l f 2 W 0 P f τ p + l s 8 W 0 2 W s 2 g 8 W 0 2 W g 2 = 0 .
X 2 ( l f 4 g ω g τ f W 0 W f ) X 1 16 ( l s g g g W s 2 W s 2 ) W 0 2 W s 2 = 0 ,
X = l f 8 g ω g τ f W 0 W f × { 1 + [ 1 + 4 g l s l f 2 ( 1 g l s W s 2 W g 2 ) W f 2 ω g 2 τ f W s 2 ] 1 / 2 } .
4 g l s l f 2 ( 1 g l s W s 2 W g 2 ) W f 2 ω g 2 τ f 2 W s 2 1 .
X l f 4 g ω g τ f W 0 W f ,
X 1 4 [ l s g g g ( W s W g ) 2 ] 1 / 2 W 0 W s .
exp [ α ( I ) L QW ] l = 1 α 0 L QW { 1 2 τ f τ s I I s I d t / W s + 1 / 2 ( I d t / W s ) } l ,
l s = l f = α 0 L QW , P f = τ s I s / 2 τ f ,
l 0 = l α 0 L QW .
1 ω g τ p = l f 8 g ω g τ f W 0 W f { 1 + [ 1 + g l f × [ 1 g l f ( W s W g ) 2 ] 1 ω g 2 τ f 2 ] 1 / 2 } l f 8 g ω g τ f W 0 W f [ 1 + ( 1 + g l f 1 ω g 2 τ f 2 ) 1 / 2 ] ,
τ f τ p = 1 8 k W 0 W f ( 1 + 1 + k ) ,
= 1 x ,
= 1 y .
g + g ω g 2 τ p 2 l s ( 1 W 0 2 W s + W 0 2 4 W s 2 ) l f l 0 = 0 .
g l s l f l 0 < 0 .
l sred = l s ( 1 W 0 W s + 1 2 W 0 2 W s 2 ) .
g l sred l f l 0 = g ω g 2 τ p 2 + 1 2 l s W 0 W s 1 4 l s ( W 0 W s ) 2 .
W 0 W f > 1 5 16 + 1 8 1 k ( 1 + 1 + k ) ,
τ f τ p > 1 1 + 5 2 k 1 + 1 + k .
1.76 τ p < 1.5 psec .
W 0 / W f > 2.07 ,
[ g ( t ) l ( t ) + g ω g 2 d 2 d t 2 + Δ T d d t ] ν ( t ) = 0 .
g 1 ( Ω ) l 1 ( Ω ) Ω 2 ω g 2 g ν 1 ν 0 + j Ω Δ T ν 1 ν 0 = 0 .
Re [ g 1 ( Ω ) l 1 ( Ω ) ] Ω 2 ω g 2 g ν 1 ν 0 > 0 .
d g d t = g g 0 τ g g τ g I I g .
g s = g 0 1 + I 0 / I g ,
g 1 = g s 1 + I 0 I g + j Ω τ g I 1 I g ,
0 = 1 2 τ f τ s I 0 I s 1 + I 0 I s 1 1 + I 0 / I s .
x 1 + y 1 = 0 2 I 1 c I s 1 + j Ω τ s + c 2 ( 1 + 2 I 0 c I s + j Ω τ f ) ( 1 + j Ω τ s ) + I 0 I s ,
1 0 I 1 I s 1 ( 1 + j Ω τ s ) + I 0 I s .
g 0 ( 1 + I 0 I g ) 2 + Ω 2 τ g 2 + α 0 L QW I g I s 1 ( 1 + I 0 I s ) 2 + Ω 2 τ s 2 g ω g 2 Ω 2 I g 2 I 0 > 0
g 0 ( 1 + I 0 I g ) 2 + α 0 L QW I g I s 1 ( 1 + I 0 I s ) 2 < 0 .
g 0 / I g ( 1 + I 0 I g ) 2 + Ω 2 τ g 2 < α 0 L QW / I s ( 1 + I 0 I s ) 2 + Ω 2 τ s 2 .
α 0 L QW g 0 I g I s < 1 .
( τ s τ g ) 2 < α 0 L QW g 0 I g I s .
τ s 2 Ω 2 > 1 α 0 L QW g 0 I g I s α 0 L QW g 0 I g I s τ g 2 τ s 2 1 .

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