Abstract

We present an auxiliary differential equation Finite-difference Time-domain (ADE-FDTD) approach to numerically model the wave propagation within a gain or absorbing medium such as quantum well structures. Start from traditional quantum electronics theory, the macroscopic susceptibility of the semiconductor is derived and expressed by a multiple-Lorentz-like model based on Prony’s method. With the auxiliary differential equation method each Lorentz-like model can be simulated in the time domain and the induced polarization is then determined by summing all the models. By incorporating the induced polarization into the time-domain Maxwell’s equations, electromagnetic wave propagation in the quantum well medium can be accurately modeled using the FDTD method.

© 2006 Optical Society of America

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  1. H. P. Sardesai and A. P. Weiner, “Nonlinear fiber-optic receiver for ultra short pulse code division multiple access communications,” IEEE Electron. Lett. 33, 610–611 (1997).
    [Crossref]
  2. C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
    [Crossref]
  3. L. E. M. Brackenbury, “Multiplexer as a universal computing element for electro-optic logic systems,” IEEE Proc. J. Optoelectron. 137, 305–310 (1990).
    [Crossref]
  4. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Tran. Antennas Propag. 46, 334–340 (1998).
    [Crossref]
  5. R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
    [Crossref] [PubMed]
  6. M. Sargent, M. O. Scully, and W. E. lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974).
  7. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).
  8. F. B. Hildebrand, Introduction to Numerical Analysis (Dover, New York, 1974).
  9. D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
    [Crossref]
  10. R. Dingle and I. Festkorperprobleme, Advances in solid state physics (Braunschweig, Pergamon-Vieweg).
  11. P. S. Zory, “Quantum Well Lasers,” (Academic Press INC, CA, 1993).
  12. N. K. Dutta, “Calculated threshold current of GaAs quantum well lasers.,” J. Appl. Phy. 53, 7211–7214 (1983).
    [Crossref]
  13. R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1279–1301 (1992).
    [Crossref]
  14. D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. 44, 792–797 (1996).
    [Crossref]
  15. D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antennas Propag. 40, 1223–1230 (1992).
    [Crossref]
  16. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [Crossref]

1999 (1)

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

1998 (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Tran. Antennas Propag. 46, 334–340 (1998).
[Crossref]

1997 (1)

H. P. Sardesai and A. P. Weiner, “Nonlinear fiber-optic receiver for ultra short pulse code division multiple access communications,” IEEE Electron. Lett. 33, 610–611 (1997).
[Crossref]

1996 (1)

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. 44, 792–797 (1996).
[Crossref]

1995 (1)

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[Crossref] [PubMed]

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

1992 (2)

D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antennas Propag. 40, 1223–1230 (1992).
[Crossref]

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1279–1301 (1992).
[Crossref]

1990 (1)

L. E. M. Brackenbury, “Multiplexer as a universal computing element for electro-optic logic systems,” IEEE Proc. J. Optoelectron. 137, 305–310 (1990).
[Crossref]

1983 (2)

D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
[Crossref]

N. K. Dutta, “Calculated threshold current of GaAs quantum well lasers.,” J. Appl. Phy. 53, 7211–7214 (1983).
[Crossref]

Arnold, J. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[Crossref] [PubMed]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Brackenbury, L. E. M.

L. E. M. Brackenbury, “Multiplexer as a universal computing element for electro-optic logic systems,” IEEE Proc. J. Optoelectron. 137, 305–310 (1990).
[Crossref]

Dapkus, P. D.

D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
[Crossref]

Dingle, R.

R. Dingle and I. Festkorperprobleme, Advances in solid state physics (Braunschweig, Pergamon-Vieweg).

Dutta, N. K.

N. K. Dutta, “Calculated threshold current of GaAs quantum well lasers.,” J. Appl. Phy. 53, 7211–7214 (1983).
[Crossref]

Festkorperprobleme, I.

R. Dingle and I. Festkorperprobleme, Advances in solid state physics (Braunschweig, Pergamon-Vieweg).

Gogny, D. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[Crossref] [PubMed]

Hafizi, B.

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (Dover, New York, 1974).

Hong, C. S.

D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
[Crossref]

Hubbard, R. F.

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

Hunsberger, F.

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1279–1301 (1992).
[Crossref]

Kapetanakos, C. A.

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

Kasemset, D.

D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
[Crossref]

Kelley, D. F.

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. 44, 792–797 (1996).
[Crossref]

lamb, W. E.

M. Sargent, M. O. Scully, and W. E. lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974).

Luebbers, R. J.

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. 44, 792–797 (1996).
[Crossref]

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1279–1301 (1992).
[Crossref]

Milehberg, H. M.

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

Nagra, A. S.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Tran. Antennas Propag. 46, 334–340 (1998).
[Crossref]

Patel, N. B.

D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
[Crossref]

Sardesai, H. P.

H. P. Sardesai and A. P. Weiner, “Nonlinear fiber-optic receiver for ultra short pulse code division multiple access communications,” IEEE Electron. Lett. 33, 610–611 (1997).
[Crossref]

Sargent, M.

M. Sargent, M. O. Scully, and W. E. lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974).

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974).

Sprangle, P.

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

Sullivan, D. M.

D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antennas Propag. 40, 1223–1230 (1992).
[Crossref]

Ting, A.

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

Weiner, A. P.

H. P. Sardesai and A. P. Weiner, “Nonlinear fiber-optic receiver for ultra short pulse code division multiple access communications,” IEEE Electron. Lett. 33, 610–611 (1997).
[Crossref]

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

York, R. A.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Tran. Antennas Propag. 46, 334–340 (1998).
[Crossref]

Ziolkowski, R. W.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[Crossref] [PubMed]

Zory, P. S.

P. S. Zory, “Quantum Well Lasers,” (Academic Press INC, CA, 1993).

IEEE Electron. Lett. (1)

H. P. Sardesai and A. P. Weiner, “Nonlinear fiber-optic receiver for ultra short pulse code division multiple access communications,” IEEE Electron. Lett. 33, 610–611 (1997).
[Crossref]

IEEE J. Quantum Electron. (2)

C. A. Kapetanakos, B. Hafizi, H. M. Milehberg, P. Sprangle, R. F. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[Crossref]

D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-theory and experiment,” IEEE J. Quantum Electron. 19, 1025–1030 (1983).
[Crossref]

IEEE Proc. J. Optoelectron. (1)

L. E. M. Brackenbury, “Multiplexer as a universal computing element for electro-optic logic systems,” IEEE Proc. J. Optoelectron. 137, 305–310 (1990).
[Crossref]

IEEE Tran. Antennas Propag. (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Tran. Antennas Propag. 46, 334–340 (1998).
[Crossref]

IEEE Trans. Antennas Propag. (3)

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1279–1301 (1992).
[Crossref]

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. 44, 792–797 (1996).
[Crossref]

D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antennas Propag. 40, 1223–1230 (1992).
[Crossref]

J. Appl. Phy. (1)

N. K. Dutta, “Calculated threshold current of GaAs quantum well lasers.,” J. Appl. Phy. 53, 7211–7214 (1983).
[Crossref]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Phys. Rev. A (1)

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[Crossref] [PubMed]

Other (5)

M. Sargent, M. O. Scully, and W. E. lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

F. B. Hildebrand, Introduction to Numerical Analysis (Dover, New York, 1974).

R. Dingle and I. Festkorperprobleme, Advances in solid state physics (Braunschweig, Pergamon-Vieweg).

P. S. Zory, “Quantum Well Lasers,” (Academic Press INC, CA, 1993).

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Figures (5)

Fig. 1.
Fig. 1.

Band structure of the semiconductor.

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Fig. 2.
Fig. 2.

FDTD grid sketch map

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Fig. 3.
Fig. 3.

Comparison of the real part and image part of χ(ω) as computed by Prony’s method and quantum electronics theory with injection current density N= 6×1018/ cm-3.

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Fig. 4.
Fig. 4.

Comparison of the field absorption factor and phase shift factor as computed by ADE-FDTD and analytical result with injection current density a) N= 1×1018/ cm-3 , b) N= 6×1018/ cm-3 , and c) N= 12×1018/ cm-3.

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Fig. 5.
Fig. 5.

(a) Multiple quantum well structure, (b) Snapshots of pulse shape propagating in the MQW structure , (c) Absorption factor and phase shift of the MQW structure.

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Tables (1)

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Table 1. Numerical Values Used in the validation Calculations

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Equations (45)

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df e ( k ) dt = − j γE ( t ) ħ [ ρ eh ( k ) − ρ eh * ( k ) ] − f e ( k ) − f e 0 ( k ) τ e
df h ( k ) dt = − j γE ( t ) ħ [ ρ eh * ( k ) − ρ eh ( k ) ] − f h ( k ) − f h 0 ( k ) τ h
dρ eh ( k ) dt = − j γE ( t ) ħ [ f e ( k ) + f h ( k ) − 1 ] − j E k ħ ρ eh ( k ) − ρ eh ( k ) T 2
f e 0 ( k ) = 1 exp ( E e ( k ) − F e k b T ) + 1
f h 0 ( k ) = 1 exp ( E h ( k ) − F h k b T ) + 1
f e ( k ) = f e 0 ( k ) − τ e τ e + τ h [ f e 0 ( k ) + f h 0 ( k ) − 1 ] L ( ħ ω − E k ) I I s 1 + L ( ħ ω − E k ) I I s
f h ( k ) = f h 0 ( k ) − τ h τ e + τ h [ f e 0 ( k ) + f h 0 ( k ) − 1 ] L ( ħ ω − E k ) I I s 1 + L ( ħ ω − E k ) I I s
ρ eh = − jγ [ E ( ω ) ⊗ ( f e ( k ) + f h ( k ) − 1 ) ] j ( ωħ + E k ) + ħ T 2
P ( t ) = Tr [ ργ ] = γ ( ρ eh + ρ eh * )
P ( t ) = ε 0 χ ( t ) ⊗ E ( t )
χ ( ω ) = γ ε 0 E ( ω ) ( ρ eh + ρ eh * )
χ ( ω ) ≈ γ 2 ε 0 ∑ k 1 V [ f e 0 ( k ) + f h 0 ( k ) − 1 ] [ 1 j ( ωħ − E k ) + ħ T 2 − 1 j ( ωħ − E k ) + ħ T 2 ]
D ( E ′ ) = ∑ n m n 2 πħ 2 H ( E ′ − E n )
D r , in ( E ′ ) = m r , i 2 πħ 2 H ( E ′ − E g , in ) , i = h , l ; n = 1,2 , ⋯
m r , i = m n m v m c + m v
E g , in = E g + E c , n + E v , in
χ ( ω ) ≈ ∑ i , n γ i 2 ε 0 ∫ D n ( E ) ( f e 0 + f h 0 − 1 ) [ 1 j ( ωħ − E k ) + ħ T 2 − 1 j ( ωħ + E k ) + ħ T 2 ] dE
γ i 2 = e 2 m 0 2 ω 2 M b 2 δ i
M b 2 = 2 ξ m 0 E g
δ i = { 3 4 [ 1 + E c , n + E v , hn ε − E g ] , i = h 1 4 [ 5 − 3 ( E c , n + E v , ln ) ε − E g ] , i = l
χ ( ω ) ≈ ∑ i , n e 2 M b 2 m 0 2 ω 2 ε 0 ∫ δ i D n ( E ) ( f e 0 + f h 0 − 1 ) • [ 1 j ( ωħ − E k ) + ħ T 2 − 1 j ( ωħ + E k ) + ħ T 2 ] dE
{ tan ( 2 m E n ħ 2 d z 2 ) = V 0 − E n E n , for even mode ( n = 2,4,6 , ⋯ ) cot ( 2 m E n ħ 2 d z 2 ) = − V 0 − E n E n , for odd mode ( n = 1,3,5 , ⋯ )
{ ∇ × E = − μ ∂ H ∂ t ∇ × H = ε ∂ E ∂ t + ∂ P ∂ t
{ μ ∂ H y ∂ t = − ∂ E x ∂ z ε ∂ E x ∂ t + ∂ P ∂ t = − ∂ H y ∂ z
χ ( t ) ≈ 2 γ 2 ħ ε 0 ∑ i , n ∫ δ i D n ( ε ) ( f e 0 + f h 0 − 1 ) sin ( ε ħ t ) exp ( − ω T t ) U ( t ) dε
J ( t ) = ∂ P ( t ) ∂ t = ε 0 ∂ χ ( t ) ∂ t ⊗ E ( t ) = σ ( t ) ⊗ E ( t )
σ ( t ) ≈ − 2 ħ ∑ i , n γ 2 ∫ δ i D n ( ε ) ( f e 0 + f h 0 − 1 ) [ ε ħ cos ( ε ħ t ) − ω T sin ( ε ħ t ) ] • exp ( − ω T t ) U ( t ) dε
σ ( t ) = ∑ i = 1 P C i D i = ∑ i = 1 P C i exp [ ( α i + jω i ) t ] U ( t )
σ ( ω ) = 2 ∑ i = 1 P 2 jωA i − ( A i α i + B i ω i ) ( α i 2 + ω i 2 ) − 2 jωα i − ω 2 = ∑ i = 1 P 2 σ i ( ω )
σ p + E 1 σ p − 1 + E 2 σ p − 2 + ⋯ + E p σ 1 = 0
σ p + 1 + E 1 σ p + E 2 σ p − 1 + ⋯ + E p σ 2 = 0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮
σ N + E 1 σ N − 1 + E 2 σ N − 2 + ⋯ + E p σ N − p = 0
D p + E 1 D p − 1 + E 2 D p − 2 + ⋯ + E p − 1 D 1 + E p = 0
C 1 + C 2 + ⋯ + C p = σ 1
D 1 C 1 + D 2 C 2 + ⋯ + D p C p = σ 2
( D 1 ) 2 C 1 + ( D 2 ) 2 C 2 + ⋯ + ( D p ) 2 C p + σ 3
⋮ ⋮ ⋱ ⋮ ⋮
( D 1 ) N − 1 C 1 + ( D 2 ) N − 2 C 2 + ⋯ + ( D p ) N − 1 C p = σ N
J ( ω ) = ∑ i = 1 P 2 J i ( ω ) = ∑ i = 1 P 2 σ i ( ω ) E ( ω )
{ ∂ 2 J i ( t ) ∂ 2 t − 2 α i ∂ J i ( t ) ∂ t + ( α i 2 + ω i 2 ) J i ( t ) = 2 A i ∂ E x ( t ) ∂ t − 2 ( A i α i + B i ω i ) E x ( t ) J ( t ) = ∑ i = 1 P 2 J i ( t )
E x n + 1 2 ( I ) = E x n − 1 2 ( I ) − Δ t Δ zε ( I ) [ H y n ( I + 1 2 ) − H y n ( I − 1 2 ) ] − Δ t ε ( I ) J n ( I )
H y n + 1 ( I + 1 2 ) = H y n ( I + 1 2 ) − Δ t Δ zμ [ E x n ( I + 1 ) − E x n ( I ) ]
J i n + 1 ( I ) = 2 − ( α i 2 + ω i 2 ) Δ t 2 1 − α i Δ t J i n ( I ) − 1 + α i Δ t 1 − α i Δ t J i n − 1 ( I )
− ( A i α i + B i ω i ) Δ t 2 − 2 A i Δ t 1 − α i Δ t E x n + 1 2 ( I ) − ( A i α i + B i ω i ) Δ t 2 + 2 A i Δ t 1 − α i Δ t E x n − 1 2 ( I )

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