Abstract

The interaction of a short optical pulse with a medium whose optical properties are periodically modulated depends on the modulation phase at the moment of the pulse arrival. A technique of extracting the linear optical response function of such a medium to the cw optical field with a judiciously chosen series of short pulses rather than with cw optical fields is presented. An example of using the technique in numerical simulations is given, and a possible experimental arrangement is discussed. In particular, this technique may be convenient for extracting the optical properties of quantum wells modulated by a terahertz field.

© 2001 Optical Society of America

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References

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  1. J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
    [CrossRef]
  2. J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
    [CrossRef]
  3. C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
    [CrossRef]
  4. S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
    [CrossRef]
  5. T. Fromhertz, “Floquet states and intersubband absorption in strongly driven double quantum wells,” Phys. Rev. B 56, 4772–4777 (1997).
    [CrossRef]
  6. D. S. Citrin, “Optical analog for phase-sensitive measurements in quantum-transport experiments,” Phys. Rev. B 60, 5659–5663 (1999).
    [CrossRef]
  7. L. Lepetit, G. Chériax, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
    [CrossRef]

1999 (2)

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

D. S. Citrin, “Optical analog for phase-sensitive measurements in quantum-transport experiments,” Phys. Rev. B 60, 5659–5663 (1999).
[CrossRef]

1997 (3)

T. Fromhertz, “Floquet states and intersubband absorption in strongly driven double quantum wells,” Phys. Rev. B 56, 4772–4777 (1997).
[CrossRef]

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

1995 (1)

1955 (1)

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[CrossRef]

Allen, S. J.

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

Autler, S. H.

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[CrossRef]

Cerne, J.

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

Chériax, G.

Citrin, D. S.

D. S. Citrin, “Optical analog for phase-sensitive measurements in quantum-transport experiments,” Phys. Rev. B 60, 5659–5663 (1999).
[CrossRef]

Coldren, L.

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

Fromhertz, T.

T. Fromhertz, “Floquet states and intersubband absorption in strongly driven double quantum wells,” Phys. Rev. B 56, 4772–4777 (1997).
[CrossRef]

Gossard, M.

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

Inoshita, T.

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

Joffre, M.

Ko, J.

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

Kono, J.

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

Lepetit, L.

Noda, T.

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

Phillips, C.

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

Sakaki, H.

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

Sherwin, M.

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

Sherwin, M. S.

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

Su, M. Y.

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

Sundaram, M.

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

Townes, C. H.

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[CrossRef]

Appl. Phys. Lett. (2)

C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Coldren, “Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells,” Appl. Phys. Lett. 75, 2728–2730 (1999).
[CrossRef]

J. Cěrne, J. Kono, T. Inoshita, M. Sherwin, M. Sundaram, and M. Gossard, “Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells,” Appl. Phys. Lett. 70, 3543–3545 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Rev. (1)

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[CrossRef]

Phys. Rev. B (2)

T. Fromhertz, “Floquet states and intersubband absorption in strongly driven double quantum wells,” Phys. Rev. B 56, 4772–4777 (1997).
[CrossRef]

D. S. Citrin, “Optical analog for phase-sensitive measurements in quantum-transport experiments,” Phys. Rev. B 60, 5659–5663 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Diagrams illustrating the mixing of Fourier components E˜(ω) of the electric field and polarization P˜(ω) in a periodically driven medium. (a) The polarization component at ω is a result of the contributions of several components at ω±nΩ of the incident short optical pulse. (b) For long optical pulses the spectrum of P˜(ω+nΩ) is well defined by the narrow spectrum of E˜(ω) since there is no mixing of the spectral components of the pulse.

Fig. 2
Fig. 2

(a) Schematic of a three-level system consisting of the ground state |0〉 and two excited states |1〉 and |2〉.

Fig. 3
Fig. 3

Response function of the system in the absence of the THz field. (a) Absolute value (left axis) and phase (right axis) of the normalized polarization (γ/d01)P (t)/E0 induced by a short optical pulse as a function of time. The absolute value was multiplied by the indicated factor. (b) Absolute value (left axis) and phase (right axis) of (γ/d01)χ˜0(ω; ω) as a function of (ω-ω1). The dashed curve shows the broad spectrum |E˜opt(ω)|2 of the short pulse. The normalization factor of γ/d01 was chosen so that the magnitude of the normalized χ˜0(ω; ω) is unity for the lower-frequency peak.

Fig. 4
Fig. 4

Response function of the system in the frequency domain in the presence of the THz field. (a) Absolute value (left axis) and phase (right axis) of (γ/d01)χ˜0(ω; ω) as a function (ω-ω1). (b) Absolute value (left axis) and phase (right axis) of (γ/d01)χ˜+1(ω+Ω; ω) as a function (ω-ω1).

Fig. 5
Fig. 5

Response function of the system in the presence of the THz field. (a) Absolute value (left axis) and phase (right axis) of the normalized polarization (γ/d01)P (t)/E0 induced by a short optical pulse that arrives at t0=0 as a function of time. (b) Absolute value (left axis) and phase (right axis) of (γ/d01)P (ω)/E˜opt(ω) for this pulse.

Fig. 6
Fig. 6

Same as in Fig. 5, only t0=T/2.

Fig. 7
Fig. 7

Comparison of the average of the short-pulse responses with the cw optical susceptibilities. (a) Absolute value of the average of (γ/d01)P˜ (ω)/E˜opt(ω) for two pulses (dashed curve) and |(γ/d01)χ˜0(ω; ω)| (solid curve). (b) Absolute value of the average of (γ/d01)P˜ (ω+Ω)/E˜opt(ω) for two pulses (dashed curve), three pulses (dotted curve), and |(γ/d01)χ˜+1(ω+Ω; ω)| (solid curve).

Fig. 8
Fig. 8

Schematic experimental arrangement that allows measurement of the phase and the magnitude of the transmission of short optical pulses through a modulated sample.

Equations (34)

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P(t)=-tdtχ(t; t)E(t).
P˜(ω)=n=-+χ˜n(ω; ω-nΩ)E˜(ω-nΩ),
P˜l(ω)=n=-χ˜n(ω; ω-nΩ)E˜l(ω-nΩ).
1Nl=0N-1P˜l(ω)E˜l(ω-mΩ)=n=-χ˜n(ω; ω-nΩ)×E˜0(ω-nΩ)E˜0(ω-mΩ)Sm-n,N,
Sm-n,N=1Nl=0N-1 exp[i(m-n)ΩlΔt]=1N1-exp[i2π(m-n)]1-exp[i2πN(m-n)].
χ˜m(ω; ω-mΩ)=1Nl=0N-1P˜l(ω)E˜l(ω-mΩ).
|E˜0[ω-(m±N)Ω]χm±N[ω; ω-(m±N)Ω]|
|E˜0(ω-mΩ)χm(ω; ω-mΩ)|.
χ˜m(ω; ω-mΩ)=1NE˜0(ω-mΩ)l=0N-1P˜l(ω)×exp[-i(ω-mΩ)lΔt].
χ˜m(ω; ω-mΩ)=P˜0(ω)E˜0(ω-mΩ)Sm,N.
Ψ(t)=a0(t)φ0+a1(t)φ1+a2(t)φ2,
da1dt=-iω1a1+id10Eopt+id12ETHza2,
da2dt=-iω2a2+id20Eopt+id21ETHza1,
Eopt(t)=Eopt(t)d10 exp(-iω1t)+c.c.,
ETHz(t)=ETHzd12 cos(Ωt),
da¯1dt=iEopt(t)+ia¯2ETHz cos(Ωt)-γa¯1,
da¯2dt=-iω21a¯2+id20d10 Eopt(t)+ia¯1ETHz cos(Ωt)-γa¯2,
P(t)=P (t)exp(-iω1t)+c.c.
P (t)=d01a¯1(t)+d02a¯2(t).
Eopt(t)=E0 exp[-(t-t0)2/τ2],
χ˜n(ω+nΩ; ω)=P˜ (ω+nΩ)E˜opt(ω),
Es(t)exp(-iω0t)=E0(t)2+Et(t)2exp(-iω0t),
I(ω)=|E˜0(ω)|24+E˜0*(ω)E˜t(ω)22+E˜0(ω)E˜t*(ω)22+|E˜t(ω)|22.
ΔI(ω)=E˜0*(ω)E˜t(ω)22+E˜0(ω)E˜t*(ω)22.
Re[t˜(ω)]=2ΔI(ω)|E˜0(ω)|2.
Es(t)exp(-iω0t)=iE0(t)2+Et(t)2exp(-iω0t).
ΔI(ω)=-iE˜0*(ω)E˜t(ω)22+iE˜0(ω)E˜t*(ω)22,
Im[t˜(ω)]=2ΔI(ω)|E˜0(ω)|2.
P(t)=-+dω exp(-iωt)P˜(ω),
P˜(ω)=12π-+dt exp(iωt)P(t).
P˜(ω)=12π-+dt0+dτ-+dω exp[i(ω-ω)t+i(ω+ω)τ/2]χ(t+τ/2; t-τ/2)E˜(ω).
χt+τ2; t-τ2=n=-+χn(τ)exp(-inΩt)
χn(τ)=1T-T/2T/2dtχ(t+τ/2; t-τ/2)exp(inΩt).
χ˜n(ω; ω-nΩ)=0dτ exp[i(ω+ω-nΩ)τ/2]χn(τ).

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