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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  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, Jr., 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

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

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, Jr., and H. Sakaki, “Resonant terahertz optical sideband generation from confined magnetoexcitons,” Phys. Rev. Lett. 79, 1758–1761 (1997).
[CrossRef]

1995

1955

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

Allen Jr., S. J.

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

Lepetit, L.

Noda, T.

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, Jr., 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, Jr., 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, Jr., 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, Jr., 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.

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

Phys. Rev.

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

Phys. Rev. B

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.

J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin, S. J. Allen, Jr., 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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