Abstract

A new approach to far-infrared spectroscopy is described that uses extremely short far-infrared pulses to measure the dielectric properties of materials. Optical rectification of femtosecond optical pulses is used to produce a Čerenkov cone of pulsed far-infrared radiation of approximately one cycle in duration in the terahertz spectral range. The coherent detection of the electric field of these far-infrared pulses by electro-optic sampling provides a capability for measuring precise changes in the shape of the waveform following reflection or transmission from materials. This method, which is equivalent to having a tunable laser in the spectral range from 0.1 to 2 THz, is illustrated by the measurement of the dielectric response of a solid-state plasma in n-type germanium and a GaAs/GaAlAs multi-quantum-well superlattice.

© 1985 Optical Society of America

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  1. For recent reviews, see Infrared and Millimeter Waves, K. J. Button, ed.(Academic, New York, 1983), Vol. 8.
    [CrossRef]
  2. D. H. Auston, Appl. Phys. Lett. 43, 713 (1983).
    [CrossRef]
  3. D. A. Kleinman, D. H. Auston, IEEE J. Quantum Electron. QE-20, 964 (1983).
  4. D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
    [CrossRef]
  5. M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
    [CrossRef]
  6. I. Kaminow, in Handbook of Lasers, J. Pressley, ed.(Chemical Rubber Company, Cleveland, Ohio, 1971), Chap. 15.
  7. A. S. Barker, A. A. Ballman, J. A. Ditzenberger, Phys. Rev. B 2, 4233 (1970).
    [CrossRef]
  8. R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
    [CrossRef]
  9. J. A. Valdmanis, G. A. Mourou, C. W. Gabel, IEEE J. Quantum Electron. QE-19, 664 (1983).
    [CrossRef]
  10. R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
    [CrossRef]
  11. A. L. Fetter, Ann. Phys. (N.Y.) 88, 1 (1974).
    [CrossRef]
  12. S. Sas Sarma, J. J. Quinn, Phys. Rev. B 25, 7603 (1982).
    [CrossRef]

1984 (1)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

1983 (3)

D. H. Auston, Appl. Phys. Lett. 43, 713 (1983).
[CrossRef]

D. A. Kleinman, D. H. Auston, IEEE J. Quantum Electron. QE-20, 964 (1983).

J. A. Valdmanis, G. A. Mourou, C. W. Gabel, IEEE J. Quantum Electron. QE-19, 664 (1983).
[CrossRef]

1982 (1)

S. Sas Sarma, J. J. Quinn, Phys. Rev. B 25, 7603 (1982).
[CrossRef]

1981 (1)

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

1978 (1)

R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
[CrossRef]

1974 (1)

A. L. Fetter, Ann. Phys. (N.Y.) 88, 1 (1974).
[CrossRef]

1970 (1)

A. S. Barker, A. A. Ballman, J. A. Ditzenberger, Phys. Rev. B 2, 4233 (1970).
[CrossRef]

1962 (1)

M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

D. H. Auston, Appl. Phys. Lett. 43, 713 (1983).
[CrossRef]

D. A. Kleinman, D. H. Auston, IEEE J. Quantum Electron. QE-20, 964 (1983).

Ballman, A. A.

A. S. Barker, A. A. Ballman, J. A. Ditzenberger, Phys. Rev. B 2, 4233 (1970).
[CrossRef]

Barker, A. S.

A. S. Barker, A. A. Ballman, J. A. Ditzenberger, Phys. Rev. B 2, 4233 (1970).
[CrossRef]

Bass, M.

M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

Dingle, R.

R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
[CrossRef]

Ditzenberger, J. A.

A. S. Barker, A. A. Ballman, J. A. Ditzenberger, Phys. Rev. B 2, 4233 (1970).
[CrossRef]

Fetter, A. L.

A. L. Fetter, Ann. Phys. (N.Y.) 88, 1 (1974).
[CrossRef]

Fork, R. L.

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Franken, P. A.

M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Gabel, C. W.

J. A. Valdmanis, G. A. Mourou, C. W. Gabel, IEEE J. Quantum Electron. QE-19, 664 (1983).
[CrossRef]

Gossard, A. C.

R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
[CrossRef]

Greene, B. I.

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Kaminow, I.

I. Kaminow, in Handbook of Lasers, J. Pressley, ed.(Chemical Rubber Company, Cleveland, Ohio, 1971), Chap. 15.

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

D. A. Kleinman, D. H. Auston, IEEE J. Quantum Electron. QE-20, 964 (1983).

Mourou, G. A.

J. A. Valdmanis, G. A. Mourou, C. W. Gabel, IEEE J. Quantum Electron. QE-19, 664 (1983).
[CrossRef]

Quinn, J. J.

S. Sas Sarma, J. J. Quinn, Phys. Rev. B 25, 7603 (1982).
[CrossRef]

Sas Sarma, S.

S. Sas Sarma, J. J. Quinn, Phys. Rev. B 25, 7603 (1982).
[CrossRef]

Shank, C. V.

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Stormer, H. L.

R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
[CrossRef]

Valdmanis, J. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

J. A. Valdmanis, G. A. Mourou, C. W. Gabel, IEEE J. Quantum Electron. QE-19, 664 (1983).
[CrossRef]

Ward, J. F.

M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Weinreich, G.

M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Wiegmann, W.

R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
[CrossRef]

Ann. Phys. (N.Y.) (1)

A. L. Fetter, Ann. Phys. (N.Y.) 88, 1 (1974).
[CrossRef]

Appl. Phys. Lett. (3)

R. Dingle, H. L. Stormer, A. C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33, 665 (1978);H. L. Stormer, J. Phys. Soc. Jpn. 49A, 1013 (1980).
[CrossRef]

D. H. Auston, Appl. Phys. Lett. 43, 713 (1983).
[CrossRef]

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. A. Valdmanis, G. A. Mourou, C. W. Gabel, IEEE J. Quantum Electron. QE-19, 664 (1983).
[CrossRef]

D. A. Kleinman, D. H. Auston, IEEE J. Quantum Electron. QE-20, 964 (1983).

Phys. Rev. B (2)

A. S. Barker, A. A. Ballman, J. A. Ditzenberger, Phys. Rev. B 2, 4233 (1970).
[CrossRef]

S. Sas Sarma, J. J. Quinn, Phys. Rev. B 25, 7603 (1982).
[CrossRef]

Phys. Rev. Lett. (2)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984).
[CrossRef]

M. Bass, P. A. Franken, J. F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Other (2)

I. Kaminow, in Handbook of Lasers, J. Pressley, ed.(Chemical Rubber Company, Cleveland, Ohio, 1971), Chap. 15.

For recent reviews, see Infrared and Millimeter Waves, K. J. Button, ed.(Academic, New York, 1983), Vol. 8.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic of the experiment used to generate and detect short far-infrared bursts by Čerenkov radiation from femtosecond optical pulses in lithium tantalate. The Čerenkov cone of radiation propagates away from the pump pulse in a direction θc with a velocity that is determined by the ratio of the low-frequency index of refraction to the optical group velocity. A probe pulse measures the small birefringence that is due to the electro-optic effect induced by the electric field of the far-infrared transient as it moves in synchronism with the Čerenkov wave front.

Fig. 2
Fig. 2

Experimental observation of the electric-field waveform of the far-infrared transient measured by the method illustrated in Fig. 1. A lithium tantalate crystal 1 mm in length was used with 100-fsec optical pulses focused to a beam waist of approximately 7 μm. The peak amplitude of the electric field was approximately 10 V/cm. The true shape of the far-infrared pulse is faster than the result shown here because of the convolution with the probe pulse.

Fig. 3
Fig. 3

Fourier spectrum of the experimental waveform in Fig. 2.

Fig. 4
Fig. 4

Schematic of the experimental arrangement used to measure the far-infrared dielectric properties of materials by coherent time-domain spectroscopy. The material to be measured is placed in contact with the electro-optic crystal in which the far-infrared pulses are generated and measured. The Čerenkov wave front is measured by the probing optical pulse before and after being reflected from the interface. The dielectric response of the material is determined by a numerical Fourier transform of the reflected waveform. To calibrate the timing, a reference sample such as a metal film or the open interface is substituted for the sample.

Fig. 5
Fig. 5

Reflected waveform from a germanium crystal containing approximately 2 × 1016 cm−3 free electrons (solid curve). For comparison, the reflected waveform from a gold film (dashed curve) is also shown.

Fig. 6
Fig. 6

Expanded view of the reflected waveforms from (a) the n-type germanium sample and (b) the gold film shown in Fig. 5. An additional waveform (c) that is due to the reflection from the open interface is also plotted. The influence of the plasma resonance shows in the strong dispersion, which distorts the reflected waveform.

Fig. 7
Fig. 7

The real (lower curve) and imaginary (upper curve) parts of the complex dielectric response of the doped germanium sample. These results are obtained by inverting the Fresnel reflection formula using the waveform in Fig. 6 to determine the frequency-dependent reflectivity.

Fig. 8
Fig. 8

Composition of the multi-quantum-well superlattice grown by molecular beam epitaxy. The reflected waveforms from this sample are shown in Fig. 9. An effective medium approximation was used to represent the superlattice by a uniform film whose far-infrared properties are determined by the average areal density ns and momentum relaxation time τ.

Fig. 9
Fig. 9

Reflected waveforms from the multi-quantum-well-superlattice illustrated in Fig. 8 (solid curve) and a bare GaAs:Cr substrate reference (dashed curve), which has been scaled to represent an equivalent incident waveform.

Equations (11)

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E ( t ) = n 0 2 r 33 W p c 2 τ 2 ( 2 cot θ c υ τ r ) 1 / 2 × U ( 2 , 2 t τ ) exp ( t 2 2 τ 2 ) ,
τ = ( τ p 2 + w 2 tan 2 θ c υ 2 ) 1 / 2 ,
E ( ω ) = ω 3 / 2 exp ( ω 2 τ 2 4 ) .
f m = 3 / ( 2 π τ ) .
r ( ω ) = + d t E ( t ) e i ω t + d t E ( t ) e i ω t r ( ω ) ,
r ( ω ) = cos θ i [ 2 ( ω ) / 1 sin 2 θ i ] 1 / 2 cos θ i + [ 2 ( ω ) / 1 sin 2 θ i ] 1 / 2 ,
2 ( ω ) 1 = sin 2 θ i + cos 2 θ i [ 1 r ( ω ) 1 + r ( ω ) ] 2
( ω ) = s n e 2 m * 1 ω ( ω + i τ ) ,
J ( t ) = s E ( t ) t + n e 2 m * t d t E ( t ) exp [ ( t t ) / τ ] .
( ω , k ) = s n s e 2 k m * 1 ω ( ω + i τ ) × sinh ( k a ) cosh ( k a ) cos ( k a ) ,
r ( ω ) = r 12 + r 23 exp ( i 2 d 2 cos θ 2 ) 1 + r 12 r 23 exp ( i 2 d 2 cos θ 2 ) ,

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