- Email: [email protected]
Measurement of the shape of picosecond light pulses using stimulated Raman scattering
Volume
3, number 4
OPTICS
MEASUREMENT
OF USING
THE
June 1971
COMMUNICATIONS
SHAPE
STIMULATED
OF
PICOSECOND
RAMAN
LIGHT
PULSES
SCATTERING
D. VON DER LINDE and A. LAUBEREAU Physik-Department
der
Technischen Received
Universitiit 26 April
Miinchen,
Munich,
Germany
19’71
The vibrational amplitude and the intensity of the Stokes light generated by transient stimulated Raman scattering rise and decay very rapidly. The time duration of the Stokes pulse and of the phonon pulse is much shorter than the pump pulse duration. It is shown that both phonon and Stokes pulses can be used to probe the shape of ultrashor; light pulses.
In the past the approximate duration of picosecond light pulses was determined from measurements of the intensity autocorrelation function [l]. It is well known, however, that intensity autocorrelation measurements do not allow to recover uniquely the detailed pulse shape [2]. In this paper a novel method for the experimental determination of the shape of ultrashort light pulses [3] is discussed in detail. In addition we present another improved experimental technique which allows to the pulse shape to be determined from a single measurement (e.g. from a single laser shot) The techniques described here are particularly well suited for the investigation of single picosecond light pulses, e.g. pulses obtained from pulse trains of lasers simultaneously mode-locked and Qswitched by a saturable dye. The basic principle of the method is illustrated in fig. 1. An intense ultrashort light pulse generates optical phonons by stimulated Raman scattering in a Raman active sample. If the damping time T of the vibrational field is much shorter than the width t of the pump pulse a very sharply rising and c?ecaying “phonon pulse” is coherently excited in the Raman sample. The light pulse to be studied is split from the original light pulse by decoupling a small fraction of the pump beam with the help of a beam splitter. This weak light pulse is delayed (or advanced) with respect to the pump pulse and interacts with the vibrational field in the Raman sample to give coherently scattered Stokes and anti-Stokes light. By measuring - as a function of delay time t D - the Raman light scattered from the vibrational field the extremely steep phonon pulse serves as a probe for the light pulse under investigation.
Fig. 1. Schematic of the experiment. Beam splitter BS; two glass prisms provide a variable optical delay VD; fixed optical delay FD; filter transmitting Raman scattered light FI; photomultiplier PM.
The magnitude and time dependence of the amplitude of the vibrational field can be calculated from the theory of transient stimulated Raman scattering [4]. As an example the square of the vibrational amplitude, Q2, is plotted as a function of time t = t’ - z/up in fig. 2a assuming a sech-shaped pump pulse, i.e. IL(t)/lL(O) z sech(t/0.382 tP) and tp/r = 32. ,z is the position in the Raman sample and up the velocity of propagation of the pump pulse. Note the position of the maximum of Q2(t) in fig. 2a: as a consequence of the transient nature of the stimulated Raman scattering the maximum is shifted to t N t p/4 [4]. The pump pulse is also shown in fig. 2a for comparison (broken line). It can readily be seen that the width (fwhm) of the phonon pulse is approximately l/4 of the width of the original pump pulse, and - most important - the wings of the phonon pulse are considerably steeper than the wings of the pump pulse. During a time interval At =tp the phonon pulse rises and decays by three orders of ten, while the intensity of the pump pulse varies only by a factor of two. This shortening of the phonon pulse (as compared to the original pulse) does not strongly depend on the original pulse 279
Volume 3, number 4
OPTICS COMMUNICATIONS
ZAS(Z) cc Q2(t) ZL(Z - ZD) .
June 1971
(1)
The signal S (t D) obtained from the photomultiplier (PM in fig. 1) is given by the time integrated Raman intensity:
tm S(fD) = const J Q 2(t) ZL(Z - tD)dt . -03
(2)
Because of the short duration and the steep wings of Q2(t) the last expression represents a very good approximation of the original pulse shape ZL( t), i.e. the function Q2(Z) can be regarded as an approximation of the g-function: S(ZD) ” ZL( -tD, .
Fig. 2. (a) Calculated vibrational amplitude: (Q/Q,,,)” versus time t/tp (full curve). The broken curve shows the original light pulse IL(t)/ZL(O) = sech(t/0.382 tp) (lo) Calculated intensity of the Stokes pulse: Is(t)/ls max versus time l/t, (full curve). The Stokes pulse is generated by the second harmonic. The original pulse (broken curve) has the same shape as in fig. 2a.
shape. On account of the very rapid (approximately exponential) growth of the vibrational amplitude Q as a function of the pump intensity, sharply rising and decaying phonon pulses are always generated, regardless of the specific choice of the input pulse shape. It is important to emphasize that this situation does not hold for the saturation region, where depletion of the pump pulse occurs. In this case the amplitude Q(t) tends to follow the pump pulse, i.e. phonon pulses very similar in shape to the pump pulses are observed. In the experiment the intensity of the Raman scattered light (e.g. the anti-Stokes intensity) is observed; it is proportional to the square of the vibrational amplitude Q-2(t) times the intensity ZL(~-~D) of the delayed laser pulse:
280
(3)
It should be noted that for positive delay b > 0 portions of the original pulse with t < 0 are probed, e.g. the time scale t and the delay time scale tD have opposite sign. As a specific example the Raman scattered intensity has been calculated for an asymmetric pulse shape, e.g. a pulse rising with a gaussian function and decaying with a sech2-function. It is assumed that the probing phonon pulse is generated by stimulated Raman scattering of an intense pump pulse with the same time dependence. The normalized scattered intensity S(b) (full curve) and the intensityZ~(t) of the original pulse (broken curve) are compared in fig. 3a. Delay time tD/tp is plotted at the upper scale and time Z/Zp at the lower scale. It can be seen from fig, 3a that the S(tD) curve represents to a good approximation the original pulse shape: In the region of the maximum the S(ZD) and IL(t) curves are nearly identical, e.g. the width (fwhm) Of S(ZD) iS 1.05 tp, where tp is the half width of the original pulse. The maximum of S(tD) is positioned at 1D = 0.2 tp. A factor of 103 below the maximum the relative deviation [S(tD) -ZD(t )]/ZL@) is approximately 10%. On the other hand the slope of the exponential branch of S(tD) is the same as the slope of the corresponding part of IL(~), i.e. the decay time of the pulse is reproduced with very high accuracy. It should be emphasized that the method described here allows the reconstruction of the pulse shape for a wide class of possible pulse forms. It does not depend on the specific choice of pulse forms, provided that no pronounced additional amplitude modulation is present. (The time resolution is approximately given by the width of the phonon probe pulse which is typically 0.2 t p -0.3 tp. ) In conclusion we can say that it is possible to determine the detailed shape of ultrashort light pulses by measuring - as a function of
Volume 3, number 4
1
OPTICS COMMUNICATIONS -DELAY 0
TlJ~Ei& I
s i= w
10-l
B
E 10-Z =
w 9
10-a
June 1971
mental value of the contrast ratio (2.95 l O.Z), i.e. the peak to background ratio is estimated to be less than 10-3. It is important to note that these experimental observations exclude a significant substructure of the pulses. In the Raman experiment we observed the antiStokes scattering from molecular vibrations in ethyl alcohol [3]. The molecular vibrations were pumped by an intense green light pulse (X = 0.53 b) of the same shape as the pulse under study. In fig. 4 the experimentally observed anti-Stokes intensity has been plotted as a function of the delay time tD (upper scale). The experimental points are averages of about ten individual measurements. The anti-Stokes intensity increases eXpOIIentially between TV = - 15 psec and tD = = - 2 psec (full curve). The maximum is observed at Q) = 1.5 psec. For larger delays (1.5 psec < < tD < 20 psec) the experimental points are well approximated by a guassian function (full curve). We conclude from fig. 4 that our light pulses rise sharply with a gaussian function and decay more slowly proportional to an exponential function, i.e. there is a weak asymmetry in our light pulses. The pulse width tp (fwhm) inferred from this measurement is 8 psec in excellent agreement with the TPF-experiments. Note the logarithmic scale in fig. 4: We have measured the average temporal profile of our pulses down to a level of less than 1O-3 below the maximum both for the rising and decaying portion of the pulse. The pulse asymmetry de-
Fig. 3. (a) Calculated intensity of the Raman scattered light: S(~D) versus delay time t&tp (full curve, upper scale). Original laser pulse for comparison: IL(~) versus time (broken curve, lower scale). Ordinate scale: maxima normalized to unity. (b) Calculated intensity of fluorescence for the TPF-experiment with Stokes and fundamental laser beam: F(tD) versus delay time t&t, (full curve, upper scale). Original light pulse Q,(t) m sin2@t/tp)/(&/tp)2 (broken curve). Ordinate scale: maxima normalized to unity.
the delay time - the intensity of the Raman light scattered from the phonon (probe) pulse. The method has been tested experimentally using frequency doubled single picosecond pulses switched from the output of an Nd-glass laser mode-locked by a saturable dye. The pulses have been thoroughly investigated in previous experiments [5]. Measurements of the spectral intensity distribution and TPF- experiments indicate: (i) the frequency width of the single pulses are well accounted for by the observed pulse duration; (ii) good mode-locking is inferred from the experi-
IIEMV
L h I(
TIME t [ps]
TIME t&s] -5
0
-10
-15
-20
-
Fig. 4. Experimentally measured anti-Stokes intensity normalized to unity: S(tD) as a function of delay time tD (upper scale).
281
Volume
3, number
4
OPTICS
tected in this way can be well accounted for regarding the finite relaxation time (T = 10 psec) of the saturable dye in the mode-locked laser generator. During the round trip in the resonator the leading edge of the pulse always interacts with the dye molecules in the ground state. On the other hand the trailing edge interacts with the dye molecules in a partially bleached state, if the life time T of the excited state of the dye molecules is of the order of the pulse duration tp or less, Tr’tp. In this case the leading edge becomes considerably steeper than the trailing one [6]. We now turn to the discussion of a modification of the technique described above. It will be shown that the shape of an individual picosecond pulse can be determined from a single measurement. This technique takes advantage of the Stokes light pulse which is generated simultaneously with the phonon pulse in the stimulated Raman scattering. Consider the following two-photon fluorescence experiment: the fundamental laser beam and a Stokes pulse generated by the second harmonic of the fundamental pulse are made to collide in a fluorescent dye. Let us assume that the energy difference AE between the lowest excited levels and the ground state of the dye is larger than 2ttwL (AE > ~RwL). Then two photon absorption of the fundamental laser frequency does not occur. If the Stokes intensity IS is sufficiently small (IS <
COnSt
Jrn Is(t) --co
282
zL(f-tD)
dk
;
June 1971
COMMUNICATIONS
(ZS, ZL intensity of the Stokes and the fundamental laser pulse, respectively; z spatial coordinate in the propagation direction of the laser pulse; z?S, !lL group velocities in the dye.) By analogy with eq. (2) it can be shown that the right hand side of eq. (4) also represents a good approximation of the original pulse shape: E(tD) = ZL( -tD) .
(5)
Clearly this approximation does not hold for the saturation region of stimulated Raman scattering. The advantages of the two-photon fluorescence experiment with the Stokes pulse probing the fundamental laser pulse are summarized as follows: (i) the pulse shape is obtained from a single laser shot. It is possible to detect fluctuations or substructures of the pulse shape, which might be averaged out in an experiment requiring a large number of laser shots; (ii) the time resolution is improved. Using the second harmonic for the Stokes generation Stokes pulse durations oft s = 0.1 tp to 0.2 Zp are readily obtained. As an example we have calculated the intensity of fluorescence E( ZD) for a fundamental laser pulse with the time dependence ZL(Z) IL($
sm2 (nt/;p )_ =
(Ft/t ) P i.e. a laser pulse exhibiting secondary intensity maxima and zero points. In fig. 3b the calculated fluorescence profile (full curve) and the original pulse shape (broken curve) are shown. It is seen that the secondary maxima are clearly resolved. On account of the logarithmic scale in fig. 3b the small deviations of F (TV) from IL(t) near the zero points are readily shown, e.g. at t/t, = 1 the approximate curve F(tD) does not drop to zero exactly but only to a value of about 5 X 10m3. Finally, it is interesting to compare our experimental results with recent work of other authors. Treaty [8] has detected a pulse asymmetry probing the laser pulse with the compressed pulse. Photographs of the two-photon fluorescence seemed to support a pulse shape with the trailing edge considerably shorter than the leading edge, contrary to our results. Pulse asymmetry of a similar kind has been reported [9] in an experiment involving third order autocorrelation functions [lo]. No attempt has been made in refs. [8] and ]9] to explain this asymmetry. Very recently
Volume 3. number 4
June 1971
OPTICS COMMUNICATIONS
a technique involving even higher correlation functions (fifth order) [ll] has been reported*. No clear asymmetry was found in this work. We feel that the interpretation of the results in refs. [8, 9] and [ll] is difficult since: (i) the total mode-locked pulse train was used; (ii) a strong frequency chirp was reported to be present in these experiments. Our experimental situation is obviously less complex. We worked with single picosecond pulses switched from a well defined position near the maximum of the pulse train. In addition the pulses had no significant frequency modulation. Therefore, we believe that the experimental curve shown in fig. 4 indeed represents a good approximation of the pulse shape.
* It should be stressed that the integrals (2) and (4) are not simple autocorrelation functions. For instance, if we approximate IS(t) in (4) by the stationary solution IS(t) = IO exp[GIL(t)] and expand the exponential function, eq. (4) can be written as a series of higher order correlation functions.
The authors gratefully acknowledge discussions with Professor W. Kaiser.
valuable
REFERENCES [l] M.Maier,
W.Kaiser and J.A.Giordmaine, Phys. Rev. Letters 17 (1966) 1275; J.A.Armstrong,‘ Appi. Phys. Letters 10 (1967) 16;
H.P.Weber,
J.Appl.Phys.38 (1967) 2231;
J.A.Giordmaine,[email protected]&, S.L.Shapiro and K.W.Wecht, Appl.Phys.Letters 11 (1967) 216. [2] H. P. Weber, J.Appl. Phys. 39 (1968) 6041. [ 31 D. von der Linde, A. Laubereau and W. Kaiser, Phys.Rev. Letters 26 (1971) 955. .141 - R. L. Carman, F. Shimizu, C. S. Wang and N. Bloembergen, Phys. Rev. A2 (1976) 60. 151 . - D. von der Linde, 0. Bernecker and W Kaiser, Opt.Commun.2 (1970) 149. [6] V. S. Letokhov, Soviet Phys. JETP 28 (1969) 562. [7] P.M.Rentzepis and M.A.Duguay, Appl.Phys. Letters 11 (1967) 218. [8] E.B.Treacy, Phys.Letters 28A (1968) 34. [9] J.W.Shelton and Y.R.Shen, Phys.Rev.Letters 26 (1971) 538. [lo] H.P.Weber and R.D&dliker, Phys.Letters 28A (1968) 77. [ll] D.M.Auston, Appl.Phys.Letters 18 (1971) 249.
283
3, number 4
OPTICS
MEASUREMENT
OF USING
THE
June 1971
COMMUNICATIONS
SHAPE
STIMULATED
OF
PICOSECOND
RAMAN
LIGHT
PULSES
SCATTERING
D. VON DER LINDE and A. LAUBEREAU Physik-Department
der
Technischen Received
Universitiit 26 April
Miinchen,
Munich,
Germany
19’71
The vibrational amplitude and the intensity of the Stokes light generated by transient stimulated Raman scattering rise and decay very rapidly. The time duration of the Stokes pulse and of the phonon pulse is much shorter than the pump pulse duration. It is shown that both phonon and Stokes pulses can be used to probe the shape of ultrashor; light pulses.
In the past the approximate duration of picosecond light pulses was determined from measurements of the intensity autocorrelation function [l]. It is well known, however, that intensity autocorrelation measurements do not allow to recover uniquely the detailed pulse shape [2]. In this paper a novel method for the experimental determination of the shape of ultrashort light pulses [3] is discussed in detail. In addition we present another improved experimental technique which allows to the pulse shape to be determined from a single measurement (e.g. from a single laser shot) The techniques described here are particularly well suited for the investigation of single picosecond light pulses, e.g. pulses obtained from pulse trains of lasers simultaneously mode-locked and Qswitched by a saturable dye. The basic principle of the method is illustrated in fig. 1. An intense ultrashort light pulse generates optical phonons by stimulated Raman scattering in a Raman active sample. If the damping time T of the vibrational field is much shorter than the width t of the pump pulse a very sharply rising and c?ecaying “phonon pulse” is coherently excited in the Raman sample. The light pulse to be studied is split from the original light pulse by decoupling a small fraction of the pump beam with the help of a beam splitter. This weak light pulse is delayed (or advanced) with respect to the pump pulse and interacts with the vibrational field in the Raman sample to give coherently scattered Stokes and anti-Stokes light. By measuring - as a function of delay time t D - the Raman light scattered from the vibrational field the extremely steep phonon pulse serves as a probe for the light pulse under investigation.
Fig. 1. Schematic of the experiment. Beam splitter BS; two glass prisms provide a variable optical delay VD; fixed optical delay FD; filter transmitting Raman scattered light FI; photomultiplier PM.
The magnitude and time dependence of the amplitude of the vibrational field can be calculated from the theory of transient stimulated Raman scattering [4]. As an example the square of the vibrational amplitude, Q2, is plotted as a function of time t = t’ - z/up in fig. 2a assuming a sech-shaped pump pulse, i.e. IL(t)/lL(O) z sech(t/0.382 tP) and tp/r = 32. ,z is the position in the Raman sample and up the velocity of propagation of the pump pulse. Note the position of the maximum of Q2(t) in fig. 2a: as a consequence of the transient nature of the stimulated Raman scattering the maximum is shifted to t N t p/4 [4]. The pump pulse is also shown in fig. 2a for comparison (broken line). It can readily be seen that the width (fwhm) of the phonon pulse is approximately l/4 of the width of the original pump pulse, and - most important - the wings of the phonon pulse are considerably steeper than the wings of the pump pulse. During a time interval At =tp the phonon pulse rises and decays by three orders of ten, while the intensity of the pump pulse varies only by a factor of two. This shortening of the phonon pulse (as compared to the original pulse) does not strongly depend on the original pulse 279
Volume 3, number 4
OPTICS COMMUNICATIONS
ZAS(Z) cc Q2(t) ZL(Z - ZD) .
June 1971
(1)
The signal S (t D) obtained from the photomultiplier (PM in fig. 1) is given by the time integrated Raman intensity:
tm S(fD) = const J Q 2(t) ZL(Z - tD)dt . -03
(2)
Because of the short duration and the steep wings of Q2(t) the last expression represents a very good approximation of the original pulse shape ZL( t), i.e. the function Q2(Z) can be regarded as an approximation of the g-function: S(ZD) ” ZL( -tD, .
Fig. 2. (a) Calculated vibrational amplitude: (Q/Q,,,)” versus time t/tp (full curve). The broken curve shows the original light pulse IL(t)/ZL(O) = sech(t/0.382 tp) (lo) Calculated intensity of the Stokes pulse: Is(t)/ls max versus time l/t, (full curve). The Stokes pulse is generated by the second harmonic. The original pulse (broken curve) has the same shape as in fig. 2a.
shape. On account of the very rapid (approximately exponential) growth of the vibrational amplitude Q as a function of the pump intensity, sharply rising and decaying phonon pulses are always generated, regardless of the specific choice of the input pulse shape. It is important to emphasize that this situation does not hold for the saturation region, where depletion of the pump pulse occurs. In this case the amplitude Q(t) tends to follow the pump pulse, i.e. phonon pulses very similar in shape to the pump pulses are observed. In the experiment the intensity of the Raman scattered light (e.g. the anti-Stokes intensity) is observed; it is proportional to the square of the vibrational amplitude Q-2(t) times the intensity ZL(~-~D) of the delayed laser pulse:
280
(3)
It should be noted that for positive delay b > 0 portions of the original pulse with t < 0 are probed, e.g. the time scale t and the delay time scale tD have opposite sign. As a specific example the Raman scattered intensity has been calculated for an asymmetric pulse shape, e.g. a pulse rising with a gaussian function and decaying with a sech2-function. It is assumed that the probing phonon pulse is generated by stimulated Raman scattering of an intense pump pulse with the same time dependence. The normalized scattered intensity S(b) (full curve) and the intensityZ~(t) of the original pulse (broken curve) are compared in fig. 3a. Delay time tD/tp is plotted at the upper scale and time Z/Zp at the lower scale. It can be seen from fig, 3a that the S(tD) curve represents to a good approximation the original pulse shape: In the region of the maximum the S(ZD) and IL(t) curves are nearly identical, e.g. the width (fwhm) Of S(ZD) iS 1.05 tp, where tp is the half width of the original pulse. The maximum of S(tD) is positioned at 1D = 0.2 tp. A factor of 103 below the maximum the relative deviation [S(tD) -ZD(t )]/ZL@) is approximately 10%. On the other hand the slope of the exponential branch of S(tD) is the same as the slope of the corresponding part of IL(~), i.e. the decay time of the pulse is reproduced with very high accuracy. It should be emphasized that the method described here allows the reconstruction of the pulse shape for a wide class of possible pulse forms. It does not depend on the specific choice of pulse forms, provided that no pronounced additional amplitude modulation is present. (The time resolution is approximately given by the width of the phonon probe pulse which is typically 0.2 t p -0.3 tp. ) In conclusion we can say that it is possible to determine the detailed shape of ultrashort light pulses by measuring - as a function of
Volume 3, number 4
1
OPTICS COMMUNICATIONS -DELAY 0
TlJ~Ei& I
s i= w
10-l
B
E 10-Z =
w 9
10-a
June 1971
mental value of the contrast ratio (2.95 l O.Z), i.e. the peak to background ratio is estimated to be less than 10-3. It is important to note that these experimental observations exclude a significant substructure of the pulses. In the Raman experiment we observed the antiStokes scattering from molecular vibrations in ethyl alcohol [3]. The molecular vibrations were pumped by an intense green light pulse (X = 0.53 b) of the same shape as the pulse under study. In fig. 4 the experimentally observed anti-Stokes intensity has been plotted as a function of the delay time tD (upper scale). The experimental points are averages of about ten individual measurements. The anti-Stokes intensity increases eXpOIIentially between TV = - 15 psec and tD = = - 2 psec (full curve). The maximum is observed at Q) = 1.5 psec. For larger delays (1.5 psec < < tD < 20 psec) the experimental points are well approximated by a guassian function (full curve). We conclude from fig. 4 that our light pulses rise sharply with a gaussian function and decay more slowly proportional to an exponential function, i.e. there is a weak asymmetry in our light pulses. The pulse width tp (fwhm) inferred from this measurement is 8 psec in excellent agreement with the TPF-experiments. Note the logarithmic scale in fig. 4: We have measured the average temporal profile of our pulses down to a level of less than 1O-3 below the maximum both for the rising and decaying portion of the pulse. The pulse asymmetry de-
Fig. 3. (a) Calculated intensity of the Raman scattered light: S(~D) versus delay time t&tp (full curve, upper scale). Original laser pulse for comparison: IL(~) versus time (broken curve, lower scale). Ordinate scale: maxima normalized to unity. (b) Calculated intensity of fluorescence for the TPF-experiment with Stokes and fundamental laser beam: F(tD) versus delay time t&t, (full curve, upper scale). Original light pulse Q,(t) m sin2@t/tp)/(&/tp)2 (broken curve). Ordinate scale: maxima normalized to unity.
the delay time - the intensity of the Raman light scattered from the phonon (probe) pulse. The method has been tested experimentally using frequency doubled single picosecond pulses switched from the output of an Nd-glass laser mode-locked by a saturable dye. The pulses have been thoroughly investigated in previous experiments [5]. Measurements of the spectral intensity distribution and TPF- experiments indicate: (i) the frequency width of the single pulses are well accounted for by the observed pulse duration; (ii) good mode-locking is inferred from the experi-
IIEMV
L h I(
TIME t [ps]
TIME t&s] -5
0
-10
-15
-20
-
Fig. 4. Experimentally measured anti-Stokes intensity normalized to unity: S(tD) as a function of delay time tD (upper scale).
281
Volume
3, number
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OPTICS
tected in this way can be well accounted for regarding the finite relaxation time (T = 10 psec) of the saturable dye in the mode-locked laser generator. During the round trip in the resonator the leading edge of the pulse always interacts with the dye molecules in the ground state. On the other hand the trailing edge interacts with the dye molecules in a partially bleached state, if the life time T of the excited state of the dye molecules is of the order of the pulse duration tp or less, Tr’tp. In this case the leading edge becomes considerably steeper than the trailing one [6]. We now turn to the discussion of a modification of the technique described above. It will be shown that the shape of an individual picosecond pulse can be determined from a single measurement. This technique takes advantage of the Stokes light pulse which is generated simultaneously with the phonon pulse in the stimulated Raman scattering. Consider the following two-photon fluorescence experiment: the fundamental laser beam and a Stokes pulse generated by the second harmonic of the fundamental pulse are made to collide in a fluorescent dye. Let us assume that the energy difference AE between the lowest excited levels and the ground state of the dye is larger than 2ttwL (AE > ~RwL). Then two photon absorption of the fundamental laser frequency does not occur. If the Stokes intensity IS is sufficiently small (IS <
COnSt
Jrn Is(t) --co
282
zL(f-tD)
dk
;
June 1971
COMMUNICATIONS
(ZS, ZL intensity of the Stokes and the fundamental laser pulse, respectively; z spatial coordinate in the propagation direction of the laser pulse; z?S, !lL group velocities in the dye.) By analogy with eq. (2) it can be shown that the right hand side of eq. (4) also represents a good approximation of the original pulse shape: E(tD) = ZL( -tD) .
(5)
Clearly this approximation does not hold for the saturation region of stimulated Raman scattering. The advantages of the two-photon fluorescence experiment with the Stokes pulse probing the fundamental laser pulse are summarized as follows: (i) the pulse shape is obtained from a single laser shot. It is possible to detect fluctuations or substructures of the pulse shape, which might be averaged out in an experiment requiring a large number of laser shots; (ii) the time resolution is improved. Using the second harmonic for the Stokes generation Stokes pulse durations oft s = 0.1 tp to 0.2 Zp are readily obtained. As an example we have calculated the intensity of fluorescence E( ZD) for a fundamental laser pulse with the time dependence ZL(Z) IL($
sm2 (nt/;p )_ =
(Ft/t ) P i.e. a laser pulse exhibiting secondary intensity maxima and zero points. In fig. 3b the calculated fluorescence profile (full curve) and the original pulse shape (broken curve) are shown. It is seen that the secondary maxima are clearly resolved. On account of the logarithmic scale in fig. 3b the small deviations of F (TV) from IL(t) near the zero points are readily shown, e.g. at t/t, = 1 the approximate curve F(tD) does not drop to zero exactly but only to a value of about 5 X 10m3. Finally, it is interesting to compare our experimental results with recent work of other authors. Treaty [8] has detected a pulse asymmetry probing the laser pulse with the compressed pulse. Photographs of the two-photon fluorescence seemed to support a pulse shape with the trailing edge considerably shorter than the leading edge, contrary to our results. Pulse asymmetry of a similar kind has been reported [9] in an experiment involving third order autocorrelation functions [lo]. No attempt has been made in refs. [8] and ]9] to explain this asymmetry. Very recently
Volume 3. number 4
June 1971
OPTICS COMMUNICATIONS
a technique involving even higher correlation functions (fifth order) [ll] has been reported*. No clear asymmetry was found in this work. We feel that the interpretation of the results in refs. [8, 9] and [ll] is difficult since: (i) the total mode-locked pulse train was used; (ii) a strong frequency chirp was reported to be present in these experiments. Our experimental situation is obviously less complex. We worked with single picosecond pulses switched from a well defined position near the maximum of the pulse train. In addition the pulses had no significant frequency modulation. Therefore, we believe that the experimental curve shown in fig. 4 indeed represents a good approximation of the pulse shape.
* It should be stressed that the integrals (2) and (4) are not simple autocorrelation functions. For instance, if we approximate IS(t) in (4) by the stationary solution IS(t) = IO exp[GIL(t)] and expand the exponential function, eq. (4) can be written as a series of higher order correlation functions.
The authors gratefully acknowledge discussions with Professor W. Kaiser.
valuable
REFERENCES [l] M.Maier,
W.Kaiser and J.A.Giordmaine, Phys. Rev. Letters 17 (1966) 1275; J.A.Armstrong,‘ Appi. Phys. Letters 10 (1967) 16;
H.P.Weber,
J.Appl.Phys.38 (1967) 2231;
J.A.Giordmaine,[email protected]&, S.L.Shapiro and K.W.Wecht, Appl.Phys.Letters 11 (1967) 216. [2] H. P. Weber, J.Appl. Phys. 39 (1968) 6041. [ 31 D. von der Linde, A. Laubereau and W. Kaiser, Phys.Rev. Letters 26 (1971) 955. .141 - R. L. Carman, F. Shimizu, C. S. Wang and N. Bloembergen, Phys. Rev. A2 (1976) 60. 151 . - D. von der Linde, 0. Bernecker and W Kaiser, Opt.Commun.2 (1970) 149. [6] V. S. Letokhov, Soviet Phys. JETP 28 (1969) 562. [7] P.M.Rentzepis and M.A.Duguay, Appl.Phys. Letters 11 (1967) 218. [8] E.B.Treacy, Phys.Letters 28A (1968) 34. [9] J.W.Shelton and Y.R.Shen, Phys.Rev.Letters 26 (1971) 538. [lo] H.P.Weber and R.D&dliker, Phys.Letters 28A (1968) 77. [ll] D.M.Auston, Appl.Phys.Letters 18 (1971) 249.
283