- Email: [email protected]
Time-delayed laser-induced double gratings with broadband lights
1 May 1995
OPTICS COMMUNICATIONS Optics Communications 116 ( 1995) 44348
Time-delayed laser-induced double gratings with broadband lights Xin Mi, Zuhe Yu, Qian Jiang, Panming Fu
’
Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China
Received 6 January 1995
Abstract We have observed beating between two broadband light sources of linewidths 4.3 nm and 2.3 nm by a method of time-delayed laser-induced double gratings. We explain the signal modulation by relating the signal intensity to the autocorrelation function of a field which consists of two frequency components.
1. Introduction Integrated-intensity gratings are involved in many nonlinear-optical experimental techniques. The integrated-intensity grating in the form of thermal gratings [l-3], which originate from the thermally induced fluctuations in the index of refraction precipitated by light absorption, can yield efficient phase-conjugate reflection in a typical four-wave mixing (FWM) experiment. Eichler et al. [ 21 studied the coherence time of a light pulse by measuring the FWM signal intensity as a function of the relative time delay between two beams which produce the thermal grating. Martin and Hellwarth [3] demonstrated an infrared to blue-green image conversion in a nondegenerate IWM setup with significant phase-conjugate return. The negative side of the thermal grating is that it may obscure the nonlinear optical effects which we are intended to study. A time-delayed method has been proposed to suppress the thermal background [ 451. Recently, we have performed experiments of timedelayed laser-induced double gratings (TDLIDG) mediated by thermal effects to study the beat between two independent laser sources [ 6,7]. In these experi-
’ To whom correspondence should be addressed. 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00088-7
ments the spatial interference between two gratings gives rise to a modulation of the FWM signal intensity when the relative time delay between pump beams is varied. TDLIDG has also been employed to obtain beating between the blue (488 nm) and the green (514.5 nm) lines of an argon ion laser, which shows a beat note of 3 1.2 fs period [ 81. Here, we demonstrate a TDLIDG with broadband lights. It is partly motivated by the recent interests in employing broadband light for the time-delayed FWM to study the ultrafast relaxation process in the time domain [ 9-121. Our results indicate that in this double frequencies time-delayed FWM, the signal intensity exhibits modulation even when the lasers have extremely broad linewidths. In principle, there are similarities between TDLIDG and the experiments performed by Eichler et al. [ 21. The time-delayed dependence of the FWM signal is studied in both experiments. The main difference between them is that in TDLIDG the pump beams consist of two frequency components, whereas in the experiment of Eichler et al. only one frequency component exists in the pump beams. According to the theory of Eichler et al. [ 21, the time-delayed dependence of the FWM signal is proportional to the absolute square of the autocorrelation function of the pump beams. In this paper we show that, due to the similari-
X. Mi et al. /Optics
444
Communications
ties between TDLIDG and the experiment of Eichler et al., TDLIDG can also be studied in the framework of the autocorrelation function of the pump beams. More specifically, we explain the signal modulation in TDLIDG by relating the signal intensity to the autocorrelation function of a field which consists of two frequency components.
116 (1995) 443-448
methanol and glycerin), the grating relaxation time has a value of approximately 104-lo5 ns for a crossing angle of about 2 degrees [ 131. In the case that the pulse width of the pump beams is much shorter than (Dq2)-‘,thefactorexp[ -Dq2(t-t’)] inEq. (3)can be neglected and we have T(r, t) =
2. Theory
dt' Vl(t’)
X We first establish a relationship between the timedelayed dependence of the FWM signal intensity and the autocorrelation function of the pump beams which was given for the first time by Eichler et al. [ 21. Let us consider a laser-induced thermal grating experiment. The electric fields of the pump beams (beams 1 and 2) can be expressed as ~i(r, t) = Vi(t) exp(*i
V:(t’)
(4)
.
Most materials exhibit a temperature-dependent refractive index. Consequently the spatial temperature modulation will become a spatial refractive-index modulation or phase grating,
An@,t)
=(g)jz) exp@z*r)
lr) ,
l2(r, t) = V,(t) exp( ti2 or) .
(1)
Here VI(t) and k, (V*(t) and k2) are the complex electric field and wave vector of beam 1 (beam 2). Beams 1 and 2 interfere in an absorbing material. In this case the spatially sinusoidal deposition of heat gives rise to a spatial modulation of the temperature in the medium. The variation in temperature through the sample T(r, t) obeys the diffusion equation [ 31, VI(t) V,*(t)
exp(iq*r)
.
dt' V,(t’)
X
V;(t’)
.
(3
We now assume that beams 1 and 2 originate from the same laser source with a relative time delay r between them, that is V,(t) = aV( t - T) and V2( t) = V(t) with a a constant. For a pulse laser V(t) can be written as V(t) =A(t) u( t), in which u(t) is anormalized, dimensionless statistical factor that includes the phase of the field and A(t) is a real, deterministic pulse-shape function. After taking ensemble average, we have
(2) Here q = k, -k,; D = P/PC, is the diffusion coefficient; p, C,, 0, cxand n are the mass density, the specific heat per unit mass, the thermal conductivity, the loss of the medium and the refractive index, respectively. The formal solution of Eq. (2) is T(r, t) = arnc exp(iq*r) ( 4TPC* 1 dt’ Vl(t’)
X
Vz(t’)
exp[ -Dq2(t-t’)]
dt’A(t’)
X
A(t’--7)
r(t’--7,
t’) ,
(6)
-cc
where r( tl, t2) = (u( tl) u * ( t2) ) is the statistical autocorrelation function of u( t) . For a random process with at least wide-sense stationarity, r( t,, t2) is a function only of the time difference r= t2 - tl. Therefore, we have
.
-cc f
(3) (Dq ‘) - ’ is the grating relaxation time due to the thermal diffusion. For a typical organic liquid (such as
xl33
dt’A(t’)
A(t’--7)
.
(7)
X. Mi et al. /Optics Communications 116 (1995) 443448
Eq. (7) indicates that ( An(r, t) ) will diminish quickly as the relative time delay between pump beams increased so that r is longer than the laser coherence time. In the case that the laser coherence time is much shorter than the laser pulse width, (Ahn(r, t) ) can be further approximated as
445
where S, (w) and S,( w) are the power spectral densities of the light with frequencies centered at wr and w2, respectively; while 71~and r], are constant values. In this case, we have a generalized autocorrelation function r(~)=~irr(r)
exp(iwlT)+r/J2(r)
exp(iw,T). (13)
Here, m C(T) =
Here, t w(t) =a
dt’ [A(P)]*
(9)
--m
is related to the energy density which has been deposited into the absorbing material. In other words, the amplitude of the phase grating depends on the integrated incident light power. The FWM signal is the result of the diffraction of a probe beam by the phase grating. The diffraction efficiency is proportional to the absolute square of the amplitude of the spatial refractive-index modulation. Therefore, we obtain the rdependence of the signal intensity I(T) a IT(r)
( 10)
I”.
It is worth mentioning that Eq. ( 10) is based on the second-order coherence function theory. The rigorous expression for the FWM signal intensity is proportional to the ensemble average of the absolute square of the amplitude of the phase grating ( 1An(r, t) I*), which involves fourth-order coherence function. The fourthorder theory reduces to the second-order theory in the case that the laser pulse width is much longer than the laser coherence time [ 141. According to the Wiener-Kbinchin theory [ 1.51, the autocorrelation function and the power spectral density S(w) form a Fourier transform pair m r( 7) =
.
1’ dw S(w) exp(ior)
(11)
In the TDLIDG experiment, the pump beams consist of two frequency components w1 and w,. Therefore, S(w) can be expressed as S(o)
=rlr&(o)
[ dA, Si(Wi +Ai) exp(iA,T)
,
(14)
J
-co
+7-M*(@)
3
(12)
with Ai = w - wp Assuming that Si( w) is a symmetric function centered at Oi, i.e., Si( wi + Ai) = &( tii - Ai), then it can be proved easily that ri:.( 7) is real and symmetric. The T-dependence of the FWM signal intensity can be obtained from Eqs. ( 10) and ( 13)) which is
X [q
exp( -iwdr)
+v*
exp(iw,r)]
,
(1%
with q = r)*/ q1 and w, = wr - 0,. The above equation indicates that the FWM signal intensity modulates with frequency w, as 7 is varied. We note that in TDLIDG 71 is a complex value in general, where its phase depends on the phase factors introduced into the wr and w2 components of the pump beams [ 71.
3. Beating between two broadband
light sources
We have related the modulation of the FWM signal intensity to the autocorrelation function of a field which consists of two frequency components. Based on this theory, it becomes obvious that the laser linewidths are relatively irrelevant for the occurrence of the signal modulation so long as the two frequency components are separated by more than the sum of the linewidths. To go to the extreme, here we consider the case that the linewidths of the lasers are very broad so that they can be regarded as incoherent light sources. The experimental setup is basicaily the same as that in Ref. [ 71. Our sample was oxazine dye dissolved in ethanol (2 X 10m4 mole). Two broadband dye lasers (DLl and DL2) pumped by the second harmonic of a QuantaRay YAG laser were used to generate frequency at wi and C+ DLl and DL2 had linewidth 4.3 nm with wave-
446
X. Mi et al. /Optics Communications 116 (1995) 43-t&
frequency components. To confirm this, we measured the r-dependence of the FWM signal when beams 1 and 2 consisted of only one frequency component. Figs. 2a and b present the results when the pump beam frequencies are w1 and w,, respectively. The difference in the zero-time delay is obvious in these figures. It is due to the large difference between the wavelengths of DL, and DL2 so that the dispersion of the optical components becomes important. We fit our data by assuming that the power spectral density Si( w) has Gaussian lineshape
Relative Time Delay (fs) Si(0) = &/$exp[-(&Zyr], Pig. 1. FWM signal intensity versus relative time delay when the laser linewidths ofDL1 and DL2 are 4.3 nm and 2.3 nm, respectively. The solid curve is the theoretical curve with a1 = 1.1 X IV3 s-l, +=5.3X 10” s-l, &=0.598X lo-I3 s, w,=1.26X lOI s-‘, y=2.Oand +=126’.
length centered at 607.0 nm and linewidth 2.3 nm centered at 633.0 run, respectively. Both the laser outputs had pulse widths 5 ns and they were vertically polarized. A beam splitter was used to combine the w1 and o, components for the pump beams (beams 1 and 2). Beam 1 and beam 2 intersected in the sample with a small angle ( 1.5”) between them. The relative time delay 7 between beam 1 and beam 2 could be varied by an optical delay line controlled by a stepping motor. In these experiments we kept the energy of DLl and DL2 low enough (50 p,J) so that the induced gratings were not saturated. The pump beam spots had diameters of approximately 1 mm. The probe beam, which propagated along the opposite direction of beam 1, was derived from the second harmonic of the YAG laser and was horizontally polarized. It was focused to less than 0.1 mm on the sample. The FWM signal, which was also horizontally polarized, propagated along a direction almost opposite to that of beam 2. It was detected by a photodiode and then fed into an EG&G 4203 signal averager for data averaging. A computer was used for data processing and for controlling the stepping motor to vary the relative time delay. We measured the FWM signal intensity as a function of the relative time delay between beams 1 and 2. Fig. 1 presents the result. The FWM signal intensity modulates with period 50 fs. On the other hand, the temporal behavior of the FWM signal is quite asymmetric. We attribute this asymmetry to the difference in the zero time delay between beams 1 and 2 for the o1 and w2
(16)
so that
K.(r) =exp[
-(--$$r] .
(17)
1 IC Relative Time Delay Ifs)
-500
-250
0
250
500
Relative Time Delay (fs) Fig. 2. FWM signal intensity versus relative time delay when pump beams consist of only one frequency component, which is o1 for (a) and cy for (b). The solid curves are the theoretical curves with parameters LY,= 1.1 X lOI S-I for (a); a,=5.3X lOI s-’ and ST=0.598X10-13sfor(b).
X. Mi et al. /Optics Communications II4 (1995) 443-448
Here oj = Swi/2 with Swj the linewidth (fwhm) of the laser with frequency wi. The FWM signal intensities then become
(18) (19) for the cases in Figs. 2a and b, respectively. Here Sr denotes the difference in the zero time delay. The solid curves in Fig. 2 are the theoretical curves, in which we usetheparameters cr,=l.l X 1013 s-l, a,=5.3X101’ s-l and k-=0.598 X lo-l3 s. For an optical glass with refractive index IZ= 1.5, the refractive index at h 1= 607.0 nm is larger than that at h2 = 630.0 nm by approximately 0.001. A 59.8 fs delay between wi and w, corresponds to the propagation of beams in the glass (mainly the prism in the optical delay line) for a distance of about 2 cm. Taking the shift of the zero time delay into account and representing 77as y exp( - i+) , we have from Eq. ( 15) the r-dependence of the FWM signal intensity for TDLIDG
I(
exp[ - $(zr] +y2 exp[ - ip(sr)r]
+2yexp[
-(&r]
x cos( 6&Q-+ 4) .
wd
(lo”
W-1)
Fig. 3. Fourier spectrum corresponding to the data in Fig. 1.
w,= (1.27+0.0X) X 1014 s-l. We then fit our experimental result by using the following parameters: al=1.1X1013~-1,
c~~=~.~X~~‘~S-‘,~~=~.~~~X
lo-l3 s o = 1.26x 1014 s-’ y=2.0 and += 126”. Here the rel$ion 4 = w, 87 is satisfied. The solid curve in Fig. 2 is the theoretical curve. It predicts not only the modulation but also the asymmetry of the temporal behavior of the FWM signal. If we consider the discrepancy in the fit of the autocorrelation function of the pump beams (Fig. 2a), the agreement between the theoretical curve and the experimental data is quite good.
4. Conclusion
exp[ -(“:sr]
(20)
Here, &- and 4 are related by 4 = 0~87. This can be understood as follows. Consider the case that the optical paths of beams 1 and 2 are equal for the wi component. Owing to the difference between the zero time delays for the wi and w, frequency components, the optical paths between beams 1 and 2 will be different by c Sr for the w, component. As a result, there is an extra phase factor w2 87 for the 0, frequency component. The frequency difference w, can be determined by making a Fourier transformation of the TDLIDG data. Fig. 3 presents the Fourier spectrum corresponding to the data in Fig. 1, from which we deduce
In this paper, we explain the FWM signal modulation in TDLIDG by relating the signal intensity to the autocorrelation function of a field which consists of two frequency components. We performed TDLIDG experiments in which beating was observed between lights of linewidths 4.3 nm and 2.3 nm. Our results also show an asymmetry in the temporal behavior of the FWM signal. We attribute this asymmetry to the shift of the zero time delay which is due to the dispersion of the optical components.
Acknowledgements The authors gratefully acknowledge the financial support from the Chinese National Nature Sciences Foundation, the climbing program from the Chinese
448
X. Mi et al. I Optics Communications
Commission of Science and Technology, the Chinese Academy of Sciences and the Third World Academy of Sciences.
References [ 1] M.H. Garrett and H.J. Hoffman, J. Opt. Sot. Am. 73 (1983) 617. [ 21 H.J. Eichler, U. Klein and D. Langhans, Appl. Phys. 21 ( 1980) 215. [3] G. Martin and R.W. Hellwartb, Appl. Phys. Lett. 34 (1979) 371. [4] P. Fu, Z. Yu, X. Mi, Q. Jiang and Z. Zhang, Phys. Rev. A 46 (1992) 1530. [5] Z. Yu, X. Mi, Q. Jiang and P. Fu, Optics Comm. 107 (1994) 120.
116 (199.5) 443448
[61 X. Mi, Q. Jiang, Z. Yu and P. Fu, Optics Lett. 16 (1991) 1526. [7] X. Mi, Z. Yu, Q. Jiang, Z. Zhang and P. Fu, J. Opt. Sot. Am. B 10 (1993) 725. [8] X. Ding, Z. Zhang, Q. Jiang, X. Mi, Z. Yu, P. Fu, D. Shen and X. Ma, Optics Comm. 112 (1994) 55. [9] N. Morita and T. Yajima, Phys. Rev. A 30 (1984)
2525.
[lo] S. Asaka, H. Nakatsnka, M. Fujiwara and M. Matsuoka
Phys.
Rev. A 29 (1984) 2286.
[ 111 R. Beach, D. DeBeer and S.R. Hartmann, (1985)
Phys. Rev. A 32
3467.
[ 121 X. Mi, Z. Yu, Q. Jiang and P. Fu, Phys. Rev. A 48 (1993) 3203. [13] H. Eichler, G. Salje and H. Stahl, J. Appl. Phys. 44 (1973) 5383. [ 141 R. Trebino, E.K. Gustafson Am. B 3 (1986) [ 151 J.W. Goodman,
and A.E. Siegman,
J. Opt. Sot.
1295. Statistical Optics (Wiley, New York, 1985).
OPTICS COMMUNICATIONS Optics Communications 116 ( 1995) 44348
Time-delayed laser-induced double gratings with broadband lights Xin Mi, Zuhe Yu, Qian Jiang, Panming Fu
’
Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China
Received 6 January 1995
Abstract We have observed beating between two broadband light sources of linewidths 4.3 nm and 2.3 nm by a method of time-delayed laser-induced double gratings. We explain the signal modulation by relating the signal intensity to the autocorrelation function of a field which consists of two frequency components.
1. Introduction Integrated-intensity gratings are involved in many nonlinear-optical experimental techniques. The integrated-intensity grating in the form of thermal gratings [l-3], which originate from the thermally induced fluctuations in the index of refraction precipitated by light absorption, can yield efficient phase-conjugate reflection in a typical four-wave mixing (FWM) experiment. Eichler et al. [ 21 studied the coherence time of a light pulse by measuring the FWM signal intensity as a function of the relative time delay between two beams which produce the thermal grating. Martin and Hellwarth [3] demonstrated an infrared to blue-green image conversion in a nondegenerate IWM setup with significant phase-conjugate return. The negative side of the thermal grating is that it may obscure the nonlinear optical effects which we are intended to study. A time-delayed method has been proposed to suppress the thermal background [ 451. Recently, we have performed experiments of timedelayed laser-induced double gratings (TDLIDG) mediated by thermal effects to study the beat between two independent laser sources [ 6,7]. In these experi-
’ To whom correspondence should be addressed. 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00088-7
ments the spatial interference between two gratings gives rise to a modulation of the FWM signal intensity when the relative time delay between pump beams is varied. TDLIDG has also been employed to obtain beating between the blue (488 nm) and the green (514.5 nm) lines of an argon ion laser, which shows a beat note of 3 1.2 fs period [ 81. Here, we demonstrate a TDLIDG with broadband lights. It is partly motivated by the recent interests in employing broadband light for the time-delayed FWM to study the ultrafast relaxation process in the time domain [ 9-121. Our results indicate that in this double frequencies time-delayed FWM, the signal intensity exhibits modulation even when the lasers have extremely broad linewidths. In principle, there are similarities between TDLIDG and the experiments performed by Eichler et al. [ 21. The time-delayed dependence of the FWM signal is studied in both experiments. The main difference between them is that in TDLIDG the pump beams consist of two frequency components, whereas in the experiment of Eichler et al. only one frequency component exists in the pump beams. According to the theory of Eichler et al. [ 21, the time-delayed dependence of the FWM signal is proportional to the absolute square of the autocorrelation function of the pump beams. In this paper we show that, due to the similari-
X. Mi et al. /Optics
444
Communications
ties between TDLIDG and the experiment of Eichler et al., TDLIDG can also be studied in the framework of the autocorrelation function of the pump beams. More specifically, we explain the signal modulation in TDLIDG by relating the signal intensity to the autocorrelation function of a field which consists of two frequency components.
116 (1995) 443-448
methanol and glycerin), the grating relaxation time has a value of approximately 104-lo5 ns for a crossing angle of about 2 degrees [ 131. In the case that the pulse width of the pump beams is much shorter than (Dq2)-‘,thefactorexp[ -Dq2(t-t’)] inEq. (3)can be neglected and we have T(r, t) =
2. Theory
dt' Vl(t’)
X We first establish a relationship between the timedelayed dependence of the FWM signal intensity and the autocorrelation function of the pump beams which was given for the first time by Eichler et al. [ 21. Let us consider a laser-induced thermal grating experiment. The electric fields of the pump beams (beams 1 and 2) can be expressed as ~i(r, t) = Vi(t) exp(*i
V:(t’)
(4)
.
Most materials exhibit a temperature-dependent refractive index. Consequently the spatial temperature modulation will become a spatial refractive-index modulation or phase grating,
An@,t)
=(g)jz) exp@z*r)
lr) ,
l2(r, t) = V,(t) exp( ti2 or) .
(1)
Here VI(t) and k, (V*(t) and k2) are the complex electric field and wave vector of beam 1 (beam 2). Beams 1 and 2 interfere in an absorbing material. In this case the spatially sinusoidal deposition of heat gives rise to a spatial modulation of the temperature in the medium. The variation in temperature through the sample T(r, t) obeys the diffusion equation [ 31, VI(t) V,*(t)
exp(iq*r)
.
dt' V,(t’)
X
V;(t’)
.
(3
We now assume that beams 1 and 2 originate from the same laser source with a relative time delay r between them, that is V,(t) = aV( t - T) and V2( t) = V(t) with a a constant. For a pulse laser V(t) can be written as V(t) =A(t) u( t), in which u(t) is anormalized, dimensionless statistical factor that includes the phase of the field and A(t) is a real, deterministic pulse-shape function. After taking ensemble average, we have
(2) Here q = k, -k,; D = P/PC, is the diffusion coefficient; p, C,, 0, cxand n are the mass density, the specific heat per unit mass, the thermal conductivity, the loss of the medium and the refractive index, respectively. The formal solution of Eq. (2) is T(r, t) = arnc exp(iq*r) ( 4TPC* 1 dt’ Vl(t’)
X
Vz(t’)
exp[ -Dq2(t-t’)]
dt’A(t’)
X
A(t’--7)
r(t’--7,
t’) ,
(6)
-cc
where r( tl, t2) = (u( tl) u * ( t2) ) is the statistical autocorrelation function of u( t) . For a random process with at least wide-sense stationarity, r( t,, t2) is a function only of the time difference r= t2 - tl. Therefore, we have
.
-cc f
(3) (Dq ‘) - ’ is the grating relaxation time due to the thermal diffusion. For a typical organic liquid (such as
xl33
dt’A(t’)
A(t’--7)
.
(7)
X. Mi et al. /Optics Communications 116 (1995) 443448
Eq. (7) indicates that ( An(r, t) ) will diminish quickly as the relative time delay between pump beams increased so that r is longer than the laser coherence time. In the case that the laser coherence time is much shorter than the laser pulse width, (Ahn(r, t) ) can be further approximated as
445
where S, (w) and S,( w) are the power spectral densities of the light with frequencies centered at wr and w2, respectively; while 71~and r], are constant values. In this case, we have a generalized autocorrelation function r(~)=~irr(r)
exp(iwlT)+r/J2(r)
exp(iw,T). (13)
Here, m C(T) =
Here, t w(t) =a
dt’ [A(P)]*
(9)
--m
is related to the energy density which has been deposited into the absorbing material. In other words, the amplitude of the phase grating depends on the integrated incident light power. The FWM signal is the result of the diffraction of a probe beam by the phase grating. The diffraction efficiency is proportional to the absolute square of the amplitude of the spatial refractive-index modulation. Therefore, we obtain the rdependence of the signal intensity I(T) a IT(r)
( 10)
I”.
It is worth mentioning that Eq. ( 10) is based on the second-order coherence function theory. The rigorous expression for the FWM signal intensity is proportional to the ensemble average of the absolute square of the amplitude of the phase grating ( 1An(r, t) I*), which involves fourth-order coherence function. The fourthorder theory reduces to the second-order theory in the case that the laser pulse width is much longer than the laser coherence time [ 141. According to the Wiener-Kbinchin theory [ 1.51, the autocorrelation function and the power spectral density S(w) form a Fourier transform pair m r( 7) =
.
1’ dw S(w) exp(ior)
(11)
In the TDLIDG experiment, the pump beams consist of two frequency components w1 and w,. Therefore, S(w) can be expressed as S(o)
=rlr&(o)
[ dA, Si(Wi +Ai) exp(iA,T)
,
(14)
J
-co
+7-M*(@)
3
(12)
with Ai = w - wp Assuming that Si( w) is a symmetric function centered at Oi, i.e., Si( wi + Ai) = &( tii - Ai), then it can be proved easily that ri:.( 7) is real and symmetric. The T-dependence of the FWM signal intensity can be obtained from Eqs. ( 10) and ( 13)) which is
X [q
exp( -iwdr)
+v*
exp(iw,r)]
,
(1%
with q = r)*/ q1 and w, = wr - 0,. The above equation indicates that the FWM signal intensity modulates with frequency w, as 7 is varied. We note that in TDLIDG 71 is a complex value in general, where its phase depends on the phase factors introduced into the wr and w2 components of the pump beams [ 71.
3. Beating between two broadband
light sources
We have related the modulation of the FWM signal intensity to the autocorrelation function of a field which consists of two frequency components. Based on this theory, it becomes obvious that the laser linewidths are relatively irrelevant for the occurrence of the signal modulation so long as the two frequency components are separated by more than the sum of the linewidths. To go to the extreme, here we consider the case that the linewidths of the lasers are very broad so that they can be regarded as incoherent light sources. The experimental setup is basicaily the same as that in Ref. [ 71. Our sample was oxazine dye dissolved in ethanol (2 X 10m4 mole). Two broadband dye lasers (DLl and DL2) pumped by the second harmonic of a QuantaRay YAG laser were used to generate frequency at wi and C+ DLl and DL2 had linewidth 4.3 nm with wave-
446
X. Mi et al. /Optics Communications 116 (1995) 43-t&
frequency components. To confirm this, we measured the r-dependence of the FWM signal when beams 1 and 2 consisted of only one frequency component. Figs. 2a and b present the results when the pump beam frequencies are w1 and w,, respectively. The difference in the zero-time delay is obvious in these figures. It is due to the large difference between the wavelengths of DL, and DL2 so that the dispersion of the optical components becomes important. We fit our data by assuming that the power spectral density Si( w) has Gaussian lineshape
Relative Time Delay (fs) Si(0) = &/$exp[-(&Zyr], Pig. 1. FWM signal intensity versus relative time delay when the laser linewidths ofDL1 and DL2 are 4.3 nm and 2.3 nm, respectively. The solid curve is the theoretical curve with a1 = 1.1 X IV3 s-l, +=5.3X 10” s-l, &=0.598X lo-I3 s, w,=1.26X lOI s-‘, y=2.Oand +=126’.
length centered at 607.0 nm and linewidth 2.3 nm centered at 633.0 run, respectively. Both the laser outputs had pulse widths 5 ns and they were vertically polarized. A beam splitter was used to combine the w1 and o, components for the pump beams (beams 1 and 2). Beam 1 and beam 2 intersected in the sample with a small angle ( 1.5”) between them. The relative time delay 7 between beam 1 and beam 2 could be varied by an optical delay line controlled by a stepping motor. In these experiments we kept the energy of DLl and DL2 low enough (50 p,J) so that the induced gratings were not saturated. The pump beam spots had diameters of approximately 1 mm. The probe beam, which propagated along the opposite direction of beam 1, was derived from the second harmonic of the YAG laser and was horizontally polarized. It was focused to less than 0.1 mm on the sample. The FWM signal, which was also horizontally polarized, propagated along a direction almost opposite to that of beam 2. It was detected by a photodiode and then fed into an EG&G 4203 signal averager for data averaging. A computer was used for data processing and for controlling the stepping motor to vary the relative time delay. We measured the FWM signal intensity as a function of the relative time delay between beams 1 and 2. Fig. 1 presents the result. The FWM signal intensity modulates with period 50 fs. On the other hand, the temporal behavior of the FWM signal is quite asymmetric. We attribute this asymmetry to the difference in the zero time delay between beams 1 and 2 for the o1 and w2
(16)
so that
K.(r) =exp[
-(--$$r] .
(17)
1 IC Relative Time Delay Ifs)
-500
-250
0
250
500
Relative Time Delay (fs) Fig. 2. FWM signal intensity versus relative time delay when pump beams consist of only one frequency component, which is o1 for (a) and cy for (b). The solid curves are the theoretical curves with parameters LY,= 1.1 X lOI S-I for (a); a,=5.3X lOI s-’ and ST=0.598X10-13sfor(b).
X. Mi et al. /Optics Communications II4 (1995) 443-448
Here oj = Swi/2 with Swj the linewidth (fwhm) of the laser with frequency wi. The FWM signal intensities then become
(18) (19) for the cases in Figs. 2a and b, respectively. Here Sr denotes the difference in the zero time delay. The solid curves in Fig. 2 are the theoretical curves, in which we usetheparameters cr,=l.l X 1013 s-l, a,=5.3X101’ s-l and k-=0.598 X lo-l3 s. For an optical glass with refractive index IZ= 1.5, the refractive index at h 1= 607.0 nm is larger than that at h2 = 630.0 nm by approximately 0.001. A 59.8 fs delay between wi and w, corresponds to the propagation of beams in the glass (mainly the prism in the optical delay line) for a distance of about 2 cm. Taking the shift of the zero time delay into account and representing 77as y exp( - i+) , we have from Eq. ( 15) the r-dependence of the FWM signal intensity for TDLIDG
I(
exp[ - $(zr] +y2 exp[ - ip(sr)r]
+2yexp[
-(&r]
x cos( 6&Q-+ 4) .
wd
(lo”
W-1)
Fig. 3. Fourier spectrum corresponding to the data in Fig. 1.
w,= (1.27+0.0X) X 1014 s-l. We then fit our experimental result by using the following parameters: al=1.1X1013~-1,
c~~=~.~X~~‘~S-‘,~~=~.~~~X
lo-l3 s o = 1.26x 1014 s-’ y=2.0 and += 126”. Here the rel$ion 4 = w, 87 is satisfied. The solid curve in Fig. 2 is the theoretical curve. It predicts not only the modulation but also the asymmetry of the temporal behavior of the FWM signal. If we consider the discrepancy in the fit of the autocorrelation function of the pump beams (Fig. 2a), the agreement between the theoretical curve and the experimental data is quite good.
4. Conclusion
exp[ -(“:sr]
(20)
Here, &- and 4 are related by 4 = 0~87. This can be understood as follows. Consider the case that the optical paths of beams 1 and 2 are equal for the wi component. Owing to the difference between the zero time delays for the wi and w, frequency components, the optical paths between beams 1 and 2 will be different by c Sr for the w, component. As a result, there is an extra phase factor w2 87 for the 0, frequency component. The frequency difference w, can be determined by making a Fourier transformation of the TDLIDG data. Fig. 3 presents the Fourier spectrum corresponding to the data in Fig. 1, from which we deduce
In this paper, we explain the FWM signal modulation in TDLIDG by relating the signal intensity to the autocorrelation function of a field which consists of two frequency components. We performed TDLIDG experiments in which beating was observed between lights of linewidths 4.3 nm and 2.3 nm. Our results also show an asymmetry in the temporal behavior of the FWM signal. We attribute this asymmetry to the shift of the zero time delay which is due to the dispersion of the optical components.
Acknowledgements The authors gratefully acknowledge the financial support from the Chinese National Nature Sciences Foundation, the climbing program from the Chinese
448
X. Mi et al. I Optics Communications
Commission of Science and Technology, the Chinese Academy of Sciences and the Third World Academy of Sciences.
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