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The statistical behaviour of nonresonant cars intensities
Volume
56, number
2
THE STATISTICAL
15 November
OPTICS COMMUNICATIONS
BEHAVIOUR
OF NONRESONANT
1985
CARS INTENSITIES
Robert J. HALL
Received
15 July 1985
The statist& properties of CARS light generated by the mixing of incoherent pump and Stokes sources have been Investigated analytically. The intensity probability densities for CARS generation from the nonresonant susceptibility are presented for several cases. It is shown that when second harmonic generation is used to provide pump sources of the right frequency, the resulting non-gaussian intensity fluctuations give rise to large enhancements of the nonresonant signal. The magnitude of the effect is dependent on the conversion efficiency of the frequency-doubling process, and is in good agreement with experiment
1. Introduction
Intensity fluctuations associated with the use of finite bandwidth, multimode laser sources in CARS influence the efficiency and spectral distributions obtained by the method, and have been the subject of theoretical investigation [l-3]. When only one pump laser (collinear or coaxial CARS), or two, temporally correlated pump lasers (crossed-beam on BOXCARS) are employed, it has been possible to make progress analytically only by assuming gaussian statistical properties for the pumps [l-3]. While a gaussian (thermal) random process is most likely a good representation for a laser consisting of many, independent modes [4], it is a fact that CARS pump sources of the right frequency are usually formed by frequencydoubling, and it is known that the statistical properties of second-harmonic light are not generally gaussian [S ,6]. If the two pump components in BOXCARS are brought into exact temporal overlap or correlation, then there should be an overall factor of two enhancement in the CARS intensity because of the coincidence of the high intensity fluctuations [ 1,2]. Rahn et al. [S] have observed an anomalously large enhancement of the intensity emanating from the non-resonant background, however, (a factor of approximately 3.4) and have raised the possibility that the effect is associated with non-gaussian fluctuations arising from the frequency doubling of their
fundamental Nd : YAG laser output. In this paper the intensity probability distribution functions are derived for CARS, and it is shown that large relative enhancements of the nonresonant signal should indeed be observed when pump components are created by secondharmonic generation. The magnitude of the effect is shown to be a function of the second-harmonic conversion efficiency.
2. Analysis To make the problem analytically tractable, only the total CARS intensity arising from the fastresponding, nonresonant part of the third-order electric susceptibility will be considered. The contribution of the slower responding, Raman resonant position is a more complex problem whose consideration will be deferred to a future paper. It is assumed that the nonresonant susceptibility is of electronic origin and that it can therefore respond instantaneously to all intensity fluctuations. All primary or fundamental sources (pumps and Stokes) are assumed to be multimode lasers whose fluctuations are described by gaussian or thermal statistics. The fmal pump components may or may not be statistically independent, and may or may not have undergone second-harmonic generation. The Stokes source will be assumed to be statistically independent. If there is no pump deple127
Volume 56, number 2
OPTICS COMMUNICATIONS
tion in the CARS process (no saturation) then the total CARS intensity can simply be represented by
P(Z)= (2ZO))-1/2 (1)
Zas(l) cc Z;(OZ&)Z&)
15 November 1985
There will be an overall conversion constant that will be a function of the interaction length and proportional to the square of the nonresonant susceptibility, but it can be omitted without loss of generality. As for second and higher harmonic generation, the intensity probability distribution for CARS can be obtained by superimposing the statistical distributions of the primary light fields through a convolution integral [6]. If the pump beams are independent, one has therefore
.
exp [- (21/(Z))“*]
(5)
if there is negligible depletion of the primary beam and no phase-matching limitations. The effect of saturation or primary beam depletion in the frequency doubling process will be treated later in the paper. Given eqs. (4), (5), the integrals in eqs. (2), (3) can be evaluated for the various cases. While it is difficult to evaluate analytically the probability integrals directly, the first and second moments of the CARS intensity can be readily evaluated. After integrating out the 6functions in eqs. (2), (3), the results for the different cases are: correlated. gaussian pumps
P&a,)=jjj dz;,dz;;dz, xP~(z;,)P~(Z;;)P,(z,)6(z,,-z~z;;zs) (2) For correlated
PasKw) =jj
pumps the proper representation
= 7 dxx3i2 0 [-(~,,/(~,,)~(l /x)1,
X ev-d)exp
is
dZpdZ,Pp(Zp)P,(Z,)6(z,, ~ Z;Z,). (3)
The probability distribution defined here is independent of time and laser bandwidth. and defines the probability of measuring a given intensity in a time interval short in comparison with an autocorrelation time given by the inverse of the spectral bandwidth. Average or mean intensities are those measured over time intervals much longer than the field auto-correlation time. Since only the order of the nonlinear process is important, other combinations of sum- and frequency-difference processes would be treated in a similar fashion. Thus, the results could be applicable to other third order processes such as CSRS and thirdharmonic generation. The integrals in eqs. (2), (3) can be evaluated for the cases in which the pump probability distributions P, are either gaussian or those applicable to second harmonic light [6]. The Stokes distribution P, is assumed always to be gaussian, which has the general form of a thermal or Boltzmann distribution P(Z) = UP1 exp (-Z/(Z)) ,
(Z,,)P,,(Z,,/(Z,,))
(4)
where (I) denotes the ensemble - or long time averaged intensity. For second harmonic light the appropriate (non-gaussian) function is [6]
(I,,) = 2 UP)2 (I,) u;,, = 12 (Z&2 ) uncorrelated,
gaussian pumps
(Z,,)P,,(Z,,/(Z,,))
= j= idx 0 0
dyx-ly-l
X e-xebY exp [-(~,,/(Z,,))(~/XV)I (I,,) = (ZPj2 (I,) (I;,) = 8(Z,J2
,
correlated, frequency-doubled
pumps
,
(7)
(Z,,)P,,(Z,,/(Z,,))= 6 s dxxP7j4 0 X expt-x1/4)exp
[-24(z,,/(Z,,))(11~)1, (8)
(I,,,= 6(Z,)2U,)~Z,2s)= 140(Z,J2 , uncorrelated,
(Z,,P,,(Z,,l(Z,,))
frequency-doubled
= j: idx 0 0
pumps
dv
X x-3/2y-3/2exp(-&)exp(-&) X exp [-4(Z,,/~,,))(~/v)l~ (I,,) = (I,)* (Is) (I&) = 72(Z,d2
128
(6)
(9)
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15 November 1985
OPTICS COMMUNICATIONS
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100% conversion to the second-harmonic the probability distribution of the latter will also approach a gaussian. As the doubling process approaches 100% efficiency, the high intensity fluctuations or excursions of the primary beam cannot be preferentially amplified, and the statistical behaviour of the output will mirror that of the input. Thus, the relative enhancement of the non-resonant CARS signal should vary between 2 and 6, with the precise value depending on the doubling conversion efficiency. In general the relative enhancement may be expressed as:
as>
Fig. 1. The intensity probability density for CARS for the following pump statistics cases: (a) coherent pump, (b) correlated gaussians, (c) uncorrelated gaussians, (d) correlated second-harmonics, (e) uncorrelated second-harmonics. The Stokes source is gaussian in the above.
Note that for gaussian pumps the expected factor of two relative enhancement is obtained (eqs. (6), (7)). For second-harmonic pumps, however, the relative enhancement is a factor of six (eqs. (8), (9)). Note also that the use of second-harmonic pumps is accompanied by a large increase in the variance of the intensity. Effortsto express the integrations in eqs. (6)-(9) in analytic form have been to no avail, and therefore they have been evaluated by numerical quadrature. The results are shown in fig. 1. Also shown in fig. 1 is the gaussian (straight line) curve that would result from the use of coherent pumps where Pp(Zp) = 6 (Zp - (Zp)). Only for the latter case will the CARS intensity fluctuations be gaussian. From fig. 1 it can be seen that the departure from thermal statistics becomes especially marked for correlated, secondharmonic pumps.
=
(Z$q/vp>2(zs>
=
(zaJcorrl(zas)UIIC
up>-2J
dZp z;P~(~,
zp) ,
(10)
0
which is equivalent to the normal&d second moment of the pump intensity. The factor y is a coupling coefficient for frequency doubling such that the limit y -+ m corresponds to 100% conversion efficiency [7]. Churnside [7] has recently considered the effects of saturation in doubling processes, showing that the general form of the intensity probability distribution is given by
‘(‘2~) = (z’/(zF)) [(y4w/(1F))1/2 X(Z’-Zzw)
(11)
+Z2,]-1exp(-Z’/(ZF)),
3. Effects of second-harmonic saturation Eqs. (8) and (9) actually overstate the effects of non-gaussian pump fields because saturation of the second-harmonic causes the real pump intensity distribution to have an intermediate behaviour. The distributions of eqs. (3), (4) actually represent the limiting cases of, respectively, complete and negligible saturation of the doubling process [7]. Assuming a gaussian fundamental or primary beam, in the limit of
01
0
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
% Second
Harmonic
Conversion
,
100
Efflcwncy
Fig. 2. The relative enhancement of the nonresonant CARS intensity as a function of the second-harmonic conversion efficiency when the pump components are created by doubling. The enhancement is defined as the ratio of the total CARS intensity with correlated pumps to that with independent pumps.
129
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2
where I’ is the solution of the transcendental tion
Z’tanh2 [(yZ’lUr#‘2] =z2w
(z2w = zp).
equa-
(12)
and (I,) is the average intensity of the fundamental. Eqs. (11) and (12) have been solved numerically for various values of the coupling coefficient y and the relative nonresonant enhancement calculated from eq. (10). Using the definition of conversion efficiency E = (Z,)/(Z,:), the results are plotted in fig. 2. It would be reasonable to estimate that the doubling efficiency in the experiments of ref. [S] was about 40%. From fig. 2 it can be seen that the predicted relative enhancement of the back-ground is 3.8, in fair agreement with the observed value of 3.4 ? 0.2.
4. Conclusions The probability density function for the nonresonant CARS intensity has been derived for various cases. It has been shown that even with gaussian primary sources the CARS intensity distributions are super-chaotic and not gaussian, the exception being the case of a coherent pump. It is shown theoretically that the non-gaussian statistical behaviour of pumps derived from frequency-doubling processes can give rise to anomalously large enhancements (greater than a factor of two) of the nonresonant CARS signal when the two pump components are correlated. The exact value is a function of the second-harmonic conversion efficiency. While the agreement with the one published result on coincidence measurements is fair, it seems likely that the magnitude of the effect will vary depending on the make of the pump laser. In paticular, any nonlinear behaviour of the amplifier stages of typical pump laser sources will tend to pro-
130
15 November 1985
duce a subchaotic field prior to the frequency doubling, with the result that the coincidence ratio will be lower than that calculated here. These calculations are thus best compared to experiments performed with zero amplification. There is recent experimental evidence [8] which indicates that the amplifier nonlinearities can counteract the effects of the doubling to such an extent that the second harmonic field is essentially gaussian! Because this analysis has been concerned only with nonresonant CARS, it is not possible to draw from it conclusions about the relative enhancement of the vibrationally resonant signal, although intuitively it seems reasonable to believe that it would have the same enhancement as the background if the pump bandwidth were less than the Raman linewidths. The resonant response to nongaussian field fluctuations is being worked out and will be addressed in a future publication [8].
Acknowledgement Very helpful conversations with Dr. D.A. Greenhalgh and Dr. T. Whittley are gratefully acknowledged.
References [ l] M.A. Yuratich, Mol. Phys. 38 (1979) 625. [2] G.S. Agarwal and S. Singh, Phys. Rev. A 25 (1982) 3195. [ 31 R.J. Hall, Optics Comm. 52 (1985) 360. [4] L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. [5] L.A. Rahn, R.L. Farrow and R.P. Lucht, Optics Lett. 9 (1984) 223. [6] A.A. Grutter, H.P. Weber and R. Dandliker, Phys. Rev. 135 (1969) 629. [7] J.H. Churnside, Optics Comm. 51 (1985) 207. [S] R.J. Mall,D.A. Grcenhalghand M. AldCn,tobe published.
56, number
2
THE STATISTICAL
15 November
OPTICS COMMUNICATIONS
BEHAVIOUR
OF NONRESONANT
1985
CARS INTENSITIES
Robert J. HALL
Received
15 July 1985
The statist& properties of CARS light generated by the mixing of incoherent pump and Stokes sources have been Investigated analytically. The intensity probability densities for CARS generation from the nonresonant susceptibility are presented for several cases. It is shown that when second harmonic generation is used to provide pump sources of the right frequency, the resulting non-gaussian intensity fluctuations give rise to large enhancements of the nonresonant signal. The magnitude of the effect is dependent on the conversion efficiency of the frequency-doubling process, and is in good agreement with experiment
1. Introduction
Intensity fluctuations associated with the use of finite bandwidth, multimode laser sources in CARS influence the efficiency and spectral distributions obtained by the method, and have been the subject of theoretical investigation [l-3]. When only one pump laser (collinear or coaxial CARS), or two, temporally correlated pump lasers (crossed-beam on BOXCARS) are employed, it has been possible to make progress analytically only by assuming gaussian statistical properties for the pumps [l-3]. While a gaussian (thermal) random process is most likely a good representation for a laser consisting of many, independent modes [4], it is a fact that CARS pump sources of the right frequency are usually formed by frequencydoubling, and it is known that the statistical properties of second-harmonic light are not generally gaussian [S ,6]. If the two pump components in BOXCARS are brought into exact temporal overlap or correlation, then there should be an overall factor of two enhancement in the CARS intensity because of the coincidence of the high intensity fluctuations [ 1,2]. Rahn et al. [S] have observed an anomalously large enhancement of the intensity emanating from the non-resonant background, however, (a factor of approximately 3.4) and have raised the possibility that the effect is associated with non-gaussian fluctuations arising from the frequency doubling of their
fundamental Nd : YAG laser output. In this paper the intensity probability distribution functions are derived for CARS, and it is shown that large relative enhancements of the nonresonant signal should indeed be observed when pump components are created by secondharmonic generation. The magnitude of the effect is shown to be a function of the second-harmonic conversion efficiency.
2. Analysis To make the problem analytically tractable, only the total CARS intensity arising from the fastresponding, nonresonant part of the third-order electric susceptibility will be considered. The contribution of the slower responding, Raman resonant position is a more complex problem whose consideration will be deferred to a future paper. It is assumed that the nonresonant susceptibility is of electronic origin and that it can therefore respond instantaneously to all intensity fluctuations. All primary or fundamental sources (pumps and Stokes) are assumed to be multimode lasers whose fluctuations are described by gaussian or thermal statistics. The fmal pump components may or may not be statistically independent, and may or may not have undergone second-harmonic generation. The Stokes source will be assumed to be statistically independent. If there is no pump deple127
Volume 56, number 2
OPTICS COMMUNICATIONS
tion in the CARS process (no saturation) then the total CARS intensity can simply be represented by
P(Z)= (2ZO))-1/2 (1)
Zas(l) cc Z;(OZ&)Z&)
15 November 1985
There will be an overall conversion constant that will be a function of the interaction length and proportional to the square of the nonresonant susceptibility, but it can be omitted without loss of generality. As for second and higher harmonic generation, the intensity probability distribution for CARS can be obtained by superimposing the statistical distributions of the primary light fields through a convolution integral [6]. If the pump beams are independent, one has therefore
.
exp [- (21/(Z))“*]
(5)
if there is negligible depletion of the primary beam and no phase-matching limitations. The effect of saturation or primary beam depletion in the frequency doubling process will be treated later in the paper. Given eqs. (4), (5), the integrals in eqs. (2), (3) can be evaluated for the various cases. While it is difficult to evaluate analytically the probability integrals directly, the first and second moments of the CARS intensity can be readily evaluated. After integrating out the 6functions in eqs. (2), (3), the results for the different cases are: correlated. gaussian pumps
P&a,)=jjj dz;,dz;;dz, xP~(z;,)P~(Z;;)P,(z,)6(z,,-z~z;;zs) (2) For correlated
PasKw) =jj
pumps the proper representation
= 7 dxx3i2 0 [-(~,,/(~,,)~(l /x)1,
X ev-d)exp
is
dZpdZ,Pp(Zp)P,(Z,)6(z,, ~ Z;Z,). (3)
The probability distribution defined here is independent of time and laser bandwidth. and defines the probability of measuring a given intensity in a time interval short in comparison with an autocorrelation time given by the inverse of the spectral bandwidth. Average or mean intensities are those measured over time intervals much longer than the field auto-correlation time. Since only the order of the nonlinear process is important, other combinations of sum- and frequency-difference processes would be treated in a similar fashion. Thus, the results could be applicable to other third order processes such as CSRS and thirdharmonic generation. The integrals in eqs. (2), (3) can be evaluated for the cases in which the pump probability distributions P, are either gaussian or those applicable to second harmonic light [6]. The Stokes distribution P, is assumed always to be gaussian, which has the general form of a thermal or Boltzmann distribution P(Z) = UP1 exp (-Z/(Z)) ,
(Z,,)P,,(Z,,/(Z,,))
(4)
where (I) denotes the ensemble - or long time averaged intensity. For second harmonic light the appropriate (non-gaussian) function is [6]
(I,,) = 2 UP)2 (I,) u;,, = 12 (Z&2 ) uncorrelated,
gaussian pumps
(Z,,)P,,(Z,,/(Z,,))
= j= idx 0 0
dyx-ly-l
X e-xebY exp [-(~,,/(Z,,))(~/XV)I (I,,) = (ZPj2 (I,) (I;,) = 8(Z,J2
,
correlated, frequency-doubled
pumps
,
(7)
(Z,,)P,,(Z,,/(Z,,))= 6 s dxxP7j4 0 X expt-x1/4)exp
[-24(z,,/(Z,,))(11~)1, (8)
(I,,,= 6(Z,)2U,)~Z,2s)= 140(Z,J2 , uncorrelated,
(Z,,P,,(Z,,l(Z,,))
frequency-doubled
= j: idx 0 0
pumps
dv
X x-3/2y-3/2exp(-&)exp(-&) X exp [-4(Z,,/~,,))(~/v)l~ (I,,) = (I,)* (Is) (I&) = 72(Z,d2
128
(6)
(9)
Volume 56, number 2
: 12
: S 0
0.50
100
---.___ I 1.50
15 November 1985
OPTICS COMMUNICATIONS
I 2.00
I 2.50
I 3.00
1 3.50
Ill 4.00
4.50
5.00
f as
100% conversion to the second-harmonic the probability distribution of the latter will also approach a gaussian. As the doubling process approaches 100% efficiency, the high intensity fluctuations or excursions of the primary beam cannot be preferentially amplified, and the statistical behaviour of the output will mirror that of the input. Thus, the relative enhancement of the non-resonant CARS signal should vary between 2 and 6, with the precise value depending on the doubling conversion efficiency. In general the relative enhancement may be expressed as:
as>
Fig. 1. The intensity probability density for CARS for the following pump statistics cases: (a) coherent pump, (b) correlated gaussians, (c) uncorrelated gaussians, (d) correlated second-harmonics, (e) uncorrelated second-harmonics. The Stokes source is gaussian in the above.
Note that for gaussian pumps the expected factor of two relative enhancement is obtained (eqs. (6), (7)). For second-harmonic pumps, however, the relative enhancement is a factor of six (eqs. (8), (9)). Note also that the use of second-harmonic pumps is accompanied by a large increase in the variance of the intensity. Effortsto express the integrations in eqs. (6)-(9) in analytic form have been to no avail, and therefore they have been evaluated by numerical quadrature. The results are shown in fig. 1. Also shown in fig. 1 is the gaussian (straight line) curve that would result from the use of coherent pumps where Pp(Zp) = 6 (Zp - (Zp)). Only for the latter case will the CARS intensity fluctuations be gaussian. From fig. 1 it can be seen that the departure from thermal statistics becomes especially marked for correlated, secondharmonic pumps.
=
(Z$q/vp>2(zs>
=
(zaJcorrl(zas)UIIC
up>-2J
dZp z;P~(~,
zp) ,
(10)
0
which is equivalent to the normal&d second moment of the pump intensity. The factor y is a coupling coefficient for frequency doubling such that the limit y -+ m corresponds to 100% conversion efficiency [7]. Churnside [7] has recently considered the effects of saturation in doubling processes, showing that the general form of the intensity probability distribution is given by
‘(‘2~) = (z’/(zF)) [(y4w/(1F))1/2 X(Z’-Zzw)
(11)
+Z2,]-1exp(-Z’/(ZF)),
3. Effects of second-harmonic saturation Eqs. (8) and (9) actually overstate the effects of non-gaussian pump fields because saturation of the second-harmonic causes the real pump intensity distribution to have an intermediate behaviour. The distributions of eqs. (3), (4) actually represent the limiting cases of, respectively, complete and negligible saturation of the doubling process [7]. Assuming a gaussian fundamental or primary beam, in the limit of
01
0
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
% Second
Harmonic
Conversion
,
100
Efflcwncy
Fig. 2. The relative enhancement of the nonresonant CARS intensity as a function of the second-harmonic conversion efficiency when the pump components are created by doubling. The enhancement is defined as the ratio of the total CARS intensity with correlated pumps to that with independent pumps.
129
Volutne 56. number
OPTICS COMMUNICATIONS
2
where I’ is the solution of the transcendental tion
Z’tanh2 [(yZ’lUr#‘2] =z2w
(z2w = zp).
equa-
(12)
and (I,) is the average intensity of the fundamental. Eqs. (11) and (12) have been solved numerically for various values of the coupling coefficient y and the relative nonresonant enhancement calculated from eq. (10). Using the definition of conversion efficiency E = (Z,)/(Z,:), the results are plotted in fig. 2. It would be reasonable to estimate that the doubling efficiency in the experiments of ref. [S] was about 40%. From fig. 2 it can be seen that the predicted relative enhancement of the back-ground is 3.8, in fair agreement with the observed value of 3.4 ? 0.2.
4. Conclusions The probability density function for the nonresonant CARS intensity has been derived for various cases. It has been shown that even with gaussian primary sources the CARS intensity distributions are super-chaotic and not gaussian, the exception being the case of a coherent pump. It is shown theoretically that the non-gaussian statistical behaviour of pumps derived from frequency-doubling processes can give rise to anomalously large enhancements (greater than a factor of two) of the nonresonant CARS signal when the two pump components are correlated. The exact value is a function of the second-harmonic conversion efficiency. While the agreement with the one published result on coincidence measurements is fair, it seems likely that the magnitude of the effect will vary depending on the make of the pump laser. In paticular, any nonlinear behaviour of the amplifier stages of typical pump laser sources will tend to pro-
130
15 November 1985
duce a subchaotic field prior to the frequency doubling, with the result that the coincidence ratio will be lower than that calculated here. These calculations are thus best compared to experiments performed with zero amplification. There is recent experimental evidence [8] which indicates that the amplifier nonlinearities can counteract the effects of the doubling to such an extent that the second harmonic field is essentially gaussian! Because this analysis has been concerned only with nonresonant CARS, it is not possible to draw from it conclusions about the relative enhancement of the vibrationally resonant signal, although intuitively it seems reasonable to believe that it would have the same enhancement as the background if the pump bandwidth were less than the Raman linewidths. The resonant response to nongaussian field fluctuations is being worked out and will be addressed in a future publication [8].
Acknowledgement Very helpful conversations with Dr. D.A. Greenhalgh and Dr. T. Whittley are gratefully acknowledged.
References [ l] M.A. Yuratich, Mol. Phys. 38 (1979) 625. [2] G.S. Agarwal and S. Singh, Phys. Rev. A 25 (1982) 3195. [ 31 R.J. Hall, Optics Comm. 52 (1985) 360. [4] L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. [5] L.A. Rahn, R.L. Farrow and R.P. Lucht, Optics Lett. 9 (1984) 223. [6] A.A. Grutter, H.P. Weber and R. Dandliker, Phys. Rev. 135 (1969) 629. [7] J.H. Churnside, Optics Comm. 51 (1985) 207. [S] R.J. Mall,D.A. Grcenhalghand M. AldCn,tobe published.