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The solution of the inverse problem in polarization CARS-spectroscopy
Specmchimica Am. Vol. 48A. No. 5. pp.683439.1992 Printedin GreatBritain
05~8539/92 ss.00+ 0.00 @I 1992Pergamon PressLtd
The solution of the inverse problem in polarization CARS-spectroscopy A. I. FISHMAN,* S. F. MIRONOV and M. KH. Department
SALAKHOV
of Physics, Kazan State University, Kazan 420008, U.S.S.R.
(Received 18 June 1991; in&al form and accepted 27 September 1991) Ahstraet-An analysis of the solution of the ill-posed inverse problem in polarization CARS is carried out on the basis of mathematical experiments. Smoothing. determination of the number of components and the determination of parameters describing each component are considered. The use of all available experimental and theoretical information is shown to be necessary for correct solution of the problem. The efficiency of the proposed algorithm in the analysis of the complex inhomogeneously broadened lines of chlorocyclohexane is demonstrated.
INTRODUCTION
IN recent years great success has been achieved in the field of non-linear light scattering spectroscopy. Because of the optical non-linearities of a medium one can pass from conventional studies of scattering by molecules excited by thermal oscillations to investigations of scattering by molecules with oscillations phased by coherent laser beams, thus significantly changing the whole picture of scattering. Non-linear polarization, being a coherent optical response of the medium analyzed, can be described by a non-linear 3rd order susceptibility x$, [l-6]. In the adiabatic (Born-Oppenheimer) approximation this susceptibility consists of two components, the first related to a resonance coherent response of an oscillatory subsystem of the molecule (x$,~), the second caused by a non-resonance coherent response of the electron subsystem (x$iNR). In Raman spectroscopy only imaginary components of the corresponding spectral components of the response functions are considered, i.e. of cubic resonance susceptibilities J&F. In CARS spectroscopy the spectral functions associated with the intensity of the coherently scattered signal contain information on the dispersion of both imaginary and real components & as well as their relative signs [7]. Complete information can be obtained from a set of experimentally measured dispersion curves I(Aw) by means of varying the interference conditions between the resonant and non-resonant contributions to the recorded signal, thus changing the polarization of the interacting light waves and also the polarization conditions of the coherent spectra recorded. In polarization CARS (PCARS) spectroscopy [5,7] the dispersion of the signal passed through a polarization analyzer is recorded. The intensity is expressed as follows:
where w = w1- w2, w1 and w2 are frequencies of the exciting waves, the frequency w2 changing smoothly in the scanning mode, passing through the resonance frequency. n(w), a coefficient for m close resonances of the Lorenz type, has the following form:
(2) where rZRkand &,R are coefficients depending on the angle E between the normal to the polarization vector of the non-resonant source and the transmission plane of the * Author to whom correspondence
should be addressed. 683
A. I.FKHMAN etal.
684
analyzer: 3(3)RX1ul
AR*E= 22/8 COS2& + 1 - {( 1 - 3~) sin 2q.~cos E + 2(3 - (3 - p) sin2 q) sin E}
(3)
where p is the degree of depolarization of the Raman line (for a Raman tensor with no antisymmetric components). I is the HWHM of the resonance curve; QZkis the frequency value of the k-d resonance; q is the angle difference between the polarization orientations of pumping waves C, and C2. The polarization analyzer allows one to change the conditions of interference of both resonant and non-resonant components of the scattering because of distinctions between their polarization characteristics [see Eqn (3)], thus placing emphasis on the details of the complicated spectral profile. It is important in such a situation that not only the number of components of the complex Raman line can be determined but also their spectral parameters, i.e. their nonlinear susceptibilities, depolarization ratios, half-widths of half-maxima and band positions. PCARS significantly enlarges the scope of vibrational spectroscopy, in particular, in solving problems of conformational analysis [9-111. The slight differences between the vibrational spectra of rotational isomers are often not resolved by conventional IR- and Raman-methods. The partial suppression of certain complex line components and the more evident manifestation of others, the interference between them and the nonresonance background allows a more reliable identification of “abundant” lines related to conformationally inhomogeneous compounds investigated by PCARS. The intricate treatment of experimental spectra and the definition of spectral characteristics requires that the inverse spectral problem be solved using the special methods of regularization and smoothing described in [12]. In these cases the additional spectroscopic information contained in PCARS spectra becomes crucially important. The experimenter obtains a great number of various spectral “projectors” of the same set of optic resonances. Thus the inverse spectroscopic problem can be formulated and solved with significantly less error in PCARS than in spontaneous Raman spectroscopy (SR). The present paper proposes an alogorithm of step-by-step solution for the case of the inverse problem as a further development of the method described in [12]. The algorithm is verified on mathematical models and by treating the PCARS spectra of chlorocyclohexane.
ANALGORITHMFORSOLVINGTHEINVERSE
PROBLEM
Let the solution of the inverse problem be divided into two basic stages. At the stage of pre-treatment, the problem involves extraction of legitimate signals from noise and a definition of the number of components. In Ref. [5] the noise characteristics of various schemes of non-linear spectroscopy have been considered in detail. The use of cooled photomultipliers, accumulation of a certain number of pulses, etc allows one to increase the signal-to-noise ratio. The signal, however, is recorded against a background of errors which is 5-25% of the legitimate signal. A great number of methods of smoothing the experimental curves have been described in Refs [13-171. For the case of CARS-spectra we apply the method of smoothing reported in Ref. [NJ, a method which is adequate for experiments with Rutherford backward scattering.
Polarization CARS-spectroscopy
685
06
0 -6
-4
-2
0
2
4
6
w,-Ill,-s1 r
Fig. 1. Smoothing of “experimental” curves. Dots-experimental curve with random relative error of 20%. Solid line-smoothed curve after 10 iterations according to Eqn (4). Dashes-true distribution.
The algorithm is as follows: an apparatus function g(x -x0) is introduced, describing the noise characteristics of the recording circuit. Unlike many other methods the real noise distributions are taken instead of the normal one. If the spectrum f(x) observed can be written in the form: f(x) = C&X)+ N(x), where N(x) is noise, q(x) is the true spectrum, then the smoothed solution obtained after )2 iterations [17] is:
f”(X) =.6-1(x)+ (f-f,-1)*g,
(4)
where * is a convolution, f, = g *f. This algorithm is easily programmable, fast converging and gives good results. Figure 1 shows the result of smoothing an “experimental”? spectrum. Irrespective of the decomposition algorithm, the number of elementary components of the compound spectrum is to be found. In the case of spectra (IQ, - C&l> 2I) solvable by the Raleigh criterion the number of components can be found from the Raman-spectra. In other cases derivative spectroscopy [19] seems to be suitable for structure analysis. If the bands overlap little, use of the first and the second spectral derivatives is adequate. In the case of strongly overlapping bands, derivatives of higher orders must be used. A regularized differentiation [18] was used for calculating derivatives, allowing the computation of experimental curve derivatives of high order with great efficiency. The second stage concerns the defining of parameters which characterize certain resonances. Similarly to Ref. [12], substituting (2) into Eqn (1) and replacing the variable, we obtain a system of linear equations (SyLE) for xi and zi (i = 1 t 2m) for each value of frequency w (S(&, w)-n~,(&))(W*m+X,W*m-l+
* - * +X*,)=Z1W**-l+Z2Wh-*+~~
* +ZZm.
(5)
t The term “experimental” here means a mathematical curve built on a given set of parameters by Eqn (1) followed by the insertion of a normally distributed random error with a relative derivation of 20%.
686
A. I.FISHMAN etal.
The variables Xi are defined by the following equalities
XI=-~
(Wj+
W;)
j=l
X*=C
(Wi+Wy)(Wj+Wf)
lWj12+C
I
j
(6)
. . . .. x *m=niWjl’.
Roots W,, WY, of Eqns (6) are found from the equation w’-2W~j+Qf+T~=(W-Wj)(W-W~)=0.
(7)
Expressions for Zi are of no value. 2m values of the unknown xi and 2m values of the unknown zi can be found from the solution of SyLE (5). If xi are defined, Eqns (6) determine the algebraic equation of the order 2m for the variables Wj, WJ w2”+x1wzm-1+
* * * +qm=O,
(8)
and the parameters Qj= ReW, and rj=ZmWj related to them. Thus, the algorithm is the following: SyLE (5) is to be solved for certain values of wk and S(wk), where k = 192, . . . N(Na4m) and roots xi and Zi(i= 1,2,. . .2m) are found; then roots Wi are found from xi of Eqn (8), the former defining Ii and Pi. The solution based on this algorithm is reported in [12] for the case of two resonances. It is shown there that for given values of the parameters there exists a region of E values within the limits of which the parameters to be found can be determined with great accuracy, the regularization applied allowing expansion of the given region. In Ref. [12], the solution is reported for values of the angle E. However, in order to increase the stability of solution, it seems to be reasonable to use all the experimental information available at the same time, particularly, SR spectra and PCARS spectra, obtained at various E angles. Since only 2m of 4m unknown xi to be found are of interest, writing down a similar system for various E; angles and summing up all N, equations, a new system with the same xi but new z( can be obtained: 2
(S(&,, W)-&(Ei))(W*m+X,W2m-‘+.
. . +X2m)=Z;W2~-‘+Z;W~-2+~~~
+rL.
(9)
Similarly, for each ci the intensity of SR I,,(w) can be subtracted from S(.si, w), allowing a new set of equations to be obtained. The number of conditions contains a great deal of information on the stability of the system, an aspect which is important for solving SyLE [cond.(A)], the stability of SyLE decreasing with increase in cond.(A). The number of conditions is also a criterion of the approach of the matrix to the degenerate form. In Fig. 2 a sample plot of the relationship between the number of matrix conditions and the angle E (5), [S(E, w) taken instead of S(E, w) - Z,(w)] are shown. The APCARS-spectrum was simulated, containing two resonances with depolarization ratios p1 = 0.1, p2 = 0.7. The presence of peaks on the curve is due to the fact that at these values of E angles suppression of one of the resonances, i.e. a loss of information on it, occurs. At the same time, the region of stability can be defined, that which is the most suitable for spectrum measuring. For example, for a given set of parameters an optimum angle E of spectrum recording lies in the range (-10” 3 + loo). A dashed line (2) is shown in the same figure, corresponding to the number of conditions on the SyLE matrix (9) for N, = 6 with a step of he= lo”. It can be seen that spectra measured at various E angles, when treated simultaneously, significantly increase the stability of the SyLE solution.
Polarization CARS-spectroscopy
687
c(deg) Fig. 2. Number of matrix condition versus angle E plot for CARS spectrum consisting of two components, with parameters: p, = 0.1, Kr’=O.6, I-,=1, Q,=O; p,=O.7, ~;‘=0.2, r,=1.2, Q2=2; q=n13.
Mathematical experiments performed by us, show that for solving the hyperdefined (N>4m) SyLE (9) a singular expansion [20] can be used quite efficiently. Only 52 and I parameters can be defined by solving SyLE. In order to determine other parameters (K and p), methods of non-linear optimization [21] were used, particularly, the Nelder-Mead method, which is an improved version of the symplex method [22]. It should be noticed that minimizing the functional of residuals, it is reasonable to use spectra recorded for several E angles. Mathematical experiments performed have proved this method to be suitable for determining all the eight parameters of the two components. Minimization was carried out in two steps: first, by defining three values I, Q and il for each component. The fast convergence of the parameters I and Sz was significant here. Further, assuming these parameters to be known, minimization was carried out for parameters K and p.
THE ANALYSIS
PCARS SPECTRA OF CHLOROCYCLOHEXANE
The CARS spectra of chlorocyclohexane in the liquid phase were recorded by a spectrometer, which is described elsewhere [ 111. The radiation of the second harmonic of the pulse laser from a YAG:Nd3’ crystal was used as w1 with a repetition rate of 12.5 Hz. Radiation with frequency o2 was generated by a dye-laser (rhodamine 6G dye in ethanol). The purity of the polarization of the pump waves passed through the cuvette with a sample was more than 0.998. A Glan prism was used as a polarizing analyzer. For improving the signal-to-noise ratio the intensity of the anti-Stocks signal was normalized against the intensity of the dye laser and after summing over a given number of pulses and averaging, it was passed to the recording device. The vibrational spectra and structure of chlorocyclohexane are both well studied [23]. It is known that in liquid and in solutions the compound exists as a mixture of axial and equatorial conformers. The IR and Raman spectra of chlorocyclohexane are interpreted in Ref. [23] using normal coordinate calculations. The fundamental vibrations Ye,and vd2 of the axial conformation and v17and v45+ V~ of the equatorial conformation fall in the spectral region near 1030 cm -I. The Raman line with a frequency of 1030cm-’ has a complicated spectral contour and is not resolved into separate components by SR. To determine the number of components hidden under this contour, we have analyzed the fourth derivatives of these spectra. They are shown in Fig. 3. All the plots clearly
688 100 c 80 -
60 40 2 F %,
-
20_
0
2 -20
-
-40
-
-60
-
-60
-
1
-100 1020
I
I
I
I
I
I
I
IO25
1030
1035
1040
1045
1050
1055
w
km-‘)
Fig. 3. Fourth derivatives of CARS spectrum I.
show the presence of four components. The spectral parameters of these components have been determined by means of the algorithm described above: sZ1= 1024 cm-‘; x~/,~/$,
= 0.10; p1 = 0.38; r, = 2.2 cm-’
Q2 = 1028 cm-‘; x~/,,l$~, = 0.30; p2 = 0.70; r, = 4.5 cm-’ Q3 = 1033 cm-‘; x~‘,,I&~, = 0.33;p,=0.67;T,=5.0cm-’ Q4 = 1038 cm-‘; x~,‘,,I$~, = 0.15; p4 = 0.62; lY4= 3.3 cm-’ The contribution of the line with frequency 1055 cm-’ significantly influences the experimental contour in the spectral region considered and was taken into account whilst adjusting the spectral parameters. The good agreement between calculated and experimental spectra for various E angles is evidence of the uniqueness of the solution of the
Fig. 4. Three-dimensional
(I, UJ,E) PCARS spectrum of chlorcyclohexane. components are given in the paper.
Parameters
of all
Polarization CARS-spectroscopy
689
inverse problem. The number of components determined in the contour agrees with the results of the normal-coordinate analysis reported in Ref. [23]. A three-dimensional (I, o, E) spectrum of the compound I in the 1020-1040 cm-’ region calculated from the parameters obtained and showing the deformation of the spectral contour caused by rotation of the polarizing analyzer, is shown in Fig. 4.
CONCLUSION
The proposed and realized algorithm for step-by-step solution of the inverse spectral problem in PCARS using a whole set of polarized spectra allows one to determine fairly accurately the spectral parameters of non-uniformly broadened Raman lines. This investigation was supported and financed by the “Laser systems” programme of the U.S.S.R. State Education committee.
REFERENCES [l] R. W. Dewitt, A. B. Harrey and W. M. Tolles, Theoretical Development of Third Order Susceptibility as Related to Coherent Anti-Stokes Raman Spectroscopy, NRL Memorandum Report No. 3260 (1976). [2] J. J. Song, G. L. Eesly and M. D. Levenson, Appl. Phys. Lett. 29, 567 (1976). [3] H. C. Andersen and B. S. Hudson, in Molecular Spectroscopy, Vol. 5 (Edited by D. A. Long). Chem. Sot., New York (1977). [4] J. W. Nibler and G. W. Knighten, Coherent Anti-Stokes Spectroscopy, in Raman Spectroscopy of Gases and Liquids. Topics in Current Physics, Vol. 11 (Edited by A. Weber). Springer-Verlag (1979). [S] S. A. Akhmanov and N. I. Koroteev, Metod’i nelineinoi optiki v spektroskopii rasseyania sveta: Aktivnaya spektroskopiya rasseyania sveta, M. Nauka, 544 (1981). [6] R. Brake1 and F. W. Schneider, in Aduances in Non-Linear Spectroscopy (Edited by R. J. H. Clark and R. E. Hester), Chapter 4, “Polarization CARS Spectroscopy”. J. Wiley and Sons (1988). [7] N. I. Koroteev, Uspekhi Fiz. Nuuk 153(3), 493 (1987). [8] G. I. Aponin, A. A. Besshposhnikov and V. B. Voronin, Rasseyaniye sveta v diagnostike gazovikh potokov v nizkotemperaturnoi plazmi, in Diagnostika plazmi. M. Energoatomizdat Vol. 7.) p. 111 (1989). [9] S. F. Mironov, A. B. Remizov and A. I. Fishman, Z. Prikf. Spektr. 49(3), 506 (1988). [lo]M. F.Vigasina, A. A. Ivanov, R. Yu. Orlov, A. B. Remizov and A. I. Fishman, DAN SSSR 283 1394 (1985). [ll] A. A. Ivanov, N. I. Koroteev, R. Yu. Orlov and A. I. Fishman, Optika i Spektr. 66(S), 1046 (1989). [12] M. Kh. Salakhov, A. I. Fishman and N. K. Shcherbakova, Optikn i Spektr. 66, 461 (1989). [13] G. I. Marchuk, Metodi Vichislitelnoi Matematiki. M. Nauka (1980). [14] J. Alberg, E. Nilson and J. Warmy, Theory of Splines and its Applications M.: Mir (1972). [15] N. P. Korneitchuk, Spluini v Teorii Pribhzhenija. M. Nauka (1984). 1161 Yu. E. Voskoboynikov, N. G. Preobrazhenskii and A. I. Sedelnikov Matematicheskaya Obrabotka Experimenta u Molekularnoi Gazodinamike. Novosibirsk, Nauka (1984). [17] I. D. Grachev, M. Kh. Salakhov and I. S. Fishman, Stutisticheskaya Regularizatsia Pri Obrubotke Experimenta v Prikladnoi Spektroskopii. Kazan (1986). [18] H. C. Hayden, Computers in Physics. 6, 74 (1987). [19] I. M. Dubrovkin and V. G. Belikov, Proizvodnaia spektrometriya: Teoria, Tekhnika, Primenenie. Rostov (1988). [20] G. E. Forsythe, M. A. Malcolm and C. B. Moler, Computer Methods for Mathematical Computations. Prentice-Hall, Englewood Cliffs, NJ (1977). [21] B. D. Bunday, Basic Optimization Methods, M.: Radio i Svyaz (1988) 128~. [22] J. A. Nelder and R. Mead, The Camp. Journal. 7 308 (1965). [23] T. Woldbek, Actu Chem. Stand. 36A(8), 641 (1982).
05~8539/92 ss.00+ 0.00 @I 1992Pergamon PressLtd
The solution of the inverse problem in polarization CARS-spectroscopy A. I. FISHMAN,* S. F. MIRONOV and M. KH. Department
SALAKHOV
of Physics, Kazan State University, Kazan 420008, U.S.S.R.
(Received 18 June 1991; in&al form and accepted 27 September 1991) Ahstraet-An analysis of the solution of the ill-posed inverse problem in polarization CARS is carried out on the basis of mathematical experiments. Smoothing. determination of the number of components and the determination of parameters describing each component are considered. The use of all available experimental and theoretical information is shown to be necessary for correct solution of the problem. The efficiency of the proposed algorithm in the analysis of the complex inhomogeneously broadened lines of chlorocyclohexane is demonstrated.
INTRODUCTION
IN recent years great success has been achieved in the field of non-linear light scattering spectroscopy. Because of the optical non-linearities of a medium one can pass from conventional studies of scattering by molecules excited by thermal oscillations to investigations of scattering by molecules with oscillations phased by coherent laser beams, thus significantly changing the whole picture of scattering. Non-linear polarization, being a coherent optical response of the medium analyzed, can be described by a non-linear 3rd order susceptibility x$, [l-6]. In the adiabatic (Born-Oppenheimer) approximation this susceptibility consists of two components, the first related to a resonance coherent response of an oscillatory subsystem of the molecule (x$,~), the second caused by a non-resonance coherent response of the electron subsystem (x$iNR). In Raman spectroscopy only imaginary components of the corresponding spectral components of the response functions are considered, i.e. of cubic resonance susceptibilities J&F. In CARS spectroscopy the spectral functions associated with the intensity of the coherently scattered signal contain information on the dispersion of both imaginary and real components & as well as their relative signs [7]. Complete information can be obtained from a set of experimentally measured dispersion curves I(Aw) by means of varying the interference conditions between the resonant and non-resonant contributions to the recorded signal, thus changing the polarization of the interacting light waves and also the polarization conditions of the coherent spectra recorded. In polarization CARS (PCARS) spectroscopy [5,7] the dispersion of the signal passed through a polarization analyzer is recorded. The intensity is expressed as follows:
where w = w1- w2, w1 and w2 are frequencies of the exciting waves, the frequency w2 changing smoothly in the scanning mode, passing through the resonance frequency. n(w), a coefficient for m close resonances of the Lorenz type, has the following form:
(2) where rZRkand &,R are coefficients depending on the angle E between the normal to the polarization vector of the non-resonant source and the transmission plane of the * Author to whom correspondence
should be addressed. 683
A. I.FKHMAN etal.
684
analyzer: 3(3)RX1ul
AR*E= 22/8 COS2& + 1 - {( 1 - 3~) sin 2q.~cos E + 2(3 - (3 - p) sin2 q) sin E}
(3)
where p is the degree of depolarization of the Raman line (for a Raman tensor with no antisymmetric components). I is the HWHM of the resonance curve; QZkis the frequency value of the k-d resonance; q is the angle difference between the polarization orientations of pumping waves C, and C2. The polarization analyzer allows one to change the conditions of interference of both resonant and non-resonant components of the scattering because of distinctions between their polarization characteristics [see Eqn (3)], thus placing emphasis on the details of the complicated spectral profile. It is important in such a situation that not only the number of components of the complex Raman line can be determined but also their spectral parameters, i.e. their nonlinear susceptibilities, depolarization ratios, half-widths of half-maxima and band positions. PCARS significantly enlarges the scope of vibrational spectroscopy, in particular, in solving problems of conformational analysis [9-111. The slight differences between the vibrational spectra of rotational isomers are often not resolved by conventional IR- and Raman-methods. The partial suppression of certain complex line components and the more evident manifestation of others, the interference between them and the nonresonance background allows a more reliable identification of “abundant” lines related to conformationally inhomogeneous compounds investigated by PCARS. The intricate treatment of experimental spectra and the definition of spectral characteristics requires that the inverse spectral problem be solved using the special methods of regularization and smoothing described in [12]. In these cases the additional spectroscopic information contained in PCARS spectra becomes crucially important. The experimenter obtains a great number of various spectral “projectors” of the same set of optic resonances. Thus the inverse spectroscopic problem can be formulated and solved with significantly less error in PCARS than in spontaneous Raman spectroscopy (SR). The present paper proposes an alogorithm of step-by-step solution for the case of the inverse problem as a further development of the method described in [12]. The algorithm is verified on mathematical models and by treating the PCARS spectra of chlorocyclohexane.
ANALGORITHMFORSOLVINGTHEINVERSE
PROBLEM
Let the solution of the inverse problem be divided into two basic stages. At the stage of pre-treatment, the problem involves extraction of legitimate signals from noise and a definition of the number of components. In Ref. [5] the noise characteristics of various schemes of non-linear spectroscopy have been considered in detail. The use of cooled photomultipliers, accumulation of a certain number of pulses, etc allows one to increase the signal-to-noise ratio. The signal, however, is recorded against a background of errors which is 5-25% of the legitimate signal. A great number of methods of smoothing the experimental curves have been described in Refs [13-171. For the case of CARS-spectra we apply the method of smoothing reported in Ref. [NJ, a method which is adequate for experiments with Rutherford backward scattering.
Polarization CARS-spectroscopy
685
06
0 -6
-4
-2
0
2
4
6
w,-Ill,-s1 r
Fig. 1. Smoothing of “experimental” curves. Dots-experimental curve with random relative error of 20%. Solid line-smoothed curve after 10 iterations according to Eqn (4). Dashes-true distribution.
The algorithm is as follows: an apparatus function g(x -x0) is introduced, describing the noise characteristics of the recording circuit. Unlike many other methods the real noise distributions are taken instead of the normal one. If the spectrum f(x) observed can be written in the form: f(x) = C&X)+ N(x), where N(x) is noise, q(x) is the true spectrum, then the smoothed solution obtained after )2 iterations [17] is:
f”(X) =.6-1(x)+ (f-f,-1)*g,
(4)
where * is a convolution, f, = g *f. This algorithm is easily programmable, fast converging and gives good results. Figure 1 shows the result of smoothing an “experimental”? spectrum. Irrespective of the decomposition algorithm, the number of elementary components of the compound spectrum is to be found. In the case of spectra (IQ, - C&l> 2I) solvable by the Raleigh criterion the number of components can be found from the Raman-spectra. In other cases derivative spectroscopy [19] seems to be suitable for structure analysis. If the bands overlap little, use of the first and the second spectral derivatives is adequate. In the case of strongly overlapping bands, derivatives of higher orders must be used. A regularized differentiation [18] was used for calculating derivatives, allowing the computation of experimental curve derivatives of high order with great efficiency. The second stage concerns the defining of parameters which characterize certain resonances. Similarly to Ref. [12], substituting (2) into Eqn (1) and replacing the variable, we obtain a system of linear equations (SyLE) for xi and zi (i = 1 t 2m) for each value of frequency w (S(&, w)-n~,(&))(W*m+X,W*m-l+
* - * +X*,)=Z1W**-l+Z2Wh-*+~~
* +ZZm.
(5)
t The term “experimental” here means a mathematical curve built on a given set of parameters by Eqn (1) followed by the insertion of a normally distributed random error with a relative derivation of 20%.
686
A. I.FISHMAN etal.
The variables Xi are defined by the following equalities
XI=-~
(Wj+
W;)
j=l
X*=C
(Wi+Wy)(Wj+Wf)
lWj12+C
I
j
(6)
. . . .. x *m=niWjl’.
Roots W,, WY, of Eqns (6) are found from the equation w’-2W~j+Qf+T~=(W-Wj)(W-W~)=0.
(7)
Expressions for Zi are of no value. 2m values of the unknown xi and 2m values of the unknown zi can be found from the solution of SyLE (5). If xi are defined, Eqns (6) determine the algebraic equation of the order 2m for the variables Wj, WJ w2”+x1wzm-1+
* * * +qm=O,
(8)
and the parameters Qj= ReW, and rj=ZmWj related to them. Thus, the algorithm is the following: SyLE (5) is to be solved for certain values of wk and S(wk), where k = 192, . . . N(Na4m) and roots xi and Zi(i= 1,2,. . .2m) are found; then roots Wi are found from xi of Eqn (8), the former defining Ii and Pi. The solution based on this algorithm is reported in [12] for the case of two resonances. It is shown there that for given values of the parameters there exists a region of E values within the limits of which the parameters to be found can be determined with great accuracy, the regularization applied allowing expansion of the given region. In Ref. [12], the solution is reported for values of the angle E. However, in order to increase the stability of solution, it seems to be reasonable to use all the experimental information available at the same time, particularly, SR spectra and PCARS spectra, obtained at various E angles. Since only 2m of 4m unknown xi to be found are of interest, writing down a similar system for various E; angles and summing up all N, equations, a new system with the same xi but new z( can be obtained: 2
(S(&,, W)-&(Ei))(W*m+X,W2m-‘+.
. . +X2m)=Z;W2~-‘+Z;W~-2+~~~
+rL.
(9)
Similarly, for each ci the intensity of SR I,,(w) can be subtracted from S(.si, w), allowing a new set of equations to be obtained. The number of conditions contains a great deal of information on the stability of the system, an aspect which is important for solving SyLE [cond.(A)], the stability of SyLE decreasing with increase in cond.(A). The number of conditions is also a criterion of the approach of the matrix to the degenerate form. In Fig. 2 a sample plot of the relationship between the number of matrix conditions and the angle E (5), [S(E, w) taken instead of S(E, w) - Z,(w)] are shown. The APCARS-spectrum was simulated, containing two resonances with depolarization ratios p1 = 0.1, p2 = 0.7. The presence of peaks on the curve is due to the fact that at these values of E angles suppression of one of the resonances, i.e. a loss of information on it, occurs. At the same time, the region of stability can be defined, that which is the most suitable for spectrum measuring. For example, for a given set of parameters an optimum angle E of spectrum recording lies in the range (-10” 3 + loo). A dashed line (2) is shown in the same figure, corresponding to the number of conditions on the SyLE matrix (9) for N, = 6 with a step of he= lo”. It can be seen that spectra measured at various E angles, when treated simultaneously, significantly increase the stability of the SyLE solution.
Polarization CARS-spectroscopy
687
c(deg) Fig. 2. Number of matrix condition versus angle E plot for CARS spectrum consisting of two components, with parameters: p, = 0.1, Kr’=O.6, I-,=1, Q,=O; p,=O.7, ~;‘=0.2, r,=1.2, Q2=2; q=n13.
Mathematical experiments performed by us, show that for solving the hyperdefined (N>4m) SyLE (9) a singular expansion [20] can be used quite efficiently. Only 52 and I parameters can be defined by solving SyLE. In order to determine other parameters (K and p), methods of non-linear optimization [21] were used, particularly, the Nelder-Mead method, which is an improved version of the symplex method [22]. It should be noticed that minimizing the functional of residuals, it is reasonable to use spectra recorded for several E angles. Mathematical experiments performed have proved this method to be suitable for determining all the eight parameters of the two components. Minimization was carried out in two steps: first, by defining three values I, Q and il for each component. The fast convergence of the parameters I and Sz was significant here. Further, assuming these parameters to be known, minimization was carried out for parameters K and p.
THE ANALYSIS
PCARS SPECTRA OF CHLOROCYCLOHEXANE
The CARS spectra of chlorocyclohexane in the liquid phase were recorded by a spectrometer, which is described elsewhere [ 111. The radiation of the second harmonic of the pulse laser from a YAG:Nd3’ crystal was used as w1 with a repetition rate of 12.5 Hz. Radiation with frequency o2 was generated by a dye-laser (rhodamine 6G dye in ethanol). The purity of the polarization of the pump waves passed through the cuvette with a sample was more than 0.998. A Glan prism was used as a polarizing analyzer. For improving the signal-to-noise ratio the intensity of the anti-Stocks signal was normalized against the intensity of the dye laser and after summing over a given number of pulses and averaging, it was passed to the recording device. The vibrational spectra and structure of chlorocyclohexane are both well studied [23]. It is known that in liquid and in solutions the compound exists as a mixture of axial and equatorial conformers. The IR and Raman spectra of chlorocyclohexane are interpreted in Ref. [23] using normal coordinate calculations. The fundamental vibrations Ye,and vd2 of the axial conformation and v17and v45+ V~ of the equatorial conformation fall in the spectral region near 1030 cm -I. The Raman line with a frequency of 1030cm-’ has a complicated spectral contour and is not resolved into separate components by SR. To determine the number of components hidden under this contour, we have analyzed the fourth derivatives of these spectra. They are shown in Fig. 3. All the plots clearly
688 100 c 80 -
60 40 2 F %,
-
20_
0
2 -20
-
-40
-
-60
-
-60
-
1
-100 1020
I
I
I
I
I
I
I
IO25
1030
1035
1040
1045
1050
1055
w
km-‘)
Fig. 3. Fourth derivatives of CARS spectrum I.
show the presence of four components. The spectral parameters of these components have been determined by means of the algorithm described above: sZ1= 1024 cm-‘; x~/,~/$,
= 0.10; p1 = 0.38; r, = 2.2 cm-’
Q2 = 1028 cm-‘; x~/,,l$~, = 0.30; p2 = 0.70; r, = 4.5 cm-’ Q3 = 1033 cm-‘; x~‘,,I&~, = 0.33;p,=0.67;T,=5.0cm-’ Q4 = 1038 cm-‘; x~,‘,,I$~, = 0.15; p4 = 0.62; lY4= 3.3 cm-’ The contribution of the line with frequency 1055 cm-’ significantly influences the experimental contour in the spectral region considered and was taken into account whilst adjusting the spectral parameters. The good agreement between calculated and experimental spectra for various E angles is evidence of the uniqueness of the solution of the
Fig. 4. Three-dimensional
(I, UJ,E) PCARS spectrum of chlorcyclohexane. components are given in the paper.
Parameters
of all
Polarization CARS-spectroscopy
689
inverse problem. The number of components determined in the contour agrees with the results of the normal-coordinate analysis reported in Ref. [23]. A three-dimensional (I, o, E) spectrum of the compound I in the 1020-1040 cm-’ region calculated from the parameters obtained and showing the deformation of the spectral contour caused by rotation of the polarizing analyzer, is shown in Fig. 4.
CONCLUSION
The proposed and realized algorithm for step-by-step solution of the inverse spectral problem in PCARS using a whole set of polarized spectra allows one to determine fairly accurately the spectral parameters of non-uniformly broadened Raman lines. This investigation was supported and financed by the “Laser systems” programme of the U.S.S.R. State Education committee.
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