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Chapter 11 Principles of spectral measurements
235
Chapter 11 PRINCIPLES OF SPECTRAL MEASUREMENTS
11.1 Introduction The appropriate experimental quantity that is related to the transition rate, as discussed in Section 2.5, is the absorption coefficient ix(v). To determine this coefficient, one should measure the incident light intensity I0(v--), the fight intensity I(v) transmitted through a sample of interest (both at a given wavenumber v), the sample concentration CO and the optical pathlength g in the sample. Since these quantities are related via Beer's law, ix(v) is obtained as ln[Io(v)/I(v)]/Cog using natural logarithm. The integrated absorption coefficient A(v-) is obtained by integrating o~(v) over the width of the vibrational band of interest (see Eq. 2.5.6). However, the spectral intensities are commonly displayed as absorbance A(v), where A(v-) = logl0[I0(v)/I(v)] = e(v)Cos - t~(v)C0s - a(v)/2.303. The difference in absorption coefficients of left versus right circularly polarized incident light, referred to as circular dichroism, can be observed for randomly oriented chiral samples; Chiral systems are said to be optically active. Optical activity can be induced in non-chiral systems by an external magnetic field, but this phenomenon is not considered in this book. Oriented samples exhibit different absorption coefficients for polarizations parallel and perpendicular to the sample's unique axis. This difference in linear polarization absorption coefficients is referred to as linear dichroism. The applications of dichroism to molecular structural elucidation are numerous. Prior to the 1980s dispersive infrared spectrometers 1 (where different wavelength components of a polychromatic incident light beam are dispersed using a grating or a prism) have been commonly used for infrared spectral measurements. The advantages of using a Michelson interferometer with Fourier transform techniques for infrared spectroscopic measurements have been well established. In a Michelson interferometer2 the incoming light is divided into two equal amplitude components, so the Michelson interferometer can also be referred to as an amplitude division interferometer (ADI). The multiplex and throughput advantages associated with ADI made the Fourier transform infrared (FTIR) spectrometers very popular 3. Vibrational Raman and Raman optical activity (VROA) measuremems, on the contrary, were commonly performed using dispersive methods 4. Recent developments 5 made it possible to perform the vibrational Raman
236
Chapter 11
measurements with Fourier transform infrared spectrometers. Since Raman scattered intensifies depend on the inverse fourth power of the incident excitation wavelength, there is an intrinsic advantage to using visible laser sources for Raman measurements. Nevertheless different factors, such as the improvements in the quality of near-infrared detectors and the advantage of reduced fluorescence interference, gave impetus for the development of FI'-Raman spectrometers with near-infrared excitation. An extension of this approach for VROA measurements has been suggested 6, but no measurements were undertaken. For VROA measurements, one may modulate the polarization of the incident laser beam between fight and left circular polarization states and measure the synchronous difference in Raman scattered intensities. Alternately, one can use the linearly polarized incident laser light, and measure the difference in the scattered Raman light intensifies with fight versus left circular polarizations. Polarization division interferometers (PDIs) 7 provide an alternate Fourier transform approach for infrared and Raman spectral measurements, including VCD and VROA. Thus, three different instrumental techniques are available for a practicing spectroscopist. Linear dichroism measurements are commonly made by orienting a linear polarizer in two different orthogonal orientations, measuring the absorbance and taking the numerical difference between these absorbance measurements. A combination of linear polarizer and quarter-wave retarder (QWR), with the axis of the linear polarizer oriented at +45 ~ and-45 ~ to the unique axis of the QWR, generates opposite circular polarizations. Thus by orienting the QWR in a fixed orientation and placing the linear polarizer in two orthogonal orientations or by orienting the linear polarizer in a fixed orientation and placing the QWR in two orthogonal orientations, the measured absorbance difference represents circular dichroism. A variation in this theme is to rotate the linear polarizer or QWR continually around the light propagation axis and to use synchronous detection (vide infra). However the introduction of photoelastic modulator (PEM) 8, provided a means for obtaining both better accuracy in polarization modulation and high frequency polarization modulation. The details involved in polarization modulation using PEM, rotating QWR and combination of these two are discussed below. The double polarization modulation concepts presented in this chapter are new, and experiments have yet to be undertaken. Another polarization modulation approach of considerable potential, especially for the far-infrared region, is to use a polarization division interferometer as an achromatic polarization modulator 9. This approach will be discussed in Chapter 13. In this chapter the principles involved in polarization modulation and signal processing are presented first, then the dispersive spectrometers are described. Conventional infrared absorption measurements are automatically involved during the measurement of dichroism, as are the
Principles of Spectral Measurements
237
normal Raman measurements during Raman optical activity measurements. For this reason, the conventional infrared absorption and Raman measurements will not be discussed separately with special emphasis in this chapter. The next two chapters focus on amplitude and polarization division interferometers for the same measurements with a similar emphasis. The readers who are not interested in signal analysis and instrumental details can skip these chapters and go directly to Chapter 14. 11.2
Polarization m o d u l a t i o n using a photoelastic m o d u l a t o r The concept involved in the operation of a PEM is that a periodic stress applied to an isotropic crystal causes synchronous variation of the difference in the refractive indices along two mutually perpendicular axes. One of these two axes is the axis along which the stress is applied. For linearly polarized incident light with the polarization axis at 45 to the stress axis of the crystal, one can resolve the incident electric vector into two components, one parallel and another perpendicular to the stress axis. The periodic variation in the birefringence of the crystal introduces timedependent phase lag into these two electric vector components of the incident light. The phase variation in time t follows the relation
8m = 8 0 i sin COmt ' Vi where tom =2rCVm is the frequency of stress modulation,
(ll.2.1) and
~0 i the
maximum phase shift introduced for wavelength )q or wavenumber vi. When the maximum stress applied corresponds to a phase shift of 8 0 = re/2, then the radiation of wavenumber ~i is said to be circularly polarized. Alternately, one of the two electric vector components incident on and parallel to the stress axis of the PEM is said to have undergone a 90 ~ phase shift or quarter-wave retardation relative to the other. For a given maximum stress on the PEM, precisely X/4 or quarter-wave retardation is achievable at only one wavenumber, which is represented by Vq. For other wavenumbers at the same PEM setting, the maximum phase shift becomes 50 = 5 0 Vi
Vq
Vi _ Zt Vi _ rt)Vq ~q 2 ~q 2 )~i
(11.2.2)
As sincomt goes through +1, 0, and -1, 8~% goes through +rt/2, 0, and -~/2, which means that the ~q component goes through right circularly
Chapter 11
238
polarized, linearly polarized, and left circularly polarized states, respectively. For the intermediate values of sincomt, corresponding intermediate polarization states are achieved. Thus, only for a minor portion of sincomt cycle is the wavenumber component ~q circularly polarized (to be precise, only at sin Ohnt = +1), and for the remaining portion of the cycle the polarization states are those of intermediate nature. Similarly, when 8 0vi - rc the linear polarization would have been rotated by 90 ~ and for one
sincomt cycle there would be two cycles of linear polarization modulation. Again the desired linear polarizations are obtained only at sino)mt = 0 and +1. Due to the presence of different polarization states at intermediate points of the sincomt cycle, it is important to understand the nature of the signal expected at the detector 10. Two different experimental arrangements will be useful to understand the signal processing involved. In one arrangement (see Fig. 11-1), a monochromatic light of wavenumber Vi is passed through a linear polarizer (P), PEM and an optically active sample (S), in that order, finally reaching a suitable detector (D). In the second arrangement (see Fig. 11-2),
9 P
PEM
S
D
Fig. 11-1. Sample transmission configuration. P: linear polarizer; PEM" photoelastic modulator; S" sample; D: detector.
9 P
PEM
D
Fig. 11-2. Calibration configuration. P, PEM and D are as in Fig.ll-1; B: birefringent plate; A: linear polarization analyzer. the optically active sample is replaced by a birefringent plate (B) and linear polarizer (now called analyzer (A)). The former arrangement will be referred to as sample transmission configuration, and the latter arrangement
239
Principles of Spectral Measurements
will be referred to as calibration configuration. Regardless of the variation in the experimental arrangement, the z axis is the direction of light propagation and the optical axes of the PEM concide with x and y axes; the polarization direction of the linear polarizer will be considered to be at 45 ~ to the x and y axes.
11.2.1 Circular dichroism (A) Sample configuration The electric vector of monochromatic light of wavenumber Vi , after passing through the first polarizer, is given as [FS(Vi)/~/-2](u + v), where and v are the unit vectors parallel to x and y axes, respectively, and FS( Vi) is the amplitude of initial electric vector. As the PEM imparts a timedependent relative phase lag Gm at wavenumber Vi, to one of the electric Vi '
vector components, the resulting electric vector, after passing through the PEM, becomes
1/o +vei'm/
(11.2.3)
If the fight and left circularly polarized vectors are denoted 10 as [FS( Vi) / ,(-2)]x(u + iv) and [FS(Vi)/~/-2](u - iv), respectively, it can be seen that Eq. (11.2.3) is equivalent to
Ill
+ +/1+ l/u_ v,l (11.2.4)
When this electric vector passes through an optically active sample, the fight and left circularly polarized components are absorbed to different extents. The resulting electric vector is
240
Chapter 11
/ m/
+ 1 + ie 1~5~i (u - iv)e -aL (~i)/2
1 .
(11.2.5)
The intensity of light exiting the sample is calculated as the product of this electric vector with its complex conjugate. The voltage output of a linearly responding detector can be represented by the intensity falling on it, which in the present case is I ( ~ i ) - [IS(~i)/2] [(e -aR(~i) + e-aL(~i))+ (e -aR(~i) - e-aL(~i))sin8 m ]
Vi ,
(11.2.6) where IS(vi) is the initial intensity of the wavenumber component vi, aL(vi ) = 2.303AL(Vi), and aR(vi) - 2.303AR( vi)- Because ~m vi is time dependent, sin ~i-m vi can be expressed 11 as sin 8mvi --sin(80,kv sin corot) - 2 E J 2 n - 1 (~0 i ) sin[(2n -1)O~mt ] , n--1
(11.2.7)
where n is an integer and Jn(80Vi) are Bessel functions. From Eqs. (11.2.6) and (11.2.7) it can be seen that the detector signal contains a timeindependent (often referred to as dc ) signal, and a time-varying (referred to as ac) signals at frequency ram, 3ram, etc. The signal at the fundamental frequency ram, can be isolated by passing the detector signal through a band pass filter centered at mm and a lock-in amplifier tuned to COm. Similarly, the dc signal can be isolated by eliminating the signals at frequencies ram, 3ram, etc., using appropriate electronic filters. Then the ratio of the signal demodulated at mm to the dc signal becomes
Icom (Vi) Idc(Vi)
eaLi/Ol
e-aR(vi ) + e-aL(vi ) Gf
(11.2.8)
where Gl and Gf are the gains introduced by the electronics of the lock-in amplifier and filters, respectively. Multiplying the numerator and
Principles of Spectral Measurements
241
denominator of Eq. (11.2.8) by e (aR (~i)+aL(~i))/2, and noting that for small values of 13, (e~-e-~)/(e~+e-~) - tanh 13--- 13, one obtains
I0.)m(Vi)/Idc(Vi)- 2Jl(~0 i ){[aL(Vi)-
aR(Vi)]/2}(G//Gf).
= 2J1 (~i0i ){2"303[AL (Vi)- AR (~i)]/2}(G//Gf) If Vi is equal to Vq, then 80i -re/2 and
J1 (re/2)---0.57.
(11.2.9) The value of
Jl(~50vi)G//Gf can be determined from a calibration spectrum. (B) Calibration configuration Here the optically active sample is replaced by a birefringent plate 12 and analyzer together (see Fig. 11-2). These two components can be oriented with respect to the preceding PEM and first polarizer in four different ways. (a). The fast axis of the birefringent plate is parallel to that of the PEM and the direction of the analyzer is parallel to that of the first polarizer. In addition to the time dependent phase difference previously introduced by the PEM, additional time independent phase lag 8 B vi is now introduced by the birefringent plate, which is given as 2r~idAn; here d and An are respectively the thickness and birefringence of the plate. Then the electric vector exiting the analyzer becomes
F(Vi) - [uS (vi) / 2] Ill + vei(~5~m+sBi)/(11 + V) ,
(11.2.10)
from which the intensity can be found to be
I(~i)- [IS(~i)/2] (1+ COS~im COsfiBvi- sin 8mVisin~SBvi) 9
(11.2.11)
Using Eq. (11.2.7) and a similar expression 11 for cos 8~m ,
cos 8~m- J0 (80i)+ 2Z J2n (80i )cos 2nf.Omt, n=l the expression analogous to Eq. (11.2.9) becomes
(11.2.12)
242
Chapter 11
I0~m(~i) = -2Jl(fiOi)sinSB i G l Idc(Vi)
(11.2.13)
1+ J0(~Oi )cos~ BVi Gf
(b). Let the polarization direction of the analyzer be perpendicular to that of the first polarizer, and the fast axis of the birefringent plate be parallel to that of the PEM. Then (u + v) in Eq. (11.2.10) becomes ( u - v), and the expressions analogous to Eqs. (11.2.11) and (11.2.13) become I(vi)-[IS(~i)/2] (1 - cos~5m m sin ~sB vi cos ~5B vi + sin S~i vi ) '
Ic0m(Vi) = Idc(Vi)
2J1 (80i )sin 8 Bvi G l 1-- J0(~Oi)cos 8 Bvi Gf
(11.2.14)
(11.2.15)
(c). Let the polarization axis of analyzer be perpendicular to that of the first polarizer and the fast axis of birefringent plate be perpendicular to that of the
i(8~n]+~sBi)
PEM. Then e and (u + v) terms in Eq. (11.2.10) become ei(8~m -8 -B vi" and ( u - v), respectively, and the expressions analogous to Eqs. (11.2.11) and (11.2.13) become
)
I(~i) - [Is (vi)/2] (1 -- cos 8 m Vi COS8 Bvi--sin 8 mVi sin 8 BVi),
Io~m (Vi) = Idc (Vi)
-2J1 (~0i )sin fiBvi G l 1-J0(~Oi )cos~ BViGf
(11.2.16)
(11.2.17)
(d). Let the polarization direction of the analyzer be parallel to that of the first polarizer, and the fast axis of the birefringent plate be perpendicular to that of PEM. Then the expressions analogous to Eqs. (11.2.11) and (11.2.13) become (11.2.18)
243
Principles of Spectral Measurements IC0m(Vi)= 2Jl(80i)sinSBv i Idc ( V i ) 1
_Gl + Jo(8Oi)cos~i-BV i Gf "
(11.2.19)
Note that Eqs. (11.2.13) and (11.2.19) are equal but of opposite sign, as are Eqs. (11.2.15) and (11.2.17). Also Eqs. (11.2.13) and (11.2.17), like Eqs. (11.2.15) and (11.2.19), are equal to each other with a nonzero magnitude of +2J1(80V i )G//Gf at cos 8 B = 0 and sin gB _ +1 Vi Vi -
-
-
-
This is .
applicable when,
~5B =-v i r t = ( 2 n + l ) rC Vi
VB 2
-~ ,
(11.2.20)
where ~B is the wavenumber for which the birefringent plate introduces a single quarter-wave retardation and n is an integer. Furthermore, Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19) are equal to each other, with zero magnitude, when sin 8 -B - 0. This condition is met when vi m
fib = _vi = nrc. vi VB 2
(11.2.21)
In practical terms, if the maximum stress setting on the PEM corresponds to one quarter-wave retardation for the wavenumber Vq and the light components of various wavenumbers are investigated using the calibration arrangement, then one would get four curves (see Fig. 11-3) represented by Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19). The non-zero crossings of these curves provide the values of 2J1(80. vi )G//Gf. These values can be interpolated to the desired wavenumber and used in Eq. (11.2.9) to determine the CD of a given sample.
11.2.2 Linear dichroism (A) Sample configuration For considering linear dichroism, Eq. (11.2.3) can be rewritten as
F(~ i) - IF s(Vi)/~-8]
I/ / / / 1 1+ e
(u + v)+ 1 - e
(u- v).
(11.2.22)
Chapter 11
244
Denoting the base e absorbance for parallel and perpendicular polarizations as ap(~i ) and as(Vi ) respectively, Eq. (11.2.22) is modified to represent the electric vector after passing through the oriented sample as I
,,
, I ,
,,
I
,,
, I ,
,,
I , ,
,
I,
,,
!
I
1.0
m
0.5 m
m
0.0
-0.5
-1.0
2000
1800
1600
1400 1200 ~aavenumbers
1000
800
Fig. 11-3. Calibration curves represented by Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19). Birefringent plate is assumed to have a thickness of 0.25 cm and birefringence of 0.02; ~q = 1000 cm -1. The values of Jl( ~0 i ) were determined from the analytic formula.
F(vi) - [F s (Vi) / ~
ll/ / 1+ e
(u + v)e -ap (vi)
+ 1-e
~ ( u - v ) e -a~(~i) o
(11.2.23)
Principles of Spectral Measurements
245
The intensity of fight exiting the sample, determined from the product of Eq. (11.2.23) with its complex conjugate, is I ( v i ) - [IS(vi)/2][(e -ap(~i) + e-as(Vi))+ (e-ap (~i)_e-as
(Vi))cosS~ii] (11.2.24)
From the expansion of cos fvm term (see Eq. (11.2.12)), one would notice that the signal detected would contain components oscillating at frequencies 2corn, 4corn etc. The signal oscillating at 2corn, normalized to the dc signal becomes
e-a
I2a~m (Vi) Idc(Vi)
{[e-ap(Vi) +e -a~(~i)
+J0(~Oi
1
]O1
e-ap(Vi)-e -a~(~i)
1}
6f
"
(11.2.25)
When linear dichroism magnitudes are small (which is not necessarily true in general), the second term in the denominator of Eq. (11.2.25) may be ignored in relation to the first term. In such situations,
I2com(Vi)
"- 212 (~0 i )tanh[Aa(vi)/21(G//Gf), Idc (~i) where Aa( ~i)/2 =(as(~i) - ap( ~i))/2 - 2.303 AA(~i )/2.
(11.2.26)
(B) Calibration configuration Again a calibration measurement is needed to determine the value of J2( 80vi)(G//Gf) in Eq. (11.2.26). The calibration measurement can be performed just as in the case for circular dichroism, except that the calibration signal would now be extracted from lock-in amplifier tuned to 2O3m. Then the terms Icom and sin8 B Vi, in Eqs . (11 . .2.13), . (11 2 15) (11.2.17) and (11.2.19), will be replaced by I2com and cos 8 B vi respectively; 2J 1( riovi ) will be replaced by-2J2(~50vi ) in Eqs. (11.2.13) and (11.2.15) and by 2J2( 8 o. vi) in Eqs. (11.2.17) and (11.2.19). The predicted curves are
Chapter 11
246 shown in Fig. 11-4.
The m a x i m u m values of these curves give
+_.2J2(~50i ) / [ 1 + J0(~Oi )] (G//Gf) from which J2(~0i)G//Gf can be found. 1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5 2000
Fig.
11-4.
1800
Calibration
1600
curves
1400 1200 wavenumbers
for
linear
I000
dichroism
800
represented
by
___2J2(~0 i )cos~Bvi /[1 + J0 (~Oi)cos~ vi B ]" Birefringent plate is assumed to have a thickness of 0.25 cm and birefringence of 0.02", Vq = 500 cm-1 . The values of J0( ~i0vi) and J2( ~0vi ) were determined from the analytic formulae. 11.3
Polarization modulation using a quarter-wave retarder The PEM shown in Fig. 11-1 can be replaced by a rotating QWR for circular dichroism measurements. For a broad wavelength coverage, however, one needs an achromatic retarder. Such a retarder can be constructed 13 from two different birefringent plates, with their fast axes in the plane of the plates and at 90 ~ to each other. The retardance of the combined plates is given by the relation, (2~/~)[Anltl-An2t2], where An is the birefringence and t is thickness; the subscripts 1 and 2 identify the two birefringent materials. Such a retarder made from CdS and CdSe plates has
247
Principles of Spectral Measurements
been reported 14 for the mid-infrared region. The signals applicable with rotating QWR, which are different from those applicable with PEM, are given in this section. For the configuration shown in Fig. 11-5, the electric vector exiting the rotating QWR is found to be, F(~i) - [Fs (~i)/x/2] X {(u + iv)cos 2 cot + i(u -iv)sin 2 tot + (u + v)(1- i) sin mt cos rot},
(11.3.1)
where to is the rotating frequency of achromatic QWR. With an optically active sample placed after QWR, the signal at the detector becomes, I(vi) = [IS(vi)/2] [e -aR(~i) cos 2 cot+ e -aL (~i)sin 2 cot] --[IS(~i) / 2] e -(aL (~i)+aR(~i))/2 [1 + (Aa(Vi) / 2)cos2mt].(11.3.2) Note that the circular dichroism signal is modulated at 20) (i.e. at twice the frequency of QWR rotation). It is easy to see from Eq. (11.3.2) that I2dIdc gives circular dichroism of the sample. For calibration purposes, the _
V
P i
QWR
S
D
i
Fig. 11-5. Sampletransmission configuration for measurementsusing a rotating quarterwave retarder(QWR); P, S, D are as defined in Fig.11-1. configuration shown in Fig. 11-6 can be used (where the PEM in Fig. 11-2 is replaced by a rotating QWR) and the signal in this case becomes I(vi ) - [Is (Vi)/2 ]((1 + 0.5cos 8 B vi ) - s i n ~5B vi cos 2cot- 0.5cos 8 B vi cos 4cot). (11.3.3) vi )" Following the The ratio I2JIdc now gives -sin 8Bi/(l+ 0.5cos 813 procedure in Section 11.2.1B, one can easily derive similar relations for the other three orientations (of the bireffingent plate and analyzer); similarly the expressions for the linear dichroism of the sample can be obtained following
Chapter I 1
248
the procedure in Section 11.2.2. Dichroism measurements using a rotating stressed birefringent plate have been reported. 15
9 P
QWR |
B
A
D
m
Fig. 11-6. Calibration configuration for measurements using a rotating quarter-wave retarder(QWR); P, B, A and D are as defined in Fig.11-2.
11.4
Double polarization modulation
One may achieve better dichroism signal quality with double polarization modulation as shown in Fig. 11-7. Here a rotating QWR and PEM are used together. For the configuration shown in Fig. 11-7, the i,
....
|
xx
P u,
QWR
PEM
S
D
Fig. 11-7. Sample transmission configuration for measurements using double polarization modulation; P, PEM, S, D are as defined in Fig.11-1. electric vector exiting the PEM is found to be
F(v i) - IF s (v i) / 2"V~] x [(u + iv) x A + (u - iv) x B],
/ m/ / / t/ m/ / lvi/t
where
1~.
A-
l+e
" ~
~ cos 2 m t + i 1 - e ~Svi sin 2 mt +
1~.
1-e
9 m
1
-i
l+e
sin mt cos mt,
(11.4.1)
Principles of Spectral Measurements
B=
249
/ m/ /ei i/ 1 - e ~8~i cos 2 m t + i 1+
1+
~i
_ i 1- e
" sin 2Cot+
sin cot cos cot.
With an optically active sample placed after the PEM, the signal becomes, I ( v i ) - IS(vi)e-(aL(~i)+aR(~i))/2{1 + (Aa(vi)/2)[J0(80 i )cos 2t.ot + J2 (~0i) cos(o~m + 2to)t +
J2 (~50i )cos(~ m -
2tO)t + J1 (~0i)sin COmt
--0.5(sin(~ m + 4co)t + sin(~ m - 4to)t)]} . (11.4.2) It is important to note that the circular dichroism signal now appears at 2o), tom, tom +2oo and tom +4o). There are two approaches for detecting the signals at 2o), O~m, O~m+2o) and corn+ 4co sirnultaneously. One approach is to use a digital signal processing method 16 and the other is to use mutiple lock-in amplifiers. If all these components can be detected simultaneously (vide infra) then the relations 11 among the Bessel functions (see Chapter 13) indicate that the measured dichroism signal becomes -2 times larger than the signal measured with PEM or rotating QWR alone. The analogous expressions for linear dichroism and calibration signals can be derived in a similar manner.
11.5 Dispersive spectrometers For a given wavelength, the signals resulting from polarization modulation are described in the previous sections. Light sources, other than lasers, are usually polychromatic. Selection of a given wavelength (to be more precise, a narrow bandwidth of wavelengths) is achieved in dispersive spectrometers, which were widely used in the first three quarters of this century. Although they are not as popular in current times (due to a wide spread use of the Fourier transform techniques), the dispersive spectrometers are unavoidable for some applications. The central component in these spectrometers is a spectral dispersing element, such as a prism or diffraction grating. When polychromatic light is directed to a reflection grating, different wavelengths are diffracted at different angles. The relation governing this dispersion is given 17 as d(sin0 + sin q~i) - m~. Here, 0 is the angle of incidence (measured from grating's surface normal), q~i is the angle of diffraction for wavelength ~,i, d is the distance between successive grooves on the grating, and m is a integer (called the order
Chapter 11
250
number). At higher orders, not only does the spatial separation between two wavelength components increase, but a wavelength of one order may fall imo the region of another order. Thus a filter blocking the wavelengths of a different order might be necessary. When single element detectors are used, the dispersed light is passed through an exit slit (whose width determines the spectral resolution) on to the detector. In this case, a scanning mechanism is used to rotate the grating so that a different wavelength component goes through the slit to the detector. Thus dispersive spectrometers using a single element detector are called scanning spectrometers. With the availability of multi-element detectors (also called detector arrays), the monochromator is replaced by a spectrograph, where the exit slit is replaced by a wide opening to hold the detector array (see Fig.
input slit
polychromatic light
mirror
grating exit slit or detector array
Fig. 11-8. Schematic of a monochromator/spectrograph. 11-8). In this case the spectral resolution isdetermined by the width of individual detector elements (referred to as pixels), which is typically ~25 ktm. The grating needs to be rotated to a new orientation only when the spectral region of interest is not covered by the combined width of the detector elements. Various methods centered around these dispersive spectrometers are discussed below.
Principles of Spectral Measurements
251
11.5.1 Dichroism measurements For a given wavenumber vi, or wavelength ~i, selected with a monochromator the signals expected at the detector with appropriate polarization modulation can be found in the previous sections. In dispersive spectrometers it is common practice to use a light chopper to modulate the source intesity. In such cases, the signals described in the previous sections have to be multiplied 1 by (1 +sino~ct)/2, where O~cis the light chopper frequency (~ 100 Hz). A block diagram of a dispersive CD spectrometer using a PEM for polarization modulation and a light chopper for source intensity modulation is shown in Fig. 11-9. Multiplying Eq. (11.2.6) with(1 +sino~ct)/2, one notices that the CD signal varies as both sin~ m and sin~Smsino~c t, while the sample Vi Vi transmission signal varies as sino~ct. The CD signal component can be extractedl by processing the detector signal with a lock-in amplifier tuned to COm. Alternately, one can use 18 two lock-in amplifiers (the output of lockin tuned to ohn is processed by a lock-in tuned to O}c)for extracting the CD signal. The CD component extracted in either manner needs to be normalized with the sample transmission signal, which is extracted using a detector mono-
chr~176 _ source chopper o)
vi
~
PEM )
~ "~ O~m polarizer
C
o)
Lock-in Amplifier -= 100 Hz
m
I 0)
C
0)
Lock-in rn Amplifier
"
-=-30-100 KHz
Fig. 11-9. Schematic of a dispersive circular dichroism spectrometer.
I I ! ! ! I I I I | ! I I I ! ! ! I ! ! ! I ! I
252
Chapter 11
different lock-in amplifier tuned to toc. Fig. 11-10 shows the first reported VCD spectra, for their historical importance. Modem instruments provide spectra with much improved signal quality. For linear dichroism signals, Eq. (11.2.24) is multiplied with (l+sincoct)/2 and the procedure for extracting the signals is similar to the one described above for CD, except that since the linear dichroism signal varies at 2Ohn the ohn lock-in used for CD is replaced by a lock-in tuned to 2o~. I
'
!
L
,
l
30OO
9
~ cm-I Fig. 11-10. The first reported VCD measurements using a dispersive VCD spectrometer. The traces with labels (+), (___) and (-) represent respectively the VCD spectra of the liquid samples of (+)-, racemic- and (-)-2,2,2-trifluoro-l-phenylethanol, shifted from each other for clarity. The top most trace is the infrared absorption spectrum. Reproduced with permission from Ref. lb. Copyright 1974 American Chemical Society.
When polarization modulation is achieved with a QWR rotating at frequency co, Eq. (11.3.2) is multiplied with (1 +sincoct)/2 and the COrnlockin amplifier shown in Fig. 11-9 is replaced with a lock-in amplifier tuned to 2to (remember to keep co>>toc). The recent emergence of digital signal processing techniques 16 eliminates the need for lock-in amplifiers; here the detector signal for a given wavelength is digitized (at time intervals determined by the higher modulation frequency) as a function of time and the Fourier transform of this time domain signal contains the intensity at different modulating frequencies. Such digital signal processing techniques are particularly important for realizing the advantages of double polarization modulation approach described in Section 11.4.
Principles of Spectral Measurements
253
11.5.2 Raman optical activity measurements For Raman spectral measurements, the sample of interest is illuminated with a laser and the intensity of scattered light is analyzed at wavenumbers that are away from the incident laser wavenumber. The differences between the wavenumbers of scattered and incident light correspond to the differences in energies of vibrational levels. The monochromator/spectrograph discussed earlier is placed in the scattered beam for wavelength selection. Although monochromators were widely used in Raman spectroscopy, the availability of multi-element detectors resulted in a wide usage of spectrographs. Due to the nature of multielement detection, signal processing is not done using lock-in amplifiers. So the polarization modulation here takes a different form (vide infra). For vibrational Raman optical activity (VROA) measurements 19-29, the incident laser beam can be modulated between right and left circular polarization states and the synchronous difference in the scattered Raman intensities measured 4 as a function of vibrational Raman shifts. This approach is referred26, 28 to as incident circular polarization (ICP) VROA (see Fig. 1111). VROA can be measured for 90 ~ scattered Raman light (Fig. 11-11A) with an analyzer parallel or perpendicular to the scattering plane; the former i"
D
D
SG
/\ 9
"
A
L
L Las
P
M
(A)
S
P
M
(B)
Fig. 11-11. Schematic of a Raman optical activity spectrometer with 90~ scattering (A) and 180~ backscattering (B) geometries using incident circularly polarized light. P: polarizer; M: modulator; S" sample; L: lens; A: analyzer; SG: spectrograph; D: detector. See Refs. 19-23 for details. is referred to as depolarized VROA4,19, while the latter as polarized VROA 20. If the analyzer in the scattered beam is placed 21 with its axis at 54.7 ~ from the scattering plane, the resulting VROA is called magic angle
Chapter 11
254
VROA. One may also choose not to include an analyzer in the scattered beam. In back- (180 ~ (Fig. l l - l l B ) and forward (0 ~ scattered geometries22, 23 ICP-VROA is measured with no analyzer in the scattered light. A brief description of the procedures followed to make a measurement is as follows (see Fig.11-11). The incident laser light is first linearly polarized; polarization modulation between circular polarizations is achieved using an electro-optic modulator (EOM) 4, QWR 24, or a liquid crystal variable retarder 25. In the EOM case, the voltage supplied to the EOM is usually a bipolar square wave, so the positive and negative halves of the voltage correspond to the two circular polarizations. The data collected during two halves of the square wave are stored in two different files. The first reported VROA measurements are shown Fig. 11-12, for their historical importance. Modern instruments provide spectra with much improved signal quality.
H---C--OH
I
CNjL
-
_
_
_ .
~Jd
J
I
+
-~- 1~cl
.,--..
"~ <]
.1 x.~_
,.
.
9
.
-, .
.
9
9 ,,
Fig. 11-12. The first reported depolarized VROA spectra (bottom traces), using a dispersive scanning ROA spectrometer, for the enantiomers of a-phenylethanol (left half) and o~-phenylethylamine(right half). The dotted lines are for (+) enantiomers and solid lines for (-) enantiomers. The top most traces are depolarized Raman spectra. Reproduced with permission from Ref. 4. Copyright 1973 American Chemical Society. In the QWR case, the orientation of the QWR is fixed in one position to give one circular polarization and data are collected into one file; then the axis of the QWR is rotated by 90 ~ to generate opposite circular polarization
255
Principles of Spectral Measurements
and the data are collected into another file. In the case of liquid crystal retarders, the birefringence of the liquid crystal depends on the square of applied electric fried" they are driven by a--2 kHz bipolar voltage (which is needed to prevent "run-away" of liquid crystal molecules into one permanent orientation) and by changing the magnitude of this bipolar driving signal, one achieves the switching of one circular polarization to another.
D
D
S
SG
A
A
L Laser
QWR
L
p
P S
(A)
(B)
Fig. 11-13. Schematic of a SCP-ROA spectrometer (A) and DCPI-ROA spectrometer (B). P: polarizer; S: sample;QWR: quarter-wave retarder; L: lens; A: analyzer; SG: spectrograph; D: detector. See Refs.24, 26 and 28 for details. An alternative to the measurements described above is to leave the incident laser beam linearly polarized, and measure the difference between fight and left circularly polarized intensities of the scattered Raman fight (see Fig. l l-13A). Such measurements are referred to as scattered circular polarization (SCP) VROA measurements 26 and have been reported for 90 ~ scattering geometry. In principle, polarization division interferometry (see Chapter 13) is an attractive approach for this measurement 27. In another type of VROA measurement, the incident laser beam is modulated between fight and left circular polarizations and the synchronous difference in the intensities of fight and left circular polarization components of the scattered Raman light are measured (see Fig. 11-13B). These measurements are referred to as dual circular polarization (DCP) VROA measurement 28. If the scattered light was analyzed for fight circular polarization when the incident light is fight circularly polarized (and for left circular polarization when the incident light is left circularly polarized) the measurement is referred to as DCPI VROA. On the otherhand, if the scattered light is analyzed for fight circular polarization when the incident laser polarization is
Chapter 11
256
left circularly polarized (and vice versa), the measurement is referred to as DCPII VROA. The DCPI measurements in the 180 ~ backward scattering geometry have been reported 28. VROA measurements are prone to artifacts and careful control 29 of these artifacts is necessary.
11.5.3 Time resolved infrared measurements Various approaches are now available for monitoring the temporal changes using infrared spectroscopy. A summary is provided in Fig. 1114. The most straightforward approach is to use a monochromatic fight Time resolved infrared spectroscopy
ContinuOUSscanI Rapid sweep I
~troboscopic I
~
1Asynchronous ~
IC~176 Asynchronous
Fig.11-14. Variousmethods for time resolved infrared spectroscopic measurements. source (either a laser or polychromatic light dispersed by a grating) with a fixed polarization and monitor the changes in intensity at desired time intervals 30. This method however restricts the observations to a single wavelength and the entire procedure has to be repeated for each of the other wavelengths of interest. Interferometry obviously overcomes this limitation, but some other considerations need to be evaluated before undertaking time resolved measurements using interferometry (see Chapters 12, 13). (A) Measurements using boxcar integrator A block diagram for time resolved infrared spectrometer 31 is given in Fig. 11-15. A desired time dependent stimulus is given to the sample and the intensity of the light component (with fixed polarization at wavelength ~ or wavenumber vi) transmitted through the sample is determined using
Principles of Spectral Measurements
Vi
Sample A --
257
Dectector "V
[ Trigger]---B"[ Box'CarI
V Fig. 11-15.
Schematic diagram for time resolved measurements using a boxcar
integrator. a boxcar integrator. Using a reference signal, synchronized to the stimulus provided, the boxcar integrator can be configured to record the detector signal after a certain delay from the stimulus and the signal integrated for a certain time window. Both delay and integration time windows can be set as desired. The signal recorded by the boxcar represents the temporal change in intensity of light transmitted through the sample, IS(Vi) e-a(t'vi). The sample transmission at t = 0, i.e. in the absence of the external stimulus, can be obtained by collecting the data as before, but without providing the stimulus to the sample (but a reference signal representing the stimulus given in the previous measurement is needed for the operation of boxcar integrator). The signal measured now is IS(Vi)e -a(t=0'~i), but measured at different delay settings of the boxcar integrator. From the logarithmic ratio of these two signals, the time dependent changes in absorbance, for a fixed polarization, can be extracted. (B) Measurements using lock-in amplifier This approach 32 is suitable for cases when the external stimulus given to the sample is oscillatory, for example a sinusoidal perturbation. If this perturbation has a frequency roe, a chosen molecular property in response to this perturbation may also vary at this frequency, although there can be a phase lag between the applied perturbation and the chosen molecular property. Thus the temporal behaviour of that molecular property can be written as o~ = (x0 + (x(t) = o~0 + & sin ({Oet+ 13),
(11.5.1)
258
Chapter 11
where ~0 is the value of the molecular property that does not change with time, and [3 is the phase lag. The time varying portion of the molecular property can be written as o~(t) = ~ip sin ~et + O~opcos O~et,
(11.5.2)
with o~ip and O~oprepresenting respectively the in-phase and out-of-phase (also called quadrature) components of molecular property and are given as ~ i p - ~ cos ~,
(11.5.3)
0~op = & sin [~.
(11.5.4)
If Oqp and O~opcan be measured separately then the phase lag ~ (which is called the dissipation factor) can be determined from Eqs (11.5.3)-(11.5.4) as
C%p/Oqp - tan $.
(11.5.5)
The temporal characteristics of the molecular property can then be extracted through Eq. (11.5.2). The in-phase and out-of-phase components of the molecular property can be determined using a lock-in amplifier. This principle is the basis for two recently developed 32,33 novel spectroscopic methods, namely dynamic infrared linear dichroism (DIRLD) and two dimensional infrared spectroscopy. These two methods are described below.
11.5.4 Dynamic infrared linear dichroism (DIRLD) The inherent linear dichroism associated with oriented samples can be perturbed with a dynamic perturbation such as a mechanical stress or electrical voltage. In the case of oriented polymers the implications of mechanical stress are of importance. So the question to be addressed here is one of molecular reorientation as a function of the external stress modulation. As linear dichroism is related to the molecular orientation, the variation of linear dichroism with stress modulation can be related to molecular reorientational dynamics. A rheometer is commonly used to modulate the applied stress. A block diagram for DIRLD measurements 32 is given in Fig. 11-16. The source light is chopped by a mechanical chopper at frequency tOc and dispersed by a monochromator. At the wavelength component of interest, the light is polarized by a linear polarizer P and modulated between two orthogonal linear polarizations by a photoelastic modulator (PEM) at 2COm. As a result of this polarization modulation, the detector signal contains a component that represents linear
Principles of Spectral Measurements
259
dichroism of the sample. If the sample is subjected to a mechanical stress perturbation at frequency COsthe linear dichroism undergoes changes at that frequency. In order to isolate the desired signal, these perturbation
Source
Chopper H chromator Mono~
PEM ~
I1
Otor
Sample
.
|
Controller 0)c
m
|
|
I
!
Controller O~m
Controller
|
!
|
COs
al nl an an
|
i
Lock-in 0~c
t
,,
w,.
o ,.
,.
,,
Lock-in
Lock-in O~m
m. . , , ., , .
. , , ,.
. , , . i ,.,
.
.m , ,. , . .
. u.
. , ,. m
.w . a . m. m. m . . m ..
. e. m w
a
w
~
t
w
u w
w
i . .
COs
~. .m . -.-
i
,-...
_----5 ....... t ............ tai l
Fig.11-16. Schematic for dynamic infrared linear dichroism measurements.
frequencies are to be well separated from each other. The usual values are COs--- 10 Hz, COc---1000 Hz and 2O~m---75 kHz. The signals from each of these perturbations are to be demodulated in the decreasing order of frequency. The detector signal is first demodulated using a lock-in amplifier tuned to 2O~m, with minimal time constant so that the signals varying at O~c and COs are not attenuated. The output of the 2O~m lock-in amplifier (containing signals varying at O3cand COs)is then demodulated by a lock-in amplifier tuned to COc and a time constant low enough not to attenuate the COs signal. The output of the O~c lock-in amplifier is then demodulated by a COslock-in amplifier. When demodulated in-phase with the reference signal from the rheometer, the output of this lock-in is proportional to the component of linear dichroism that varies in-phase with mechanical stress. When demodulated out-of-phase with the reference signal from the rheometer, the output of the COs lock-in amplifier is proportional to the component of linear dichroism that varies out-of-phase with mechanical stress. These measured dichroism signals in conjunction with Eqs (11.5.2) to (11.5.5) provide temporal changes in dichroism. However, it should be noted that temporal resolution is dictated by the time constant of the COslock-in amplifier.
Chapter 11
260
11.5.5 Two dimensional spectroscopy For a system that responds to the time dependent external stimulus, cross correlation function for two different time varying signals can be given 33 as =
l lf
~(vl,t) ~(v2,t + x)dt.
(I 1.5.6)
T---)~, a-T/2
Here x is the time off-set (correlation time) at which time the function x(x ) is evaluated, ct(Vl) and tx(v2) are some appropriate signals monitored at wavenumbers vl and v2. At x - 0 the function x(x ) is simply the one that represents synchronous variation between the signals tx(~ 1) and ct(v2 )- If Eq.(11.5.6) is evaluated for different z values and is found to have maximum at a non-zero x value, then that particular x(z) indicates the magnitude of asynchronous variations between the signals tX(~l) and tx(~2). Then x represents the phase shift between the two signals. Thus if one were able to measure ct(Vl) and t~(v2) as a function of time, with sufficient time resolution, then such data will enable determination of synchronous and asynchronous variations among different molecular properties. An alternative to this approach is as follows. Suppose that the external stimulus is sinusoidal in nature. Then the time dependent variations in molecular properties being probed can be represented by Eq. (11.5.2) and one uses 33 the relations x('C) = t~(V1, V2) cos O~et+ ~(V1, V2) sin O~et,
(11.5.7)
where, (11.5.8) (11.5.9) The function q)(v1, V2 ) represents the synchronous correlation intensity and 9 (~1, ~2) represents the asynchronous correlation intensity. Since the molecular properties ~p(Vi) and tXop(Vi) can be determined using a lock-in amplifier (as explained earlier), the correlation intensities q)(vl, v2),
Principles of Spectral Measurements
261
9 (v1, v2) can be calculated. The plots of these correlation intensifies represent two dimensional spectra, with Vl and v2 represented along x and z axes respectively. The y-axis represents the correlation intensity. Such plots have been reported not only for infrared absorption but also for infrared linear dichroism. References
1 (a). I. Chabay and G. Holzwarth, Appl. Opt. 14 (1975) 454; (b). G. Holzwarth, E. C. Hsu, H. S. Mosher, T. R. Faulkner and A. Moscowitz, J. Am.Chem. Soc. 96 (1974) 251. 2 A.A. Michelson, "Light waves and their uses", The University of Chicago Press (1907). 3 P. R. Griffiths and J. A. de Haseth, "Fourier Transform Infrared Spectrometry", John Wiley & Sons, NY (1988). 4 L.D. Barron, M. P. B0gaard and A. D. Buckingham, J. Am. Chem. Soc. 95 (1973) 603. 5 T. Hirschfeld and D. B. Chase, Appl. Spectrosc. 40 (1986) 133; D. B. Chase, J. Am. Chem. Soc. 108 (1986) 7485. 6 P.L. Polavarapu, Spectrochim Acta 46A (1990) 171. 7 P.L. Polavarapu, Principles and Applications of Polarization Division Interferometry, John Wiley &sons (1997). 8 M. Billardon, J. Badoz, C. R. Acad. Sci. Ser. B 262 (1966) 1672; J. C. Kemp, J. Opt. soc. Am. 59 (1969) 950; L. F. Mollenauer, D. Downie, H. Engstrom and W. B. Grant, Appl. Opt. 8 (1969) 661. 9 P.L. Polavarapu, SPIE, 1166 (1989) 472. 10 K. W. Hipps and G. A. Crosby, J. Phys. Chem. 83 (1979) 555; J. C. Kemp, Polarized Light and Its Interaction With Modulating Devices, Hinds International inc., (1987). 11 A. Erdelyi, W. Magnus, F. Obeshettinger and F. G. Tricomi, Higher Transcendental Functions, McGraw Hill, New York (1953); M. C. Potter, Mathematical Methods in the Physical Sciences, Prentice Hall, Englewood Cliffs, New Jersey, 1978. 12 L. A. Nafie, T. A. Keiderling and P. J. Stephens, J. Am. Chem. Soc. 98 (1976) 2715. 13 M. Francon and S. Mallick, Polarization Interferometers, Wiley Interscience (1971). 14 D. B. Chenault and R. A. Chipman, Appl. Opt. 32 (1993) 4223. 15 P. Malon and T. A. Keiderling, Appl. Spectrosc. 50 (1996) 669; P. Malon and T. A. Keiderling, Appl. Opt. 36 (1997) 6141. 16 C. J. Manning and P. R. Griffiths, Appl. Spectrosc. 47 (1993) 1345. 17 F. A. Jenkins and H. E. White, "Fundamentals of Optics", McGraw Hill, NY (1976). 18 L. A. Nafie and M. Diem, Acc. Chem. Res. 12 (1979) 296; T. A. Keiderling, Appl. Spectrosc. Rev. 17 (1981) 189; M. Diem, G. M.
262
Chapter 11
Roberts, A. Barlow, and O. Lee, Appl. Spectrosc. 42 (1988) 20; P. Xie and M. Diem, Appl. Spectrosc. 50 (1996) 675. 19 L. Hecht and L. D. Barron, J. Raman Spectrosc. 25 (1994) 443. 20 W. Hug and H. Surbeck, Chem. Phys. Lett. 60 (1979) 186. W. Hug in "Raman spectroscopy" J. Lascombe and P. V. Huong, Eds. WileyHeyden, Chichester, UK (1982). 21 L. Hecht and L. D. Barron, Spectrochim Acta, 45A (1989) 671. 22 L. Hecht, L. D. Barron, A. R. Gargaro, Z. Q. Wen and W. Hug, J. Raman Spectrosc. 23 (1992) 401. 23 L. D. Barron, L. hecht, A. R. Gargaro and W. Hug, J. Raman Spectrosc. 21 (1990) 375. 24 L. Hecht, D. Che, and L. A. Nafie, J. Raman Spectrosc. 45 (1991)18. 25 P. L. Polavarapu (unpublished work). 26 K. M. Spencer, T. B. Freedman and L. A. Nafie, Chem. Phys. Lett. 149 (1988) 367. 27 P. L. Polavarapu, Chem. Phys. Lett. 148 (1988) 21. 28 D. Che, L. Hecht and L. A. Nafie, Chem. Phys. Lett. 180 (1991) 182; G.-S. Yu, L. A. Nafie, Chem. Phys. Lett. 222 (1994) 404. 29 W. Hug, Appl. Spectrosc. 35 (1981) 115; L. D. Barron and J. Vrbancich, J. Raman Spectrosc. 15 (1984) 47; J. R. Escribano, Chem. Phys. Lett. 121 (1985) 191; L. Hecht, B. Jordanov, and B. Schrader, Appl. Spectrosc. 41 (1987) 295; L. Hecht and L. D. Barron, Appl. Spectrosc. 44 (1990) 483; D. Che and L. A. Nafie, Appl. Spectrosc. 47 (1993) 544. 30 B. Locke, T. Lian and R. M. Hochstrasser, Chem. Phys. 158 (1991) 409; M. Lim, T. A. Jackson and P. A. Anfinrud, Science 269 (1995) 962. 31 T. Yuzawa, C. Kato, M. W. George and H. Hamaguchi, Appl. Spectrosc. 48 (1994) 684; K. Iwata and H. Yamaguchi, Appl. Spectrosc. 44 (1990) 1431. 32 I. Noda, A. E. Dowrey and C. Marcott, Appl. Spectrosc. 42 (1988) 203; B. Chase and R. Ikeda, Appl. Spectrosc. 47 (1993) 1350. 33 I. Noda, J. Am. Chem. Soc. 111 (1989) 8116; I. Noda, A. E. Dowrey and C. Marcott, Polymer News 18 (1993) 167; I. Noda, A. E. Dowrey and C. Marcott, Appl. Spectrosc. 47 (1993) 1317.
Chapter 11 PRINCIPLES OF SPECTRAL MEASUREMENTS
11.1 Introduction The appropriate experimental quantity that is related to the transition rate, as discussed in Section 2.5, is the absorption coefficient ix(v). To determine this coefficient, one should measure the incident light intensity I0(v--), the fight intensity I(v) transmitted through a sample of interest (both at a given wavenumber v), the sample concentration CO and the optical pathlength g in the sample. Since these quantities are related via Beer's law, ix(v) is obtained as ln[Io(v)/I(v)]/Cog using natural logarithm. The integrated absorption coefficient A(v-) is obtained by integrating o~(v) over the width of the vibrational band of interest (see Eq. 2.5.6). However, the spectral intensities are commonly displayed as absorbance A(v), where A(v-) = logl0[I0(v)/I(v)] = e(v)Cos - t~(v)C0s - a(v)/2.303. The difference in absorption coefficients of left versus right circularly polarized incident light, referred to as circular dichroism, can be observed for randomly oriented chiral samples; Chiral systems are said to be optically active. Optical activity can be induced in non-chiral systems by an external magnetic field, but this phenomenon is not considered in this book. Oriented samples exhibit different absorption coefficients for polarizations parallel and perpendicular to the sample's unique axis. This difference in linear polarization absorption coefficients is referred to as linear dichroism. The applications of dichroism to molecular structural elucidation are numerous. Prior to the 1980s dispersive infrared spectrometers 1 (where different wavelength components of a polychromatic incident light beam are dispersed using a grating or a prism) have been commonly used for infrared spectral measurements. The advantages of using a Michelson interferometer with Fourier transform techniques for infrared spectroscopic measurements have been well established. In a Michelson interferometer2 the incoming light is divided into two equal amplitude components, so the Michelson interferometer can also be referred to as an amplitude division interferometer (ADI). The multiplex and throughput advantages associated with ADI made the Fourier transform infrared (FTIR) spectrometers very popular 3. Vibrational Raman and Raman optical activity (VROA) measuremems, on the contrary, were commonly performed using dispersive methods 4. Recent developments 5 made it possible to perform the vibrational Raman
236
Chapter 11
measurements with Fourier transform infrared spectrometers. Since Raman scattered intensifies depend on the inverse fourth power of the incident excitation wavelength, there is an intrinsic advantage to using visible laser sources for Raman measurements. Nevertheless different factors, such as the improvements in the quality of near-infrared detectors and the advantage of reduced fluorescence interference, gave impetus for the development of FI'-Raman spectrometers with near-infrared excitation. An extension of this approach for VROA measurements has been suggested 6, but no measurements were undertaken. For VROA measurements, one may modulate the polarization of the incident laser beam between fight and left circular polarization states and measure the synchronous difference in Raman scattered intensities. Alternately, one can use the linearly polarized incident laser light, and measure the difference in the scattered Raman light intensifies with fight versus left circular polarizations. Polarization division interferometers (PDIs) 7 provide an alternate Fourier transform approach for infrared and Raman spectral measurements, including VCD and VROA. Thus, three different instrumental techniques are available for a practicing spectroscopist. Linear dichroism measurements are commonly made by orienting a linear polarizer in two different orthogonal orientations, measuring the absorbance and taking the numerical difference between these absorbance measurements. A combination of linear polarizer and quarter-wave retarder (QWR), with the axis of the linear polarizer oriented at +45 ~ and-45 ~ to the unique axis of the QWR, generates opposite circular polarizations. Thus by orienting the QWR in a fixed orientation and placing the linear polarizer in two orthogonal orientations or by orienting the linear polarizer in a fixed orientation and placing the QWR in two orthogonal orientations, the measured absorbance difference represents circular dichroism. A variation in this theme is to rotate the linear polarizer or QWR continually around the light propagation axis and to use synchronous detection (vide infra). However the introduction of photoelastic modulator (PEM) 8, provided a means for obtaining both better accuracy in polarization modulation and high frequency polarization modulation. The details involved in polarization modulation using PEM, rotating QWR and combination of these two are discussed below. The double polarization modulation concepts presented in this chapter are new, and experiments have yet to be undertaken. Another polarization modulation approach of considerable potential, especially for the far-infrared region, is to use a polarization division interferometer as an achromatic polarization modulator 9. This approach will be discussed in Chapter 13. In this chapter the principles involved in polarization modulation and signal processing are presented first, then the dispersive spectrometers are described. Conventional infrared absorption measurements are automatically involved during the measurement of dichroism, as are the
Principles of Spectral Measurements
237
normal Raman measurements during Raman optical activity measurements. For this reason, the conventional infrared absorption and Raman measurements will not be discussed separately with special emphasis in this chapter. The next two chapters focus on amplitude and polarization division interferometers for the same measurements with a similar emphasis. The readers who are not interested in signal analysis and instrumental details can skip these chapters and go directly to Chapter 14. 11.2
Polarization m o d u l a t i o n using a photoelastic m o d u l a t o r The concept involved in the operation of a PEM is that a periodic stress applied to an isotropic crystal causes synchronous variation of the difference in the refractive indices along two mutually perpendicular axes. One of these two axes is the axis along which the stress is applied. For linearly polarized incident light with the polarization axis at 45 to the stress axis of the crystal, one can resolve the incident electric vector into two components, one parallel and another perpendicular to the stress axis. The periodic variation in the birefringence of the crystal introduces timedependent phase lag into these two electric vector components of the incident light. The phase variation in time t follows the relation
8m = 8 0 i sin COmt ' Vi where tom =2rCVm is the frequency of stress modulation,
(ll.2.1) and
~0 i the
maximum phase shift introduced for wavelength )q or wavenumber vi. When the maximum stress applied corresponds to a phase shift of 8 0 = re/2, then the radiation of wavenumber ~i is said to be circularly polarized. Alternately, one of the two electric vector components incident on and parallel to the stress axis of the PEM is said to have undergone a 90 ~ phase shift or quarter-wave retardation relative to the other. For a given maximum stress on the PEM, precisely X/4 or quarter-wave retardation is achievable at only one wavenumber, which is represented by Vq. For other wavenumbers at the same PEM setting, the maximum phase shift becomes 50 = 5 0 Vi
Vq
Vi _ Zt Vi _ rt)Vq ~q 2 ~q 2 )~i
(11.2.2)
As sincomt goes through +1, 0, and -1, 8~% goes through +rt/2, 0, and -~/2, which means that the ~q component goes through right circularly
Chapter 11
238
polarized, linearly polarized, and left circularly polarized states, respectively. For the intermediate values of sincomt, corresponding intermediate polarization states are achieved. Thus, only for a minor portion of sincomt cycle is the wavenumber component ~q circularly polarized (to be precise, only at sin Ohnt = +1), and for the remaining portion of the cycle the polarization states are those of intermediate nature. Similarly, when 8 0vi - rc the linear polarization would have been rotated by 90 ~ and for one
sincomt cycle there would be two cycles of linear polarization modulation. Again the desired linear polarizations are obtained only at sino)mt = 0 and +1. Due to the presence of different polarization states at intermediate points of the sincomt cycle, it is important to understand the nature of the signal expected at the detector 10. Two different experimental arrangements will be useful to understand the signal processing involved. In one arrangement (see Fig. 11-1), a monochromatic light of wavenumber Vi is passed through a linear polarizer (P), PEM and an optically active sample (S), in that order, finally reaching a suitable detector (D). In the second arrangement (see Fig. 11-2),
9 P
PEM
S
D
Fig. 11-1. Sample transmission configuration. P: linear polarizer; PEM" photoelastic modulator; S" sample; D: detector.
9 P
PEM
D
Fig. 11-2. Calibration configuration. P, PEM and D are as in Fig.ll-1; B: birefringent plate; A: linear polarization analyzer. the optically active sample is replaced by a birefringent plate (B) and linear polarizer (now called analyzer (A)). The former arrangement will be referred to as sample transmission configuration, and the latter arrangement
239
Principles of Spectral Measurements
will be referred to as calibration configuration. Regardless of the variation in the experimental arrangement, the z axis is the direction of light propagation and the optical axes of the PEM concide with x and y axes; the polarization direction of the linear polarizer will be considered to be at 45 ~ to the x and y axes.
11.2.1 Circular dichroism (A) Sample configuration The electric vector of monochromatic light of wavenumber Vi , after passing through the first polarizer, is given as [FS(Vi)/~/-2](u + v), where and v are the unit vectors parallel to x and y axes, respectively, and FS( Vi) is the amplitude of initial electric vector. As the PEM imparts a timedependent relative phase lag Gm at wavenumber Vi, to one of the electric Vi '
vector components, the resulting electric vector, after passing through the PEM, becomes
1/o +vei'm/
(11.2.3)
If the fight and left circularly polarized vectors are denoted 10 as [FS( Vi) / ,(-2)]x(u + iv) and [FS(Vi)/~/-2](u - iv), respectively, it can be seen that Eq. (11.2.3) is equivalent to
Ill
+ +/1+ l/u_ v,l (11.2.4)
When this electric vector passes through an optically active sample, the fight and left circularly polarized components are absorbed to different extents. The resulting electric vector is
240
Chapter 11
/ m/
+ 1 + ie 1~5~i (u - iv)e -aL (~i)/2
1 .
(11.2.5)
The intensity of light exiting the sample is calculated as the product of this electric vector with its complex conjugate. The voltage output of a linearly responding detector can be represented by the intensity falling on it, which in the present case is I ( ~ i ) - [IS(~i)/2] [(e -aR(~i) + e-aL(~i))+ (e -aR(~i) - e-aL(~i))sin8 m ]
Vi ,
(11.2.6) where IS(vi) is the initial intensity of the wavenumber component vi, aL(vi ) = 2.303AL(Vi), and aR(vi) - 2.303AR( vi)- Because ~m vi is time dependent, sin ~i-m vi can be expressed 11 as sin 8mvi --sin(80,kv sin corot) - 2 E J 2 n - 1 (~0 i ) sin[(2n -1)O~mt ] , n--1
(11.2.7)
where n is an integer and Jn(80Vi) are Bessel functions. From Eqs. (11.2.6) and (11.2.7) it can be seen that the detector signal contains a timeindependent (often referred to as dc ) signal, and a time-varying (referred to as ac) signals at frequency ram, 3ram, etc. The signal at the fundamental frequency ram, can be isolated by passing the detector signal through a band pass filter centered at mm and a lock-in amplifier tuned to COm. Similarly, the dc signal can be isolated by eliminating the signals at frequencies ram, 3ram, etc., using appropriate electronic filters. Then the ratio of the signal demodulated at mm to the dc signal becomes
Icom (Vi) Idc(Vi)
eaLi/Ol
e-aR(vi ) + e-aL(vi ) Gf
(11.2.8)
where Gl and Gf are the gains introduced by the electronics of the lock-in amplifier and filters, respectively. Multiplying the numerator and
Principles of Spectral Measurements
241
denominator of Eq. (11.2.8) by e (aR (~i)+aL(~i))/2, and noting that for small values of 13, (e~-e-~)/(e~+e-~) - tanh 13--- 13, one obtains
I0.)m(Vi)/Idc(Vi)- 2Jl(~0 i ){[aL(Vi)-
aR(Vi)]/2}(G//Gf).
= 2J1 (~i0i ){2"303[AL (Vi)- AR (~i)]/2}(G//Gf) If Vi is equal to Vq, then 80i -re/2 and
J1 (re/2)---0.57.
(11.2.9) The value of
Jl(~50vi)G//Gf can be determined from a calibration spectrum. (B) Calibration configuration Here the optically active sample is replaced by a birefringent plate 12 and analyzer together (see Fig. 11-2). These two components can be oriented with respect to the preceding PEM and first polarizer in four different ways. (a). The fast axis of the birefringent plate is parallel to that of the PEM and the direction of the analyzer is parallel to that of the first polarizer. In addition to the time dependent phase difference previously introduced by the PEM, additional time independent phase lag 8 B vi is now introduced by the birefringent plate, which is given as 2r~idAn; here d and An are respectively the thickness and birefringence of the plate. Then the electric vector exiting the analyzer becomes
F(Vi) - [uS (vi) / 2] Ill + vei(~5~m+sBi)/(11 + V) ,
(11.2.10)
from which the intensity can be found to be
I(~i)- [IS(~i)/2] (1+ COS~im COsfiBvi- sin 8mVisin~SBvi) 9
(11.2.11)
Using Eq. (11.2.7) and a similar expression 11 for cos 8~m ,
cos 8~m- J0 (80i)+ 2Z J2n (80i )cos 2nf.Omt, n=l the expression analogous to Eq. (11.2.9) becomes
(11.2.12)
242
Chapter 11
I0~m(~i) = -2Jl(fiOi)sinSB i G l Idc(Vi)
(11.2.13)
1+ J0(~Oi )cos~ BVi Gf
(b). Let the polarization direction of the analyzer be perpendicular to that of the first polarizer, and the fast axis of the birefringent plate be parallel to that of the PEM. Then (u + v) in Eq. (11.2.10) becomes ( u - v), and the expressions analogous to Eqs. (11.2.11) and (11.2.13) become I(vi)-[IS(~i)/2] (1 - cos~5m m sin ~sB vi cos ~5B vi + sin S~i vi ) '
Ic0m(Vi) = Idc(Vi)
2J1 (80i )sin 8 Bvi G l 1-- J0(~Oi)cos 8 Bvi Gf
(11.2.14)
(11.2.15)
(c). Let the polarization axis of analyzer be perpendicular to that of the first polarizer and the fast axis of birefringent plate be perpendicular to that of the
i(8~n]+~sBi)
PEM. Then e and (u + v) terms in Eq. (11.2.10) become ei(8~m -8 -B vi" and ( u - v), respectively, and the expressions analogous to Eqs. (11.2.11) and (11.2.13) become
)
I(~i) - [Is (vi)/2] (1 -- cos 8 m Vi COS8 Bvi--sin 8 mVi sin 8 BVi),
Io~m (Vi) = Idc (Vi)
-2J1 (~0i )sin fiBvi G l 1-J0(~Oi )cos~ BViGf
(11.2.16)
(11.2.17)
(d). Let the polarization direction of the analyzer be parallel to that of the first polarizer, and the fast axis of the birefringent plate be perpendicular to that of PEM. Then the expressions analogous to Eqs. (11.2.11) and (11.2.13) become (11.2.18)
243
Principles of Spectral Measurements IC0m(Vi)= 2Jl(80i)sinSBv i Idc ( V i ) 1
_Gl + Jo(8Oi)cos~i-BV i Gf "
(11.2.19)
Note that Eqs. (11.2.13) and (11.2.19) are equal but of opposite sign, as are Eqs. (11.2.15) and (11.2.17). Also Eqs. (11.2.13) and (11.2.17), like Eqs. (11.2.15) and (11.2.19), are equal to each other with a nonzero magnitude of +2J1(80V i )G//Gf at cos 8 B = 0 and sin gB _ +1 Vi Vi -
-
-
-
This is .
applicable when,
~5B =-v i r t = ( 2 n + l ) rC Vi
VB 2
-~ ,
(11.2.20)
where ~B is the wavenumber for which the birefringent plate introduces a single quarter-wave retardation and n is an integer. Furthermore, Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19) are equal to each other, with zero magnitude, when sin 8 -B - 0. This condition is met when vi m
fib = _vi = nrc. vi VB 2
(11.2.21)
In practical terms, if the maximum stress setting on the PEM corresponds to one quarter-wave retardation for the wavenumber Vq and the light components of various wavenumbers are investigated using the calibration arrangement, then one would get four curves (see Fig. 11-3) represented by Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19). The non-zero crossings of these curves provide the values of 2J1(80. vi )G//Gf. These values can be interpolated to the desired wavenumber and used in Eq. (11.2.9) to determine the CD of a given sample.
11.2.2 Linear dichroism (A) Sample configuration For considering linear dichroism, Eq. (11.2.3) can be rewritten as
F(~ i) - IF s(Vi)/~-8]
I/ / / / 1 1+ e
(u + v)+ 1 - e
(u- v).
(11.2.22)
Chapter 11
244
Denoting the base e absorbance for parallel and perpendicular polarizations as ap(~i ) and as(Vi ) respectively, Eq. (11.2.22) is modified to represent the electric vector after passing through the oriented sample as I
,,
, I ,
,,
I
,,
, I ,
,,
I , ,
,
I,
,,
!
I
1.0
m
0.5 m
m
0.0
-0.5
-1.0
2000
1800
1600
1400 1200 ~aavenumbers
1000
800
Fig. 11-3. Calibration curves represented by Eqs. (11.2.13), (11.2.15), (11.2.17), and (11.2.19). Birefringent plate is assumed to have a thickness of 0.25 cm and birefringence of 0.02; ~q = 1000 cm -1. The values of Jl( ~0 i ) were determined from the analytic formula.
F(vi) - [F s (Vi) / ~
ll/ / 1+ e
(u + v)e -ap (vi)
+ 1-e
~ ( u - v ) e -a~(~i) o
(11.2.23)
Principles of Spectral Measurements
245
The intensity of fight exiting the sample, determined from the product of Eq. (11.2.23) with its complex conjugate, is I ( v i ) - [IS(vi)/2][(e -ap(~i) + e-as(Vi))+ (e-ap (~i)_e-as
(Vi))cosS~ii] (11.2.24)
From the expansion of cos fvm term (see Eq. (11.2.12)), one would notice that the signal detected would contain components oscillating at frequencies 2corn, 4corn etc. The signal oscillating at 2corn, normalized to the dc signal becomes
e-a
I2a~m (Vi) Idc(Vi)
{[e-ap(Vi) +e -a~(~i)
+J0(~Oi
1
]O1
e-ap(Vi)-e -a~(~i)
1}
6f
"
(11.2.25)
When linear dichroism magnitudes are small (which is not necessarily true in general), the second term in the denominator of Eq. (11.2.25) may be ignored in relation to the first term. In such situations,
I2com(Vi)
"- 212 (~0 i )tanh[Aa(vi)/21(G//Gf), Idc (~i) where Aa( ~i)/2 =(as(~i) - ap( ~i))/2 - 2.303 AA(~i )/2.
(11.2.26)
(B) Calibration configuration Again a calibration measurement is needed to determine the value of J2( 80vi)(G//Gf) in Eq. (11.2.26). The calibration measurement can be performed just as in the case for circular dichroism, except that the calibration signal would now be extracted from lock-in amplifier tuned to 2O3m. Then the terms Icom and sin8 B Vi, in Eqs . (11 . .2.13), . (11 2 15) (11.2.17) and (11.2.19), will be replaced by I2com and cos 8 B vi respectively; 2J 1( riovi ) will be replaced by-2J2(~50vi ) in Eqs. (11.2.13) and (11.2.15) and by 2J2( 8 o. vi) in Eqs. (11.2.17) and (11.2.19). The predicted curves are
Chapter 11
246 shown in Fig. 11-4.
The m a x i m u m values of these curves give
+_.2J2(~50i ) / [ 1 + J0(~Oi )] (G//Gf) from which J2(~0i)G//Gf can be found. 1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5 2000
Fig.
11-4.
1800
Calibration
1600
curves
1400 1200 wavenumbers
for
linear
I000
dichroism
800
represented
by
___2J2(~0 i )cos~Bvi /[1 + J0 (~Oi)cos~ vi B ]" Birefringent plate is assumed to have a thickness of 0.25 cm and birefringence of 0.02", Vq = 500 cm-1 . The values of J0( ~i0vi) and J2( ~0vi ) were determined from the analytic formulae. 11.3
Polarization modulation using a quarter-wave retarder The PEM shown in Fig. 11-1 can be replaced by a rotating QWR for circular dichroism measurements. For a broad wavelength coverage, however, one needs an achromatic retarder. Such a retarder can be constructed 13 from two different birefringent plates, with their fast axes in the plane of the plates and at 90 ~ to each other. The retardance of the combined plates is given by the relation, (2~/~)[Anltl-An2t2], where An is the birefringence and t is thickness; the subscripts 1 and 2 identify the two birefringent materials. Such a retarder made from CdS and CdSe plates has
247
Principles of Spectral Measurements
been reported 14 for the mid-infrared region. The signals applicable with rotating QWR, which are different from those applicable with PEM, are given in this section. For the configuration shown in Fig. 11-5, the electric vector exiting the rotating QWR is found to be, F(~i) - [Fs (~i)/x/2] X {(u + iv)cos 2 cot + i(u -iv)sin 2 tot + (u + v)(1- i) sin mt cos rot},
(11.3.1)
where to is the rotating frequency of achromatic QWR. With an optically active sample placed after QWR, the signal at the detector becomes, I(vi) = [IS(vi)/2] [e -aR(~i) cos 2 cot+ e -aL (~i)sin 2 cot] --[IS(~i) / 2] e -(aL (~i)+aR(~i))/2 [1 + (Aa(Vi) / 2)cos2mt].(11.3.2) Note that the circular dichroism signal is modulated at 20) (i.e. at twice the frequency of QWR rotation). It is easy to see from Eq. (11.3.2) that I2dIdc gives circular dichroism of the sample. For calibration purposes, the _
V
P i
QWR
S
D
i
Fig. 11-5. Sampletransmission configuration for measurementsusing a rotating quarterwave retarder(QWR); P, S, D are as defined in Fig.11-1. configuration shown in Fig. 11-6 can be used (where the PEM in Fig. 11-2 is replaced by a rotating QWR) and the signal in this case becomes I(vi ) - [Is (Vi)/2 ]((1 + 0.5cos 8 B vi ) - s i n ~5B vi cos 2cot- 0.5cos 8 B vi cos 4cot). (11.3.3) vi )" Following the The ratio I2JIdc now gives -sin 8Bi/(l+ 0.5cos 813 procedure in Section 11.2.1B, one can easily derive similar relations for the other three orientations (of the bireffingent plate and analyzer); similarly the expressions for the linear dichroism of the sample can be obtained following
Chapter I 1
248
the procedure in Section 11.2.2. Dichroism measurements using a rotating stressed birefringent plate have been reported. 15
9 P
QWR |
B
A
D
m
Fig. 11-6. Calibration configuration for measurements using a rotating quarter-wave retarder(QWR); P, B, A and D are as defined in Fig.11-2.
11.4
Double polarization modulation
One may achieve better dichroism signal quality with double polarization modulation as shown in Fig. 11-7. Here a rotating QWR and PEM are used together. For the configuration shown in Fig. 11-7, the i,
....
|
xx
P u,
QWR
PEM
S
D
Fig. 11-7. Sample transmission configuration for measurements using double polarization modulation; P, PEM, S, D are as defined in Fig.11-1. electric vector exiting the PEM is found to be
F(v i) - IF s (v i) / 2"V~] x [(u + iv) x A + (u - iv) x B],
/ m/ / / t/ m/ / lvi/t
where
1~.
A-
l+e
" ~
~ cos 2 m t + i 1 - e ~Svi sin 2 mt +
1~.
1-e
9 m
1
-i
l+e
sin mt cos mt,
(11.4.1)
Principles of Spectral Measurements
B=
249
/ m/ /ei i/ 1 - e ~8~i cos 2 m t + i 1+
1+
~i
_ i 1- e
" sin 2Cot+
sin cot cos cot.
With an optically active sample placed after the PEM, the signal becomes, I ( v i ) - IS(vi)e-(aL(~i)+aR(~i))/2{1 + (Aa(vi)/2)[J0(80 i )cos 2t.ot + J2 (~0i) cos(o~m + 2to)t +
J2 (~50i )cos(~ m -
2tO)t + J1 (~0i)sin COmt
--0.5(sin(~ m + 4co)t + sin(~ m - 4to)t)]} . (11.4.2) It is important to note that the circular dichroism signal now appears at 2o), tom, tom +2oo and tom +4o). There are two approaches for detecting the signals at 2o), O~m, O~m+2o) and corn+ 4co sirnultaneously. One approach is to use a digital signal processing method 16 and the other is to use mutiple lock-in amplifiers. If all these components can be detected simultaneously (vide infra) then the relations 11 among the Bessel functions (see Chapter 13) indicate that the measured dichroism signal becomes -2 times larger than the signal measured with PEM or rotating QWR alone. The analogous expressions for linear dichroism and calibration signals can be derived in a similar manner.
11.5 Dispersive spectrometers For a given wavelength, the signals resulting from polarization modulation are described in the previous sections. Light sources, other than lasers, are usually polychromatic. Selection of a given wavelength (to be more precise, a narrow bandwidth of wavelengths) is achieved in dispersive spectrometers, which were widely used in the first three quarters of this century. Although they are not as popular in current times (due to a wide spread use of the Fourier transform techniques), the dispersive spectrometers are unavoidable for some applications. The central component in these spectrometers is a spectral dispersing element, such as a prism or diffraction grating. When polychromatic light is directed to a reflection grating, different wavelengths are diffracted at different angles. The relation governing this dispersion is given 17 as d(sin0 + sin q~i) - m~. Here, 0 is the angle of incidence (measured from grating's surface normal), q~i is the angle of diffraction for wavelength ~,i, d is the distance between successive grooves on the grating, and m is a integer (called the order
Chapter 11
250
number). At higher orders, not only does the spatial separation between two wavelength components increase, but a wavelength of one order may fall imo the region of another order. Thus a filter blocking the wavelengths of a different order might be necessary. When single element detectors are used, the dispersed light is passed through an exit slit (whose width determines the spectral resolution) on to the detector. In this case, a scanning mechanism is used to rotate the grating so that a different wavelength component goes through the slit to the detector. Thus dispersive spectrometers using a single element detector are called scanning spectrometers. With the availability of multi-element detectors (also called detector arrays), the monochromator is replaced by a spectrograph, where the exit slit is replaced by a wide opening to hold the detector array (see Fig.
input slit
polychromatic light
mirror
grating exit slit or detector array
Fig. 11-8. Schematic of a monochromator/spectrograph. 11-8). In this case the spectral resolution isdetermined by the width of individual detector elements (referred to as pixels), which is typically ~25 ktm. The grating needs to be rotated to a new orientation only when the spectral region of interest is not covered by the combined width of the detector elements. Various methods centered around these dispersive spectrometers are discussed below.
Principles of Spectral Measurements
251
11.5.1 Dichroism measurements For a given wavenumber vi, or wavelength ~i, selected with a monochromator the signals expected at the detector with appropriate polarization modulation can be found in the previous sections. In dispersive spectrometers it is common practice to use a light chopper to modulate the source intesity. In such cases, the signals described in the previous sections have to be multiplied 1 by (1 +sino~ct)/2, where O~cis the light chopper frequency (~ 100 Hz). A block diagram of a dispersive CD spectrometer using a PEM for polarization modulation and a light chopper for source intensity modulation is shown in Fig. 11-9. Multiplying Eq. (11.2.6) with(1 +sino~ct)/2, one notices that the CD signal varies as both sin~ m and sin~Smsino~c t, while the sample Vi Vi transmission signal varies as sino~ct. The CD signal component can be extractedl by processing the detector signal with a lock-in amplifier tuned to COm. Alternately, one can use 18 two lock-in amplifiers (the output of lockin tuned to ohn is processed by a lock-in tuned to O}c)for extracting the CD signal. The CD component extracted in either manner needs to be normalized with the sample transmission signal, which is extracted using a detector mono-
chr~176 _ source chopper o)
vi
~
PEM )
~ "~ O~m polarizer
C
o)
Lock-in Amplifier -= 100 Hz
m
I 0)
C
0)
Lock-in rn Amplifier
"
-=-30-100 KHz
Fig. 11-9. Schematic of a dispersive circular dichroism spectrometer.
I I ! ! ! I I I I | ! I I I ! ! ! I ! ! ! I ! I
252
Chapter 11
different lock-in amplifier tuned to toc. Fig. 11-10 shows the first reported VCD spectra, for their historical importance. Modem instruments provide spectra with much improved signal quality. For linear dichroism signals, Eq. (11.2.24) is multiplied with (l+sincoct)/2 and the procedure for extracting the signals is similar to the one described above for CD, except that since the linear dichroism signal varies at 2Ohn the ohn lock-in used for CD is replaced by a lock-in tuned to 2o~. I
'
!
L
,
l
30OO
9
~ cm-I Fig. 11-10. The first reported VCD measurements using a dispersive VCD spectrometer. The traces with labels (+), (___) and (-) represent respectively the VCD spectra of the liquid samples of (+)-, racemic- and (-)-2,2,2-trifluoro-l-phenylethanol, shifted from each other for clarity. The top most trace is the infrared absorption spectrum. Reproduced with permission from Ref. lb. Copyright 1974 American Chemical Society.
When polarization modulation is achieved with a QWR rotating at frequency co, Eq. (11.3.2) is multiplied with (1 +sincoct)/2 and the COrnlockin amplifier shown in Fig. 11-9 is replaced with a lock-in amplifier tuned to 2to (remember to keep co>>toc). The recent emergence of digital signal processing techniques 16 eliminates the need for lock-in amplifiers; here the detector signal for a given wavelength is digitized (at time intervals determined by the higher modulation frequency) as a function of time and the Fourier transform of this time domain signal contains the intensity at different modulating frequencies. Such digital signal processing techniques are particularly important for realizing the advantages of double polarization modulation approach described in Section 11.4.
Principles of Spectral Measurements
253
11.5.2 Raman optical activity measurements For Raman spectral measurements, the sample of interest is illuminated with a laser and the intensity of scattered light is analyzed at wavenumbers that are away from the incident laser wavenumber. The differences between the wavenumbers of scattered and incident light correspond to the differences in energies of vibrational levels. The monochromator/spectrograph discussed earlier is placed in the scattered beam for wavelength selection. Although monochromators were widely used in Raman spectroscopy, the availability of multi-element detectors resulted in a wide usage of spectrographs. Due to the nature of multielement detection, signal processing is not done using lock-in amplifiers. So the polarization modulation here takes a different form (vide infra). For vibrational Raman optical activity (VROA) measurements 19-29, the incident laser beam can be modulated between right and left circular polarization states and the synchronous difference in the scattered Raman intensities measured 4 as a function of vibrational Raman shifts. This approach is referred26, 28 to as incident circular polarization (ICP) VROA (see Fig. 1111). VROA can be measured for 90 ~ scattered Raman light (Fig. 11-11A) with an analyzer parallel or perpendicular to the scattering plane; the former i"
D
D
SG
/\ 9
"
A
L
L Las
P
M
(A)
S
P
M
(B)
Fig. 11-11. Schematic of a Raman optical activity spectrometer with 90~ scattering (A) and 180~ backscattering (B) geometries using incident circularly polarized light. P: polarizer; M: modulator; S" sample; L: lens; A: analyzer; SG: spectrograph; D: detector. See Refs. 19-23 for details. is referred to as depolarized VROA4,19, while the latter as polarized VROA 20. If the analyzer in the scattered beam is placed 21 with its axis at 54.7 ~ from the scattering plane, the resulting VROA is called magic angle
Chapter 11
254
VROA. One may also choose not to include an analyzer in the scattered beam. In back- (180 ~ (Fig. l l - l l B ) and forward (0 ~ scattered geometries22, 23 ICP-VROA is measured with no analyzer in the scattered light. A brief description of the procedures followed to make a measurement is as follows (see Fig.11-11). The incident laser light is first linearly polarized; polarization modulation between circular polarizations is achieved using an electro-optic modulator (EOM) 4, QWR 24, or a liquid crystal variable retarder 25. In the EOM case, the voltage supplied to the EOM is usually a bipolar square wave, so the positive and negative halves of the voltage correspond to the two circular polarizations. The data collected during two halves of the square wave are stored in two different files. The first reported VROA measurements are shown Fig. 11-12, for their historical importance. Modern instruments provide spectra with much improved signal quality.
H---C--OH
I
CNjL
-
_
_
_ .
~Jd
J
I
+
-~- 1~cl
.,--..
"~ <]
.1 x.~_
,.
.
9
.
-, .
.
9
9 ,,
Fig. 11-12. The first reported depolarized VROA spectra (bottom traces), using a dispersive scanning ROA spectrometer, for the enantiomers of a-phenylethanol (left half) and o~-phenylethylamine(right half). The dotted lines are for (+) enantiomers and solid lines for (-) enantiomers. The top most traces are depolarized Raman spectra. Reproduced with permission from Ref. 4. Copyright 1973 American Chemical Society. In the QWR case, the orientation of the QWR is fixed in one position to give one circular polarization and data are collected into one file; then the axis of the QWR is rotated by 90 ~ to generate opposite circular polarization
255
Principles of Spectral Measurements
and the data are collected into another file. In the case of liquid crystal retarders, the birefringence of the liquid crystal depends on the square of applied electric fried" they are driven by a--2 kHz bipolar voltage (which is needed to prevent "run-away" of liquid crystal molecules into one permanent orientation) and by changing the magnitude of this bipolar driving signal, one achieves the switching of one circular polarization to another.
D
D
S
SG
A
A
L Laser
QWR
L
p
P S
(A)
(B)
Fig. 11-13. Schematic of a SCP-ROA spectrometer (A) and DCPI-ROA spectrometer (B). P: polarizer; S: sample;QWR: quarter-wave retarder; L: lens; A: analyzer; SG: spectrograph; D: detector. See Refs.24, 26 and 28 for details. An alternative to the measurements described above is to leave the incident laser beam linearly polarized, and measure the difference between fight and left circularly polarized intensities of the scattered Raman fight (see Fig. l l-13A). Such measurements are referred to as scattered circular polarization (SCP) VROA measurements 26 and have been reported for 90 ~ scattering geometry. In principle, polarization division interferometry (see Chapter 13) is an attractive approach for this measurement 27. In another type of VROA measurement, the incident laser beam is modulated between fight and left circular polarizations and the synchronous difference in the intensities of fight and left circular polarization components of the scattered Raman light are measured (see Fig. 11-13B). These measurements are referred to as dual circular polarization (DCP) VROA measurement 28. If the scattered light was analyzed for fight circular polarization when the incident light is fight circularly polarized (and for left circular polarization when the incident light is left circularly polarized) the measurement is referred to as DCPI VROA. On the otherhand, if the scattered light is analyzed for fight circular polarization when the incident laser polarization is
Chapter 11
256
left circularly polarized (and vice versa), the measurement is referred to as DCPII VROA. The DCPI measurements in the 180 ~ backward scattering geometry have been reported 28. VROA measurements are prone to artifacts and careful control 29 of these artifacts is necessary.
11.5.3 Time resolved infrared measurements Various approaches are now available for monitoring the temporal changes using infrared spectroscopy. A summary is provided in Fig. 1114. The most straightforward approach is to use a monochromatic fight Time resolved infrared spectroscopy
ContinuOUSscanI Rapid sweep I
~troboscopic I
~
1Asynchronous ~
IC~176 Asynchronous
Fig.11-14. Variousmethods for time resolved infrared spectroscopic measurements. source (either a laser or polychromatic light dispersed by a grating) with a fixed polarization and monitor the changes in intensity at desired time intervals 30. This method however restricts the observations to a single wavelength and the entire procedure has to be repeated for each of the other wavelengths of interest. Interferometry obviously overcomes this limitation, but some other considerations need to be evaluated before undertaking time resolved measurements using interferometry (see Chapters 12, 13). (A) Measurements using boxcar integrator A block diagram for time resolved infrared spectrometer 31 is given in Fig. 11-15. A desired time dependent stimulus is given to the sample and the intensity of the light component (with fixed polarization at wavelength ~ or wavenumber vi) transmitted through the sample is determined using
Principles of Spectral Measurements
Vi
Sample A --
257
Dectector "V
[ Trigger]---B"[ Box'CarI
V Fig. 11-15.
Schematic diagram for time resolved measurements using a boxcar
integrator. a boxcar integrator. Using a reference signal, synchronized to the stimulus provided, the boxcar integrator can be configured to record the detector signal after a certain delay from the stimulus and the signal integrated for a certain time window. Both delay and integration time windows can be set as desired. The signal recorded by the boxcar represents the temporal change in intensity of light transmitted through the sample, IS(Vi) e-a(t'vi). The sample transmission at t = 0, i.e. in the absence of the external stimulus, can be obtained by collecting the data as before, but without providing the stimulus to the sample (but a reference signal representing the stimulus given in the previous measurement is needed for the operation of boxcar integrator). The signal measured now is IS(Vi)e -a(t=0'~i), but measured at different delay settings of the boxcar integrator. From the logarithmic ratio of these two signals, the time dependent changes in absorbance, for a fixed polarization, can be extracted. (B) Measurements using lock-in amplifier This approach 32 is suitable for cases when the external stimulus given to the sample is oscillatory, for example a sinusoidal perturbation. If this perturbation has a frequency roe, a chosen molecular property in response to this perturbation may also vary at this frequency, although there can be a phase lag between the applied perturbation and the chosen molecular property. Thus the temporal behaviour of that molecular property can be written as o~ = (x0 + (x(t) = o~0 + & sin ({Oet+ 13),
(11.5.1)
258
Chapter 11
where ~0 is the value of the molecular property that does not change with time, and [3 is the phase lag. The time varying portion of the molecular property can be written as o~(t) = ~ip sin ~et + O~opcos O~et,
(11.5.2)
with o~ip and O~oprepresenting respectively the in-phase and out-of-phase (also called quadrature) components of molecular property and are given as ~ i p - ~ cos ~,
(11.5.3)
0~op = & sin [~.
(11.5.4)
If Oqp and O~opcan be measured separately then the phase lag ~ (which is called the dissipation factor) can be determined from Eqs (11.5.3)-(11.5.4) as
C%p/Oqp - tan $.
(11.5.5)
The temporal characteristics of the molecular property can then be extracted through Eq. (11.5.2). The in-phase and out-of-phase components of the molecular property can be determined using a lock-in amplifier. This principle is the basis for two recently developed 32,33 novel spectroscopic methods, namely dynamic infrared linear dichroism (DIRLD) and two dimensional infrared spectroscopy. These two methods are described below.
11.5.4 Dynamic infrared linear dichroism (DIRLD) The inherent linear dichroism associated with oriented samples can be perturbed with a dynamic perturbation such as a mechanical stress or electrical voltage. In the case of oriented polymers the implications of mechanical stress are of importance. So the question to be addressed here is one of molecular reorientation as a function of the external stress modulation. As linear dichroism is related to the molecular orientation, the variation of linear dichroism with stress modulation can be related to molecular reorientational dynamics. A rheometer is commonly used to modulate the applied stress. A block diagram for DIRLD measurements 32 is given in Fig. 11-16. The source light is chopped by a mechanical chopper at frequency tOc and dispersed by a monochromator. At the wavelength component of interest, the light is polarized by a linear polarizer P and modulated between two orthogonal linear polarizations by a photoelastic modulator (PEM) at 2COm. As a result of this polarization modulation, the detector signal contains a component that represents linear
Principles of Spectral Measurements
259
dichroism of the sample. If the sample is subjected to a mechanical stress perturbation at frequency COsthe linear dichroism undergoes changes at that frequency. In order to isolate the desired signal, these perturbation
Source
Chopper H chromator Mono~
PEM ~
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I
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i
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,,
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,.
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. , , ,.
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.
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. u.
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,-...
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Fig.11-16. Schematic for dynamic infrared linear dichroism measurements.
frequencies are to be well separated from each other. The usual values are COs--- 10 Hz, COc---1000 Hz and 2O~m---75 kHz. The signals from each of these perturbations are to be demodulated in the decreasing order of frequency. The detector signal is first demodulated using a lock-in amplifier tuned to 2O~m, with minimal time constant so that the signals varying at O~c and COs are not attenuated. The output of the 2O~m lock-in amplifier (containing signals varying at O3cand COs)is then demodulated by a lock-in amplifier tuned to COc and a time constant low enough not to attenuate the COs signal. The output of the O~c lock-in amplifier is then demodulated by a COslock-in amplifier. When demodulated in-phase with the reference signal from the rheometer, the output of this lock-in is proportional to the component of linear dichroism that varies in-phase with mechanical stress. When demodulated out-of-phase with the reference signal from the rheometer, the output of the COs lock-in amplifier is proportional to the component of linear dichroism that varies out-of-phase with mechanical stress. These measured dichroism signals in conjunction with Eqs (11.5.2) to (11.5.5) provide temporal changes in dichroism. However, it should be noted that temporal resolution is dictated by the time constant of the COslock-in amplifier.
Chapter 11
260
11.5.5 Two dimensional spectroscopy For a system that responds to the time dependent external stimulus, cross correlation function for two different time varying signals can be given 33 as =
l lf
~(vl,t) ~(v2,t + x)dt.
(I 1.5.6)
T---)~, a-T/2
Here x is the time off-set (correlation time) at which time the function x(x ) is evaluated, ct(Vl) and tx(v2) are some appropriate signals monitored at wavenumbers vl and v2. At x - 0 the function x(x ) is simply the one that represents synchronous variation between the signals tx(~ 1) and ct(v2 )- If Eq.(11.5.6) is evaluated for different z values and is found to have maximum at a non-zero x value, then that particular x(z) indicates the magnitude of asynchronous variations between the signals tX(~l) and tx(~2). Then x represents the phase shift between the two signals. Thus if one were able to measure ct(Vl) and t~(v2) as a function of time, with sufficient time resolution, then such data will enable determination of synchronous and asynchronous variations among different molecular properties. An alternative to this approach is as follows. Suppose that the external stimulus is sinusoidal in nature. Then the time dependent variations in molecular properties being probed can be represented by Eq. (11.5.2) and one uses 33 the relations x('C) = t~(V1, V2) cos O~et+ ~(V1, V2) sin O~et,
(11.5.7)
where, (11.5.8) (11.5.9) The function q)(v1, V2 ) represents the synchronous correlation intensity and 9 (~1, ~2) represents the asynchronous correlation intensity. Since the molecular properties ~p(Vi) and tXop(Vi) can be determined using a lock-in amplifier (as explained earlier), the correlation intensities q)(vl, v2),
Principles of Spectral Measurements
261
9 (v1, v2) can be calculated. The plots of these correlation intensifies represent two dimensional spectra, with Vl and v2 represented along x and z axes respectively. The y-axis represents the correlation intensity. Such plots have been reported not only for infrared absorption but also for infrared linear dichroism. References
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