J. Phys. Chem. 1994,98, 3601-3611
3601
Optical Analysis of an Ellipsometric Technique for Time-Resolved Magnetic Circular Dichroism Spectroscopy Diping Che, Robert A. Goldbeck,' Steven W. McCauley,? and David S. Kliger Department of Chemistry and Biochemistry, University of California at Santa Cruz, Santa Cruz, California 95064 Received: September 13, 1993; In Final Form: December 17, 1993"
The Mueller matrix calculus is used to derive signal equations for a technique for nanosecond time-resolved magnetic circular dichroism (TRMCD) spectroscopy that is based on detection of the polarization state of highly eccentric, elliptically polarized light. These equations link the observed signal to the magnetic circular dichroism of the sample in the presence of measurement artifacts associated with imperfect optical devices and the birefringence induced in laser-photolyzed samples by photoselection. The results provide a methodical basis for optimization of optics and artifact reduction. An experimental example of the most important artifact, arising from the second-order coupling of birefringence and Faraday rotation, is presented for the case of laser-photolyzed (carbonmonoxy)myoglobin. The first-order linear birefringence artifact previously identified in ellipsometric measurements of natural C D cancels from magnetic CD spectra obtained as the difference of opposed field-polarity measurements, an important advantage for fast and ultrafast TRMCD studies of photoinitiated molecular processes.
Introduction Magnetic circular dichroism (MCD) spectroscopy has proven to be a very useful technique for acquiring information about molecular electronic states and their magnetic properties.' Although its origin differs from that of natural circular dichroism (NCD), MCD can be measured in the same way as NCD. Because both MCD and NCD are weak effects, a measurement of the difference between the absolute absorption of right and left circularly polarized incident light by the sample is not commonly practiced. Instead, the incident beam is typically modulated by a photoelastic modulator (PEM) between the two circular polarization states at a high frequency (1-100 kHz), in either square or sinusoidal wave form, and the transmitted light is measured with phase-sensitive detection. Several years ago, this laboratory developed an ellipsometric technique for circular dichroism measurements.* In this technique, the incident probe light alternates between right and left elliptical polarization states of very small ellipticity and the intensity of the light transmitted along the minor axis of the polarization ellipse is measured for each state. The ratio of the intensity differenceto the intensity sum with respect to polarization defines a signal that is proportional to the circular dichroism. The new method provides a sensitivity comparable to conventional methods using PEMs and phase-sensitive detection, with greater instrumental simplicity and stability. The ellipsometric method is thus well suited to time-resolved spectral applications in the nanosecond to microsecond regime, a time resolution previously unavailable for MCD studies. Time-resoIved MCD (TRMCD) spectra have been obtained with this technique for electronicallyexcited porphyrins' and photolysis intermediates of CO complexes of hemeproteins, including mammalian4sand bacterial6 cytochrome oxidases, and cytochrome c3.' Fast TRMCD techniques and applications have been recently reviewed.839 As in other polarization spectroscopy techniques, ellipsometric CD measurements are susceptible to artifacts arising from imperfections in optical devices and interference by other optical properties of the sample. Recently, the artifacts in ellipsometric t Permanent address: Department of Physics, California State PolytechNcal University at Pomona, Pomona, CA 91768. Abstract published in Aduance ACS Abstracts, March 1, 1994.
0022-3654I9412098-3601604.50lO
NCD measurements were analyzed using the Jones matrix calculus.1° Major sourcesof artifacts were identifiedand discussed in that work, and methods to reduce or eliminate them were developedand experimentally implemented. Most of these results also apply to ellipsometric MCD measurements. However, additional effects associated with the magnetic field and added optical components are unique to the ellipsometric MCD instrument and have not been addressed before. In the present work, ellipsometric MCD signals are analyzed using the Mueller matrixcalculus in order to characterize artifacts particular to ellipsometricCD measurementson dissolved samples in the presence of an applied magnetic field. It is found that the major interference to MCD measurements arises from the combined action of Faraday rotation (by the cell windows and solvent in the applied field) and linear birefringence (inadvertently present in cell windows or induced in the sample by photoselection) on the polarization state of the probe beam. The coupling of photoselection-induced birefringence and Faraday rotation can be particularly pernicious for time-resolved MCD measurements because the time evolution, spectral band shape, and field polarity dependence of this artifact echo those of a bona fide MCD signal. An experimental example of this artifact and its amelioration is presented below. Several other features are new to the present analysis. Because the Mueller matrix formalism can deal with unpolarized and partially polarized light, we are able to include the effect of finite polarizer extinction ratio more rigorously than in the previous NCD artifact treatment. Many of the signal equations are also reduced to simpler and more useful forms through the use of suitable approximations, and the wavelength dependence of the CD signals observed in the presence of different optical imperfections is discussed. Finally, an equation that describes the collective effects of many stray linear birefringences (LB) is derived. Many of these results are general to CD as an optical phenomenon irrespective of its physical origin and are thus applicable to both MCD and NCD measurements. We point out that there can be an additional effect of laser photoselection on MCD measurements due to orientation of the photolyzed chromophores with respect to the polarization vector of the actinic laser and the applied magnetic field. In contrast to the artifacts discussed in this paper, this orientation effect is 0 1994 American Chemical Society
3602 The Journal of Physical Chemistry, Vol. 98, No. 14, 1994
U.l"rl
Figure 1. Optical components of ellipsometric MCD instrument: LPI, first polarizer; SP, strain plate; LBI and LB2, front and back windows of sample cell; LB3 and LB4. front and back windows of rotator cell; LPZ, second polarizer.
a genuine feature of the anisotropy of MCD as a molecular property, and as such it will be discussed elsewhere." *ry
Mwller Matrix Calculus. The versatility and power of the Mueller and Jones matrix calculi12-'6 have been demonstrated in many papers in the field of polarization spectroscopy. Although they are equivalent in many applications, the Mueller matrix calculus can explicitly treat measurement artifacts arising from imperfectly polarized light and imperfectly polarizing optical devices, whereas the Jones calculus describes only comdetelv . . polarized light." In the Mueller matrix calculus. the intensitv darization properties of a collimated light bCBm are char&erized by a coordinatedependent one-dimensional array of four elements called the Stokes vector,
rs. i
Che et al. After passing through the sample, the transmitted intensity of the prohe beam polarized along the minor axis of the polarization ellipse is selected by another prism polarizer and detected. In addition to inducing circular dichroism in the absorbing solute, the magnetic field induces optical rotation in the transparent solvent and samplecell. (In most MCDapplications, the contribution of the solute to the magnetic optical rotation is negligible compared to that of the solvent because of the large disparity in concentrations.) For single wavelength detection, the second polarizer can be rotated to match the minor axis of thepolarizationellipssofthebeam. Becausethemagneticoptical rotation isgenerally wavelength dependent, however, thismethod cannot be used in multichannel applications where the spectrum over a wide wavelength range is to be measured. In this case, the opticalrotationmay becompensated withanoptical rotatorhaving a suitable wavelength dispersion. One way to provide a compensating rotation is to apply a magnetic field of identical strength hut opposite field direction to pure solvent contained in a second cell identical to the sample Another approach is to takeadvantageofthe fact that theoptical rotatorydispersions (ORD) oftransparent materialscan bevery similar, whether the ORD is of magnetic or natural origin, so that the natural optical rotationofsuitahlychosenchiralsuhstancescanbeusedtocancel the magnetic optical rotation of the sample and the cell.l9 Our signal analysis considers the effects of finite polarizer extinctionratio, alignment of thepolarizersandstrainplate,optical rotation of the sample and the rotator, stray linear birefringence in cell windows, and the presence of linear birefringence and dichroism in the sample on a CD measurement. The Mueller matrices for these optical components are shown Wow. Linear Polarizers.
rr
t MLP=MIx
2
wheresois the totallight intensity,SI is thedifferenceofintensities polarized along the X and Y axes, S2 is the difference of the intensities polarized along the directions + 4 5 O and - 4 5 O to the
Xaxis,andS3isthedifferenceofrightandleftcircularlypolarized intensities. For unpolarized light, SI= S2= S, = 0. (It should be noted that there are several notations for the Stokes vector, the one used here being the most common.) A 4x4 Mueller matrix describes the optical properties of a device:
( I - r ) uls 2 p (I - F)' ms' 2 p + 2#/' ( I - r ) sin ZP ( I - r'l')' sin Z P ms 2 p (1
- r) ms 2 p
0
(I - 7 ) sin 2 p (I
- r1f2)'sin 2 p ms 2 p + ZI"'
'1 O
( I - r'/')' sin' 2 p
0
0
22/2
(5) where fm.. is the maximum transmission and r (=fmi./fm) is the extinction ratio. The angle p is the azimuthal angle between the transmitting and X axes. Without losing generality, we select the transmitting axis of the first polarizer, LPI, as our spacefixed X axis. The matrix for the first polarizer then simplifies to n+r
I-r
0
b
o
0
1
0
TheStokesvector ofthe light emergingfrom theopticaldevice,
Sr,is calculated by the matrix product of the Mueller matrix of the device and the Stokes vector of the light entering the device, Si:
S' = Msi
(3)
The effect of an array of N optical devices is the product of the Mueller matrices for all the constituent devices, with the last device placed at the far left:
sf= MMAlSi = -N-l...MIsi
(4)
MuellerMatricesfor EllipsometricMCD OpticalComponents. The ellipsometric TRMCD instrument, based on a nanosecond laser photolysis apparatus, has k e n previously described in detail.3.8.9 As shown schematically in Figure 1, the probe beam from an unpolarized light source is given linear polarization by a prism-type polarizer. A strain plate is then used to make the beam left or right elliptically polarized. The sample solution is placed in a magnetic field with field lines oriented parallel or antiparallel to the propagation direction of the probing beam.
Because the total beam intensity emerging from the second polarizer is detected, only the first row of the Mueller matrix for the second polarizer is needed. Assuming the two polarizers are of the same quality, we have =
%L"
'-. [l+r 2
( 1 - r ) c o s 2 p 2 ( 1 - r ) s i n 2 p 2 01 (7)
Sfrain Plate and Sfray Linear Birefringence. =
r
0
0
1
0
"usm + sin'2suma
o
s i n 2 & I m a 2 & , ( ~ - ~ b ) a i n ' ~ ~ + m r ' ~ 8 ~rm m 2a8 ~w i i n b
0 sin28,rinb
9. ain us"
Us"
CIIi u s d l
ain 6,
-cm 9,) -in 2&
dn 6,
mb
(8)
is the general matrix for linear birefringence where BLBand ~
L B
The Journal of Physical Chemistry, Vol. 98, No.14, 1994 3603
Optical Analysis of Ellipsometric MCD are the azimuthal angle and retardation, respectively. The azimuthal angle is defined as the angle between the fast axis of the birefringent plate and the laboratory X axis. A strain plate is simply an adjustable linear birefringenceplate. In the experiment, its azimuthal angle is rotated back and forth by 90' to switch the light beam entering the sample between left and right elliptical polarization states. Using eq 8, the Mueller matrix sum and difference for the two orientations of the strain plate are derived as
@, = -(-2LDl e-A 2
+ CB-LD, - CD-LB,)
Mz= 2e-A( - 2 L D ,
+ CD*LB, - CB*LD,)
M3= ye-A( 2 C D + LDyLB, - LDl.LBJ
+
AMSP= MLB(@) - MLB(@ 90°) 0 0 0 -cos 20 sin 6
28 sin 6
~ ~ =0c2(0 2 + C D +~ L D , ~+ LD;)
1
0
-sin 28 sin 6 cos 2fl sin 6
0
(9)
M,,= 2e-A( - 2 L D 1 + CD-LB, - CB-LDJ Ml = 2(2 e-A
- CB, + LD,, - LB;)
M,= $(2CB
+ LD,*LD, + LB,.LB,)
M3= 2e-A( 2 L B , - CB*LB, - CD*LD,) 0
Sample. In the most general case, an optical device or sample can exhibit linear dichroism, linear birefringence, circular dichroism,and circular birefringence. The general Mueller matrix can be expressed 8814-16 MS
eo= -(-2LD2 e-A + CB*LDl - CD-LB,) 2
= e-' = e-A(I - F
+ '/,FZ - ...)
(1 1)
where I is the unit matrix and
e3= -e-A 2 ( 2 L B , + CB*LB, + CD-LDJ e-A eo= $2CD - LD,-LB, + LDl.LBJ
el= H=F+AI
(13)
with
+ CBnLB, + CD-LD,)
+(2LB,
e,
- CB-LB, - CDsLD,)
= $(2LB,
A = 1.15 1(A,
+ A,)
LD, = 1.15 1(A, - A,) LD, = 1,151(A450-Al35e) LB, = 27r(n,-
n,)l/A
LB, = 2 ~ ( n , ~-. nls5.)I/A CD = 1.15 1(AL- A,) CB = 27r(nL - nR)l/A
(mean absorbance)
(X-Ylinear dichroism) (45-135O linear dichroism)
(X-Ylinear birefringence) (45-135' linear birefringence) (circular dichroism)
e-A e3= 2(2
(15)
as derived previously by other workers.16.20 Isotropic samples or uniaxial samples probed along the unique axis possess only circular effects. The exact matrix for this simple case is given by FoshCD
0
0
sinh C D 7
cos CB
sin CB cosCB
0 0
-sinCB 0
binhCD 0 cosh CDJ Circular Birefringence (Optical Rotation). As stated above, a rotator is used to null the optical rotation of the sample solution. The matrix for such a device is
(circular birefringence) (14)
where A, A, n, and I are absorbance, wavelength, refractive index, and pathlength, respectively. In general, each of the 16 matrix elements for the sample is a complicated function of all of the linear and circular effects. Here, we assume that LBI, LBz, LD1, LDz, CB, and CD are small compared to the mean absorbance of the sample. From eq 12, and keeping the first three terms of eq 11, we get
+ CD, - L B , -~ LB;)
r1 MCBR
ILo
= 0 0
o cosCB,
-sinCB, 0
0 sinCBR cosCB, 0
01 0
Angles associated with the devices introduced above are illustrated in Figure 2. Signal Analysis. The experimentally detected signal in ellipsometric MCD measurements is the ratio of the difference and
3604
Che et al.
The Journal of Physical Chemistry, Vol. 98, No. 14, 1994
to examine a functional form for &A). We use a classical dispersion equation for the index of refraction of the transparent plate,
Probe
Beam
n=l+C-
x2 x2 - x,z
where Cand X,areconstants.2’ From this follows an approximate expression for the X dependence of the retardation 6, showing that 6 is approximately proportional to the reciprocal of the wavelength : LP2
I
L
6=
Figure 2. Schematic illustration of some orientation angles for devices and light bgams in laser-photolysis ellipsometric-MCD apparatus. Magnetic field lines (not shown) are parallel (or antiparallel)to probe
beam propagationvector. The probe beam propagationand polarization vectors (single- and double-headed arrows, respectively) define the XZ plane. The probe beam is shown after emerging from the first polarizer with linear polarization along thelaxis. SP,thestrain plate (retardance 6) has an azimuthal angleb ( 4 5 O ) between the fast retardanceand Xaxis. Crossed ( 9 0 O ) excitation is shown with the laser polarization vector perpendicular to the probe beam propagation. The photoselection axis in this case has an azimuthal angle of x if the laser propagatesat an angle x - 90° to the XZ plane. (Other excitation geometries discussed in text are crossed beams with laser polarization parallel to probe propagation and collinear excitation,the latter having a photoselectionazimuth of x . ) An optical rotation (R) of (YR compensates for Faraday rotation by the sample cell and solvent before the probe beams encountersan analyzing polarizer (LP2) of polarization azimuth p = 90’. the sum of the detected intensities for right and left elliptically polarized incident beams:
Using eq 4 and the Mueller matrices for the optical components presented above, the general signal for the optical arrangement shown in Figure 1 can be calculated from M,LP~MLB‘MCBRMLB’MLB~M~MLBI AMSPMLPIS S = M,LP’MLB~MCBRMLB’MLB~M~MLBI ZMSPMLPIS
(19)
where the MLBi matrices represent the effects of stray LBs present in the sample and optical rotator cell windows. For the ideal case, the inadvertent birefringents LB1, LB2, LB3, and LB4 in Figure 1 vanish, the sample optical rotation is canceled by the rotator at any given wavelength (CBR(X)= -CB(A) for all A), the azimuthal angle of the strain plate is 45O, the linear polarizers LPl and LP2 are perfectly crossed, and the sample exhibits only circular birefringence and dichroism. Using matrices appropriate to these ideal conditions, the signal from eq 19 is S =
sin 6 sinh C D cosh C D - cos 6 cos(CB
+ CB,)
=-2CD 6
(20)
where the dependence on cancellation of CB and CBR is made explicit and the approximations are made that sinh(CD) = CD, cosh(CD) = 1, and sin(6) = 6, Le., CD and MCD are small effects and 6 is chosen to be small. These approximations will also be used in the signal equations that follow. In using eq 20 to obtain an NCD or MCD spectrum, it is necessary to know 6 at each wavelength since the strain plate retardation has its own (slowly varying) dispersion. Although this can be experimentally determined from the dispersion of the strain plate transmission (measured by removing the sample and rotator from the optical configuration in Figure l), it is useful
2 d ( n 0- ne) =K-
x
h
x2 - h,Z
where no and ne are the refractive indices of the ordinary and extraordinary rays, the thickness of the strain plate is d, and K is a constant. The CD spectrum (under ideal conditions) can be written as
where the constants K and X,may be empirically determined for a particular strain plate. Real experimental situations can deviate significantly from the ideal represented by eq 20. One should, in principle, consider all the possible imperfections in the optical components and use eq 19 to find their combined effect on the signal. However, multiplication of the nine matrices gives an extremely complicated result. We believe a more focused approach to the problem can be more illuminating. Therefore, the following analysis will treat special cases wherein the effects of particular imperfections are emphasized. Photoselected Samples. An important application of the ellipsometric approach to MCD spectroscopy is in time-resolved studies of samples that have been photolyzed by a brief pulse of light. Photons from the pump beam are absorbed by a portion of the sample molecules, initiating photoconversion to new states or new chemical species. Because the transition probability of a molecule is a function of the scalar product of its transition dipole moment and the electric field of the pump photons, a photoconverted molecule has a preferential initial orientation relative to the polarization of the pump beam. This gives rise to photoselection-induced partial alignment in an initially isotropic, fluid sample if the rotational relaxation is slow compared to the delay of the probe pulse. Photoselection in an optically thin sample is uniaxial. The unique axis is the polarization direction of a linearly polarized pump beam or the propagation direction of an unpolarized or circularly polarized pump beam. Subsequently, rotational diffusion causes the extent of alignment to decrease exponentially with time.22 For an optically thicksample in a magnetic field, photoselection by a linearly polarized pump beam can be affected by the orientation of the beam propagation vector with respect to the applied field. Faraday rotation is a longitudinal effect, varying with the cosine of the angle between the light propagation vector and the field lines; therefore, two limiting cases can be considered for the excitation geometry, crossed and collinear. In the case of crossed excitation, pump beam propagation is perpendicular to the axis of the probe beam and applied field, and the sample is uniaxial throughout the optical path of the probe beam as it traverses the photolyzed sample. In the collinear case, the pump propagates in parallel to the probe and field axis, and Faraday rotation of the pump polarization results in a photoselected sample that is not simply uniaxial. We first consider the case of crossed excitation and come back to the collinear case later.
Optical Analysis of Ellipsometric MCD
The Journal of Physical Chemistry, Vol. 98,No. 14,I994 3605
Crossed Excitation. From eq 19, and consideringly only interfering optical effects originating from the sample itself (Le., MLBi= I and CBR = -CB), we get
+ CD-LD, CB*CD*LD2}{62( 1 + LD,) + CD2 + LD? + LB: 2CD*LB2+ CB(LB,*LB, + LD,.LD2 - CD-LB,)]-' (24)
s = 6(2CD - 2LB2 - CB*LB,
where terms fourth order in sample properties are dropped and the fact that LDl-LB, = LD2.LBl has been used. If the pump beam is linearly polarized perpendicular to the probedirection and formsan anglex with theXaxis (theazimuth of the first polarizer), as illustrated in Figure 2, then
LD, = LD cos 2x
LD, = LD sin 2%
LB, = LB cos 2x
LB, = LB sin 2x
(25)
where LD and LB are the linear dichroism and birefringence, respectively, along the laser polarization axis. (The signs of LD and LB are determined by the absorptive and refractivedifferences between the initial and photoconverted molecules.) Substituting eqs 25 into eq 24 gives s = 6{2CD - (2LB - CBCD-LD) sin 2x
+
(CDeLD - CB-LB) COS 2x]{S2(1 + LD COS 2 ~ +) CD2 + (LB2 + LD2)(sin 2x + CB cos 2x) sin 2x CD-LB(2 sin 2%+ CB cos 2x)}-' (26) We further simplify eq 26 by noting that the CB corresponding to the Faraday effect of the cell windows and solvent is typically several orders of magnitude larger than the CD (NCD + MCD) of the sample, justifying the neglect of all terms higher than first order containing factors of CD. If we assume that the azimuthal angle x for LB and LD has been adjusted to lie close enough to a multiple of ?r/2 that all other attenuating terms in the denominator of eq 26 can be neglected compared to 62, and recalling that LD
