Power broadening of the spectral hole width in an optically thick sample

Chemical Physics ELSEVIER

Chemical Physics 193 (1995) 173-180

Power broadening of the spectral hole width in an optically thick sample Eric S. Maniloff, Felix R. Graf, Hansruedi Gygax, Stefan B. Altner, Stefan Bernet, Alois Renn, Urs P. Wild Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH-Zentrum, CH-8092, Ziirich, Switzerland Received 22 August 1994

Abstract

Under the assumption of negligible absorption, spectral holes are known to broaden in frequency proportional to the square root of the recording intensity. In this paper the standard expression for power broadening is compared with more general results for the case of non-zero absorption. Analytic results are derived for the upper and lower limits of the hole width, corresponding to the cases of low and high absorption, respectively. These results are compared with numerical solutions, and used to fit experimental data for an optically thick sample of chlorin in PVB, in a temperature range of 450 mK to 3 K. The temperature dependence and zero kelvin value of the dephasing time, /'2, are evaluated. 1. Introduction

Spectral hole-burning has become a well-established method for determining spectroscopic parameters of a substance, used for example in analyses of the homogeneous absorption line width, FHom. This is of great interest because it allows insight into various material properties, such as the kinetics of a system or other molecular parameters as the dipole moment. Investigations of dye molecules in polymers have been strongly emphasized recently, because of their high potential data storage capabilities

[1]. Dye-doped polymers are interesting due to their large ratio of inhomogeneous to homogeneous line widths. The inhomogeneous line width is defined as the absorption line width from the ensemble of the absorption centers, whereas the homogeneous line width is the absorption line width for each individual dye molecule. A large ratio of these two widths is indicative of a sample in which many unique subsets

of molecules can be independently addressed using a spectrally narrow light source. In amorphous materials, the inhomogeneous broadening of an absorption line arises from the differing environmental conditions of the absorbing centers. The homogeneous line width, FHom, of an absorbing center is given by 1 1 1 FHo m -- 2"rrTl + ,rrT2,

,rrT2 ,

(1)

where T~ is the lifetime and /'2* is the dephasing time [2]. At room temperature, the value of T2* dominates the line width, leading to mainly homogeneously broadened transitions. However, at low temperatures the dephasing process becomes sufficiently long that a highly monochromatic source, such as a laser, incident on an inhomogeneously broadened sample will excite only a subset of the molecules from the inhomogeneous distribution, producing a population hole in the absorption line. If holes are burned at low power and fluence, the

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E.S. Maniloff et al. / Chemical Physics 193 (1995) 173-180

homogeneous line width corresponds to half the measured absorption hole width [3] apart from contributions that can derive from spectral diffusion or concentration dependent interaction. There is an inherent problem of detection which sets a practical limit on the minimum useful fluence, or energy. This problem can be partly overcome using holography due to the considerable improvement in signal-tonoise ratio as compared to transmission measurements, because holographic readout is a backgroundfree method. If transmission measurements are performed, the signal-to-noise ratio is decreased, which increases the minimum practical hole-burning intensity. Previous analyses of spectral hole-burning have concentrated on the effects arising in the case of small initial absorption coefficients [3]. Analyses of optically thick samples have already been published for saturable absorbers [4,5] and for temporal effects of spectral hole-burning [6]. In this paper we present a theoretical framework for analyzing spectral holeburning data in both optically thin and thick samples, which leads to analytic solutions for the upper and lower limits of the observed hole width. In addition a discussion of numerical solution methods is included. This analysis is then applied to temperature dependent data for hole burning in chlorin doped polyvinylbutyral (PVB) films.

2.

Theory

The kinetic schemes that occur in a hole-burning sample involve: Two electronic levels: A pure ground state-electronically excited state (S0-S 1) transition is assumed. Three electronic levels: An additional triplet state lying below the electronically excited state is added. Usually this state acts as a bottleneck due to the slow intersystem crossing rate. Four or many electronic levels: Several intermediate and at least one product state are present, enabling persistent spectral hole-burning. We have investigated temperature, fluence, and power dependence of the line width of chlorin in PVB in an optically thick sample. Much work has already been done on this system [7], including

measurements at temperatures below 1 K [8]. An overview of the work performed on chlorin can be found in Ref. [2]. Chlorin in PVB can be classified as a four level system, with a product state lying approximately 50 nm blue shifted from the initial ground state. At liquid helium temperatures the product state is stable for several days, whereas at room temperature it rapidly decays into the ground state of the educt. The hole-burning mechanism present in chlorin is called a photochemical persistent spectral hole-burning process [9]. It was shown that the total quantum efficiency is of the order of 10-5-10 -6 [10]. Two different types of hole broadening are discussed here, namely fluence and power broadening. Light induced spectral diffusion has been investigated in an optically thin sample of chlorin in PVB by Al'shits et al. [7] showing that no additional broadening can be associated with it. Other types, such as that arising from concentration dependent broadening are not analyzed here. Using small power and short exposures, depletion of the educt state can be ignored. Under this assumption, it is possible to treat chlorin as a three-level system, thus simplifying the remaining analysis. 2.1. Fluence broadening

Meixner et al. [11] showed that the line widths of the holographic and absorption holes are approximately linear with burning fluence until the onset of saturation. The theoretical treatment in Ref. [11] shows that both the holographic and the transmission widths are identical for zero fluence, but that the holographic signal broadens more rapidly than the transmission signal as the exposure progresses. This yields an easy method to determine experimentally whether the fiuence is close to zero, by measuring and comparing the line widths of the holographic and transmission holes. 2.2. Power broadening

From power broadening studies, the dependence of the hole width on the optical intensity is determined. For the case of shallow holes, the total number of absorbing centers is approximately constant in time, allowing us to treat the system as a

175

E.S. Maniloff et al. / Chemical Physics 193 (1995) 173-180

three-level system. As was already mentioned the persistent spectral hole is then assumed to be proportional to the steady state transient hole. The analysis begins with the Bloch equations in the form given in Ref. [3], do-12 dt dp22 dt dpn dt dP33 dt

( 1 1.2 t0"12 21X( , iA

+

1022 -- P l l ) '

(2)

1. -(k21 + k23 )j022 + ~1)((021 - Or12),

(3)

k21 t022 +k31/)33 nt- ½ix(°'12 - °'2l),

(4)

k23 P22 - k31 P33,

(S)

1 = Pn + P22 + P33,

2.2.1. Optically thin samples

Spectral hole-burning in optically thin materials is treated by solving the Bloch equations with the assumption of spatially constant burning intensity [3]. Since the Rabi frequency is proportional to the optical intensity, in this case the Rabi frequency is constant through the sample depth. Solving the equations for the ground and excited state populations in steady state allows the determination of the frequency dependence of the absorption coefficient:

[

ol= olo 1

1+¢1+K2

X

(6)

where Pii is the ith diagonal element of the density matrix, 13, representing the population of state i, o'ij(~j) is equal to the coherence between states i and j, i.e. the off-diagonal elements of the density matrix, /3, multiplied with e x p ( - i w t ) . This additional factor in the coherence comes from the transformation into the rotating frame. In this representation of the Bloch equations, X is the Rabi frequency, k~j is the transition constant between levels i and j, A is the detuning from the burning laser frequency, and T2 is the total dephasing time as defined in Eq. (1). Power broadening was treated in Ref. [3] for the case of constant hole-burning intensity. However, while this analysis is sufficient for samples with low starting absorption, in most samples the burning light decreases in intensity as it traverses the sample. The spectral hole is therefore burned with a range of intensities, which results in a range of hole widths through the sample depth. Since the spectral hole is measured by observing the intensity of a beam transmitted through the entire sample, the hole width observed can be considered an "effective hole width". This effective hole width arises from a superposition of both holes near the entry and narrower holes closer to the leaving surface. In the next sections we review the low absorption (optically thin) case, and additionally present analytic solutions for the hole width in the presence of absorption. Finally, numerical solutions to obtain exact values are discussed,

¢I + K2T~

(71

)2 A~ + (1/T22)(1 + ¢1 + K 2 T 2

'

with g 2 = X2(2 + k23/k31 ) 2(k21 + k31)T2 ' where ap represents the detuning of the probe frequency from the burning frequency, and % is the starting absorption coefficient. This absorption hole has a FWHM line width k v l v w m ~ : 1 (1+¢1+(KT2)2).

(8)

In the case where low power is assumed Eq. (8) is approximated by: 2 2 + kz3/k31 X 2 my IFWHM~ ,rrT2 + 4,rr(k21 + k23) ,

(9)

and the extrapolated value for zero power results to the well-known k v I FwnM= 2 Fno m. This solution assumes that the molecules have their transition dipole moments, /x, aligned parallel to the exciting polarization. In an amorphous matrix, however, there is a random orientation of the molecules, and the Rabi frequency for each molecule is given by [1 1] ([~*IIEI) x =

h

I cos 01,

(10)

with the angle between the molecules' transition dipole moment and the optical polarization given by 0. Since there is no angular dependence in the Bloch equations, it is possible to replace X 2 with an

176

E.S. Maniloff et aL / Chemical Physics 193 (199.5) 173-180

ensemble average, defined j,2, which is averaged over all possible orientations. ,{,2 is therefore determined by

,~2 = ~

h

COS2 0 dl2,

(11)

where /2 represents a solid angle. This results in the relation ,~ 2 = )(2//3. This factor of one-third needs to be taken into consideration if an attempt is made to determine physical parameters from hole width data. Note that the assumption of zero absorption corresponds to the largest possible intensity through the sample depth. Therefore, Eq. (8) is actually an upper limit for the hole width, since decreasing the burning intensity always leads to narrower holes.

2.2.2. Exponential intensity decay A second limiting case, which has the advantage of being analytically solvable, can be obtained by assuming an unperturbed burning absorption coefficient a(z), where z is the position in the sample in the direction of the propagating beam and solving for the probe absorption. In this case the burning intensity decays exponentially through the sample depth, which corresponds to the lowest possible burning intensity distribution. This analysis therefore results in a lower limit for the hole width. The analysis begins with Eq. (7), since the solution of the Bloch equations to this point is performed only in the frequency dimension. However, X 2 is proportional to the burning intensity, I, therefore K20t I includes the depth, or z, dependence in Eq. (7). The reading intensity, Ir at the end of the sample is It(d) = I t ( 0 ) e x p ( - - £ d a ( z ) d z ) ,

(12)

where d is the sample thickness and a is given by Eq. (7). This equation can be solved as shown in the Appendix to obtain the depth and frequency dependence of the readout beam: l,(d)

lr(0)

= exp( - otod) (T2~p2~ -+-(2 + CA + (52 K°)2 )2 ×(r2ap)2 + [1 + ¢1 + (r2Ko): exp(-,~od)

]2' (13)

where K 0 has been defined as the value of K in the absence of absorption and d is the sample thickness. In order to compare this result with the optically thin result, it is necessary to determine the spectral hole width observed on the transmitted beam. However, for an accurate comparison, the FWHM must be determined for ln(Ir/Io), since the solution in Eq. (8) is for the width of the absorption hole. This solution is straightforward, and results in a width A v I FW,M 1

-- , ; r r 2 { [ l + ¢ l + ( r 2 K o )

2]

×[l+¢l+(T2Ko)2exp(-aod)]}

1/2 (14)

Eq. (14) gives the lower limit of the spectral hole width in the presence of absorption. In general then, the observed hole width is bounded by Eq. (8) and Eq. (14). Two interesting limiting cases of Eq. (14) are for small and large absorptions. It is simple to show that for zero absorption, Eq. (14) equals Eq. (8), as expected. As the absorption becomes large the limit of the hole width is A , , I FW.M =

~/2

[l+¢l+(T2Ko)2] 1/2,

(15)

which is proportional to the square root of the width obtained for the case of no absorption. Comparing Eq. (8) with Eq. (15), it is apparent that in optically thick samples the observed hole width increases more slowly with increasing intensity than in optically thin samples. In the same manner as in the thin film case Eq. (15) can be linearized, yielding

2

2 + k23/k31

X 2.

(16)

A/"FWHM ~' "rr~-f -']'- 8"IT(k21 -~ k23 ) Eq. (16) shows that both approaches yield the same line width when interpreted to zero power, but that in optically thick samples the width increases only half as fast with increasing intensity as in optically thin samples. This indicates that even in the linear power regime, trying to obtain kinetic constants from the slope of the line width versus power, the thick film model needs to be applied, otherwise the determined values will be incorrect. Again if random orienta-

E.S. Maniloff et al. / Chemical Physics 193 (1995) 173-180 1.0

tions are considered one needs to replace X 2 by ,~ 2, using the definitions as before.

.

.

.

.

i

.

.

.

177

.

i

.

.

.

.

i

.

.

.

.

i

.

.

.

.

i

.

.

.

.

0.8

2.2.3. Numerical solutions

o.6

The two previous cases set upper and lower limits on the hole width, however to obtain an exact solution numerical methods must be employed. The solutions for the thin case do not apply when the sample becomes optically thick, whereas the exponential decay solutions do not apply if the intensity is large enough to excite a significant percentage of the guest molecules. Analytically this corresponds to the case where the coupled differential equations between the absorption and the intensity cannot be decoupled for separate solutions. Numerically this problem can be solved by cutting the direction of propagation z into slices of length d z. The intensity is calculated through an infinitesimal Lambert-Beer law

l(z+dz) =/(z)[1-a(v,

z)dz],

(17)

and the absorption at each plane, o~(~,, z + dz), is calculated as in the thin sample case. The following assumptions were made: - The burning is done with a monochromatic beam. For the readout, the power is attenuated and a scan over the hole width is made. Burning and reading light of the same polarization are assumed. - The persistent hole has the same shape as the hole that was burned into the transient state. For weak burning this is certainly true. The only difference lies in the change of the amplitude, which is scaled by the kinetic c o n s t a n t k34 and the exposure time. K and Tz were determined from the measurements, and an optically thick sample of OD 2.3 corresponding to the absorbance of our sample was assumed. Fig. 1 shows the intensity variation throughout the sample. As expected, going to higher powers the decay becomes less and less exponential, as the absorption at the front of the sample is decreased by the exciting light. In Fig. 2 a comparison of the power dependence of the line width at the beginning and the end of the sample is shown, along with the calculated value for the observed hole. In addition several points calculated using Eq. (15) are added.

.~_ N 0.4 t~

z

0.2

0

0.5

Relative thickness of the sample Fig. 1. Simulation of the intensity decay through a sample with optical thickness 2.3 for powers of 1 i x W / c m 2 (dotted line (a)), 100 ~tW/cm 2 (solid line (b)), and 200 i x W / c m 2 (dashed line (c)).

The line width at the front corresponds to the thin sample case. It can be clearly seen that at higher powers (i.e., 200 ixW/cm 2) the front of the sample is power broadened whereas the back side of the sample nearly does not change in hole width over the whole power range. A comparison between the thin film, thick film and the simulated observed line width plot indicates that for this optical density and power range, the exponential intensity decay model is much more appropriate.

600

. . . , . . . ,

560 "~' 520 -I-

~ 440 ..I

40O

................................................................

360 320 '

'

';,0'

'

'

'

' 1'20'

'

't60'

'

'200

Intensity/[ i.tW/crr~ ] Fig. 2. A simulation of the line widths in the front of the sample (thin film case long dashed (a)), at the end of the sample (short dashed (b)) and of the hole as it would be observed behind the sample (solid (c)). The dots indicate the obtained values applying the intensity decay model.

E.S. Maniloff et al. / Chemical Physics 193 (1995) 173-180

178

3. Experiments

.350 / N I

3.1. Experimental setup

~.

The sample was prepared as described in an earlier publications [11]. The setup is illustrated in Fig. 3. The light source used was an argon pumped dye laser (Coherent 899-29), which was under computer control. The dye used during all of the experiments was sulforhodamine B, which is tunable from 600 to 650 nm. This corresponds well to the range of the inhomogeneous band of chlorin (622-640 nm with the center at 633 nm). The line width of the laser was approximately 1 MHz, which is much narrower than the line width observed in chlorin. The stabilizer (Cambridge Research & Instrumentation, Ls 100) stabilizes the laser down to 5% power fluctuation. The beam is expanded with a long focal length telescope in front of the isolation box (not shown). The maximum power obtained behind the telescope was around 600 ixW/cm 2. Filters or the power control of the stabilizer were used to reduce the power to the desired values. Shutters were installed in each beam and in front of the photomultipliers. An Oxford H e 3 / H e 4 cryostat was used, which allows experiments of up to one hour to be performed at temperatures as low as 450 mK. Below 1 K the temperature was determined with a carbon resistor. To measure the temperature dependence of the line width, it was desirable to have more than one data point below 1.3 K. This was achieved by Power Stabilizer

BS

Dye Laser

I

Argon Laser

~¢i

Is

Reference Wave / ~ . . Sample

M. / " ~ /

, S

Object

/

ToP ToPM

x t

Wave

/

. z

Isolation Box BS: Beam Splitter M: Mirror S: Shutter Fig. 3. The setup used for the experiments.

300

o

,7::2-

./

/./0

/

//

a~ 2 5 0

_~-

200 0

. . . . . . . . . . . . . . 50 1oo

150

Exposure t i m e / [ s ] Fig. 4. Experimentally determined line widths at 450 mK keeping the burning power constant at 25 ixW/cm 2. The upper line is a fit to the holographic data, the lower line corresponds to the transmission data. It can be seen that the interception yields the same value for both fits of approximately 220 + 40 MHz.

heating up the active carbon sponge and monitoring the temperature, using the resistor.

3.2. Fluence broadening measurements Preliminary experiments were done to test the regime of fluence broadening. For these experiments holograms were recorded at constant power, at a variety of input fluences. Since the power broadening is thus kept constant, it is then possible to isolate the effects of fluence broadening. It was determined experimentally that the maximum fluence values used produced holograms that were far from their saturation values. This results in exposures in a regime for which linear fits to the line widths versus time can be performed. The data are shown in Fig. 4, with solid lines indicating the linear curve fit. At lower temperatures the lines become narrower, which causes saturation to occur for lower fluence. Since it was our intention to remove the effects of fluence broadening from all subsequent measurements, data were taken at a constant power of 25 ixW/cm: and at a temperature of 450 mK, which corresponds to the lowest temperature experimentally available. It can be seen in Fig. 4 that the holographic and absorption data yield the same homogeneous line width for zero fluence. Additionally the slope of the holographic plot is steeper, as expected. Additional data were taken at 1.7 K to confirm that no saturation occurs, and that the assumption of the three-level system is accurate to within the error margins of the experiments [12].

E.S. Maniloff et al. / Chemical Physics 193 (1995) 1 7 3 - 1 8 0 1600

800

%~ 14oo

"~ 6oo

. . . .

,

179

,

,

. . . .

,

,

. . . .

4

I

A

!200 400 1ooo

c

•~ 200

soo 600

.

0

10 20 3o AFrequency/[OHz]

Fig. 5. Plot of the recorded transmission holes illustrated with a frequency offset of 473595 GHz (633.45 nm, 15786.5 c m - 1). The chosen temperature was 450 mK. Lorentzians were fitted to these data to determine the line width. The total fluence was chosen to be 600 ixJ/cm 2. The powers were (from left to right in units of txW/cm2): 10, 20, 40, 80, 175 and 600.

3.3. Power broadening measurements

45O 400 N

350 ~: 3 0 0

250 2O0 150

,

0

,

L

200

,

,

,

i

400

,

,

,

,

6oo

,

,

,

soo

Intensity/[ /.~W/cm']

Fig. 6, Plot of the power dependence of the line width of the absorption holes at constant fluence taken at 450 mK. The plotted solid line corresponds to a fit assuming exponential decay where as the dashed line illustrates a fit with the thin film solution. The interception at 0 power yields 90 + 30 MHz and 100 + 40 MHz, respectively. It can be seen that the thick sample fit is better.

.

.

.

0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1.0 1.5 2.0 Tempereture/[K]

.

L

2.5

~

5.0

Fig. 7. Plot of the temperature dependence of the line width of chlorin in PVB. The standard error was determined by applying a weighted linear regression to the logarithmic data. Every data point was obtained from a regression to zero power and fluence. All given values are given with a Student-t accuracy of 95%. The data are fitted using y = a o + al Taz yielding the following values: a line width at zero kelvin of a0=120-+_60 MHz, the constant a 1 = 135_+50 MHz and a temperature dependence of T1.2

To isolate the effects of power broadening, holograms and transmission holes were recorded at constant fluence value of 600 ixW/cm 2, for a variety of powers. During the experiments the windows of the cryostat were foggy, which reduced the quality of the holograms. Therefore only transmission holes could be fitted. The line widths were determined by fitting the data to Lorentzians. The data are shown in Fig. 5. The power dependence of the line width is illustrated in Fig. 6. One can clearly observe the nonlinear dependence. The plotted solid line corresponds to a best fit to the data using Eq. (14). The dashed line corresponds to the

.c

0.0

4o

±

0.2.

thin film fit Eq. (8). As expected, it can be seen that the fit is better assuming exponential intensity decay. Measurements were performed between 0.45 and 3.0 K at nine different temperatures. Holes were burned with intensities between 20 and 80 ~ W / c m 2. Comparing with the power range used for the measurements illustrated in Fig. 5 and with the simulations, we determined that the data are linear in this intensity range. The exposure time was kept fixed at 60 s yielding a maximum fluence of 2400 txW/cm 2. In Fig. 7 the extrapolated line widths to zero power and fluence are plotted versus the temperature. The fit yields a homogeneous line width of 60 ( + 30) MHz at zero kelvin and a temperature dependence of T12 ± 0.2. This value within its error bars is in agreement with the empirical T 13 law reported in Ref. [13]. The large error bars are due to the errors resulting from the extrapolation to zero power keeping the temperature constant. A comparison of the homogeneous line width to lifetime measurements [14] for chlorin in PVB indicates that the lifetime limited value is nearly reached. It needs to be emphasized again that during the whole treatment other type than fluence or power broadening were neglected. It cannot be excluded that differences between the lifetime limit value and the one obtained here are due to other types of broadening such as concentration effects or to a different temperature dependence in the regime under 1 K [15].

180

E.S. Maniloff et aL / Chemical Phys&s 193 U995) 173-180

4. Conclusion

In this paper data of cw-measurements of the homogeneous line width of chlorin in PVB are given. A theory is developed, which shows how the experimental data should be evaluated, using optically thick samples in order to obtain an estimate of the homogeneous line width. The analysis from Ref. [3] was extended to include effects due to randomly oriented molecules and to optically thick samples. The evaluation of the line widths measurements yields a value close to the one obtained from fluorescence lifetime. An analytic lower limit for the power dependence of the linewidth showing a 4th root power dependence is given. This and the thin film limit can be combined to obtain insight into the intensity dependence of spectral hole-burning.

where K = K2(z) with K as defined in Eq. (7), the z dependence coming from )(2(Z)(XI(Z). In this form this equation can only be solved iteratively but a lower limit can be given by putting K = K 2 exp(-ot0z). This method corresponds to the wellknown first order perturbation theory. Defining q=(1

q._~ ) 2 ,

(19)

we obtain

[ I~(d) I

In(/-~

d

) = - ol0d- fz

dq

=o (APT2) 2 + q

,

(20)

which then leads directly to Eq. (13).

References Acknowledgement

This work was performed in cooperation with the Material Research Department of Ciba-Geigy AG, Marly (Heinz Spahni) and supported by the KWF (Kommission zur F/Srderung der wissenschaftlichen Forschung) and SPP Optik (Schwerpunkt-programm Optische Wissenschaften, Anwendungen und Technologie).

Appendix

The intensity dependence expressed in Eq. (12) results in the equation

= - a o d + c~ofz d

X A~ + . ( i / T ~ ) [ 1

K

+

~ ] z

dz,

(is)

[1] B. Kohler, S. Bernet, A. Renn and U. Wild, Opt. Letters 18 (1993) 2144. [2] K. Holliday and U.P. Wild, in: Molecular luminescence spectroscopy, ed. S.G. Schulman (Wiley, New York, 1993). [3] H. de Vries and D,A. Wiersma, J. Chem. Phys. 72 (1980) 1851. [4] R.W. Keyes, IBM J. 7 (1963) 336. [5] M. Hechter, Appl. Opt. 6 (1967) 947. [6] F.W. Deeg, L. Madison and F. Fayer, J. Chem. Phys, 94 (1985) 265. [7] E.L. Al'shits, B. Kharlamov and N. Ulitsky, J. Opt. Soc. Am. B 9 (1992) 950. [8] F. Burkhalter, G.W. Suter, U.P. Wild, V.D. Samoilenko, N.V. Razumova and R.I. Personov, Chem. Phys. Letters 94 (1983) 483. [9] W.E. Moerner, ed., Persistent spectral hole-burning: science and applications (Springer, Berlin, 1988). [10] S. Bernet, PhD Thesis, ETH Federal Institute of Technology (1993). [11] A.J. Meixner, A. Renn and U.P. Wild, J. Chem. Phys. 91 (1989) 67. [12] F.R. Graf, Master's Thesis, ETH Zurich, (1993). [13] H. Thijssen, R.V. den Berg and S. Voelker, Chem. Phys. Letters 92 (1982) 7. [14] S. Voelker and R. Macfarlane, IBM J. Res. Develop. 23 (1979) 547. [15] S. Jahn, K.-P. Mueller and D. Haarer, Phys. Rev. Letters 66 (1991) 2344.