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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 13 — Jun. 30, 2003
  • pp: 1520–1530
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Model of temperature grating relaxation times in distributed feedback dye lasers

Nasrullah Khan, Tom A. Hall, and Norman Mariun  »View Author Affiliations


Optics Express, Vol. 11, Issue 13, pp. 1520-1530 (2003)
http://dx.doi.org/10.1364/OE.11.001520


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Abstract

A theoretical model and its experimental realization for the temperature grating relaxation time constant and its impact on the temporal and the spectral profiles of a Q-switched and modelocked Nd:YAG laser pumped distributed feedback dye laser (DFDL) is reported. Boundary conditions for different types of excitation pulses have been established to predict the effect of temperature phase gratings on laser gain build-up and temporal elongation of the DFDL pulses. The proposed transient grating method is useful in measuring grating relaxation time constants for lasing dye solutions. The proposed mathematical model is demonstrated by measurement of the relaxation time constant of R6G in ethanol at 10-3M. The measured relaxation time constant of 16±0.2 ns is very close to the tabulated values determined using other techniques.

© 2003 Optical Society of America

1. Introduction

A temperature grating model has been developed and the results have been applied to experimentally measured results. The influence of a continuous temperature accumulation on the amplitude of the temperature grating in a dye cell was studied with a passively Q-switched and mode locked Nd:YAG laser. It is found theoretically as well as experimentally that the peak intensity pulse of the Nd:YAG laser and DFDL do not correspond in time. Rather, the peak pulse of the DFDL pulses is delayed from the peak pulse of the pump laser depending upon the interpulse period of the excitation laser. This increase in the amplitude of the temperature grating was simulated using computer program written in Fortran 77. Several other researchers also have reported similar results [1

1. V. Yu. Kurstak and S. S. Anufrick “Influence of thermal phase lattice on ultrashort pulses characteristics generated by DFDL”, LFNM, Kharkiv, Ukraine, 3–5 (2002).

,2

2. J. Liang, H. Sun, and Y. Hu et.al, “The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,” Opt. Commun. 216, 173 (2003). [CrossRef]

,3

3. A. A. Afanas’ev et al. “Effect of a thermal lattice on the generated line width of a dye laser with distributed feedback,” J. Appl. Spectr. 37, 899 (1982) [CrossRef]

]. They assumed that this peak-shifting phenomenon does not depend on the temperature or reflection of the Bragg grating, rather it depends upon some kind of gain mechanism or re-absorption within the lasing medium. Some researchers have reported similar observations in different disciplines of science and engineering [4

4. A. W. Broerman, D.C. Venerus, and J. D. Schiebler, “Evidence of the stress thermal rule in an elastomer subjected to simple elongation,” J. Chem. Phys. 11, 6955 (1999).

5

5. R. Y. Choie, T.H. Barnes, and W.J. Sandle et al, “Observation of a thermal phase grating contribution to diffraction in erythrosine doped gelatin films,” Opt. Commun. 186, 43 (2000). [CrossRef]

]. This study is focused on a theoretical and experimental investigation of a temperature-grating model and a demonstration of its practical efficacy in distributed feedback dye lasers. Before starting a description of the modeling algorithm and its implementation on DFDL we will review the basic theory of a temperature grating in a distributed feedback dye laser.

2. Theory of temperature gratings

Tt=.[Dth(T)T]+αI(r,t)QC
(1)

If the thermal diffusivity is temperature independent then Eq. (1) simplifies to

Tt=Dth2T+αI(r,t)QC
(2)

Where T is the temperature and ∇the nabla operator. The thermal diffusivity is given by

Dth=sQCp
(3)

Where s is heat conductivity, Q is the density, and Cp is the specific heat at constant pressure. The temperature dependence of Dth can usually be neglected but cannot be ignored on phase transition and at low temperatures. Further, s and Dth are second-rank tensors, α is the absorption coefficient at pump frequency, and I(r,t) is their intensity. In the treatment of thermal gratings, it is assumed that the two pump beams have the same polarization as the opposite polarizations cannot cause interference patterns. The intensity may be given by

I=Iav+2ΔIcosqx,
(4)

where Iav=IA+IB and ΔI=(IAIB)1/2. Furthermore it is assumed that the pump beam interaction region is larger than the grating wavelength. Under these circumstances, the temperature response to the absorbed pump radiation can be conveniently split into a slowly varying average Tav and a grating structureΔT cos qx. The two contributions are almost uncoupled under most experimental conditions. It is therefore possible to split Eq. (2) into two parts which can be solved independently

ΔTt+Dthq2ΔT=2αΔIQC,
(5)
TavtDth2T=αIavQC.
(6)

There are two relevant times associated with [7

7. A. Bosh, M. Brodin, N. Orchair, S. Odulov, and S. Soskin, Soviet Physics, JEPT Lett.18, 397 (1973).

8

8. H. J. Eichler, Ch. Hartig, and J. Knof, Phys Status Solidi (a)45, 433, 1978. [CrossRef]

] are

τq1Dthq2,
(7)
τww28Dth,
(8)

where w is the beam waist. Solution to Eq. (6) is simple when the pump beams are rectangular pulses, short spikes or modulated train of pulses. Rectangular pulses with duration tP

ΔT=ΔTst(1etτq),t<tpwhere
(9)
ΔT=2αΔIsq2
(10)

is the steady-state value of the thermal grating amplitude. After the end of the pump pulse, the grating amplitude decays exponentially with the same time constant as before

ΔT=ΔT(tp)e(ttp)τq,ttp.
(11)

Thus, by recording the temporal dependence of the scattering, the thermal diffusivity can be determined using standard techniques [2

2. J. Liang, H. Sun, and Y. Hu et.al, “The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,” Opt. Commun. 216, 173 (2003). [CrossRef]

]. The response to a short pump pulse with duration tp226A;interpulse period, τ, is

ΔTp=(2αΔItpQC)etτq.
(12)

Amplitude modulation of the pump beam at frequency Ω/2π results in a square root of a Lorentzian for ΔT

ΔT(Ω)=ΔTst(1+Ω2τq2)12.
(13)

When Eq. (13) is squared then it is identical to the respective constitutive equation.

Let us turn to the discussion of the average temperature rise. The steady-state solution to Eq. (6) strongly depends on boundary conditions. The response to a pulse with durationtpτw, on the other hand, can easily be calculated and helps understanding the total temperature development process. For such a short pulse, the heat dissipation by diffusion out of the pumped zone is negligible while the pump pulse is impinging on it. Integration of Eq. (6) thus simply yields

Tav(x,t)=[αIav(x)QC]·t,ttp.
(14)

T(x,t)(14πDtht)exp(x24Dtht).
(15)

This is a Gaussian distribution for width wth

wth22=4Dtht
(16)

increases with time, while the peak height decreases inversely.

T(0,t)14πDtht.
(17)

The temperature distribution Eq. (15) can obviously be matched to that at the end of the pump pulses if the time origin is adjusted properly. The condition wth=w is satisfied at time τw, justifying Eq. (8). Note that the width of the temperature distribution is that of the pump intensity, the width of which is w/√2. When the time zero is assumed at the beginning of the pump pulses, the resulting expression for the average temperature distribution is

Tav(x,t)=[T0(τwt+τwtp)]·exp[(2xw)2(τwt+τwtp)2]fort>tp.
(18)

Finally for strong absorption a fairly simple analytic solution to Eq. (2) exists for strong pump-light absorption and short pulse excitation. The product

T=T(x,t)·g(z,t)
(19)

allows a separation that yields for T the above-discussed plane grating solution. The explicit form of g(z,t), a combination of exponentials and Erfc functions, represents the temperature response to absorption of an unmodulated plane wave. Since the diffracted probe light integrates over the irradiated dye solution, in general, the exact shape of g(z,t) is not very important in a Forced Raleigh Scattering (FRS) experiment with heavily absorbing dye solutions. The time profile of ps-ns regime pumping pulses is often studied by pump probe technique and for ns-µs regime can be measured in real time.

3. Modeling thermal gain accumulation

  1. It is assumed that some r% amplitude of the previous temperature grating is retained in period 2Lc/c and is added up in the newly created temperature gratings by next pulse provided the interpulse period is less than the thermal relaxation time of the constant of the temperature gratings.
  2. Gradual increase in the amplitude of the temperature grating results in higher reflectivity and lower threshold. This may reinforce a weak pump pulse at the end of the envelope to exceed the threshold, which otherwise it would not.
  3. Computer simulation based on this model can create the experimentally observed delay between the peak pulses to lase at shifted central peaks. It does not take into consideration gain of the medium. It provides with percentage value of the overall increase in amplitude of the temperature gratings by creating a peak pulse delay and envelope profile as observed. The maximum reduction in threshold value may cause 1, 2 or 3 extra pulses to appear depending upon the interpulse period or pulse power in q.switched lasers [9

    9. Zs. Bor, A. Muller, and B. Racz, “UV and blue ps pulse generation by a N-laser pumped DFDL,” Optic. Commun. 40, 294 (1982). [CrossRef]

    ].

3.1 Model for modelocked laser

(na)n=n1(1rn)(1r)
(20)

where n1 is the amplitude of single temperature grating. If distribution of amplitude in the temperature fringe is Gaussian i.e. n1=noz1z2e[(zzod)2za]dz . then Eq. (20) modifies to

(na)n=no(1rn)2π(1r)z1z2e[(zzod)2za]dz
(21)

where no is the peak amplitude of the temperature fringe with Gaussian distribution and zod=Δz/2 and za is a constant. In CW modelocked lasers “n” is very large. Increase in amplitude of the temperature fringe represents the percentage of the amplitude of a previously produced temperature fringe. Let us assume it tends to be infinite to be generalized as

(na)nlimn=no2π(1r)z1z2(Fringeamplitudedistribution)dz
(22)

where (-1<r<1) condition is satisfied. Theoretically the efficiency of a DFDL after a long time becomes 1/(1-r) times more than initial value. However the maximum efficiency cannot exceed a certain limit where the threshold for the extra pulse is satisfied. It can happen with short pulse modelocked and long pulse Q.switched laser as was demonstrated by Bor et al [9

9. Zs. Bor, A. Muller, and B. Racz, “UV and blue ps pulse generation by a N-laser pumped DFDL,” Optic. Commun. 40, 294 (1982). [CrossRef]

]. It is more important for repetitively pulsed lasers particularly when r has high values. A sketch of this scheme is illustrated in Fig. 1.

Fig.1. Intensity profile of modelocked laser pump and the DFDL output.

The intensity of last pulse may be twice the first pulse and its pulse length may be 1.5 to 2 times the first pulse.

3.2 Model for Q-switched and modelocked laser

If the dye cell is pumped by a passively Q-switched and modelocked laser then the amplitude of the accumulated temperature grating depends upon the interpulse period 2Lc/c. If the period 2Lc/c is more than τ (thermal relaxation time constant) then thermal relaxation of the medium does not affect the steady state operation of the DFDL. On the other hand, if the interpulse time is less than the relaxation time, the situation becomes complex. Let the amplitude of a fringe in the temperature grating be n1 and the accumulated amplitude (na)k where k=1, 2, 3…..n. The accumulation of temperature grating in the dye medium will enhanced due to the rising intensities of the excitation pulses in the early part of the envelope. This will tend to increase the effective amplitude of a temperature grating and the subsequent intensity of the DFDL output pulses. The effective increase for the nth pulse will be given by [10

10. M. Fogiel, ‘Handbook of Mathematical Scientific and Engineering Formulas, Tables, Functions, Graphs and Transforms,’ TEA New York, 304, 7 (1986).

].

(na)n=k=1nn1(nk+1)r(k1)
(23)

Where n1 is the amplitude of the temperature grating.

In Eq. (24) the energy of each pulse is different from each other pulses. There are three possible cases of interest based on the interrelation of interpulse period 2Lc/c, temperature relaxation time constant τq and pulse length τp. These cases are

Case 1 2Lcc<τqand2Lccτp

Case 2 2Lccτqor2Lccτp

Case 3 2Lccτp

Case 1 2Lcc<τqand2Lccτp

Discrete peak value of the amplitude of the temperature grating, in a Q-switched and modelocked laser, can be expressed by a Gaussian distribution profile as follows

(no)n=nop2πk=1ne[(tktop)2ta]
(24)

where top=[(te/2)+(τp/2)], te=(n2Lc/c+nτp) and nop is the peak amplitude of the q.switching envelope. The total accumulated amplitude can be written as

(na)n=(no)kr(k1)2π(k=1ne[(tktop)2ta])z1z2e[(zzop)2zb]dz
(25)

where tk=[(n-1)2Lc/c+nτp] varies from 0 to the envelope length which is usually from 50 to 200 ns. The generalized equation in terms of pulse and the envelope profile shapes can be expressed as

(na)n=r(k1)(no)kk=1n(Theenvelopeprofile)z1z2(Tempfringeprofile)dz
(26)

Case 2 2Lccτqor2Lccτp

(na)n=(n1)1(1rn)(1r)+r(no)x[1nr(n1)+(n1)r(n1)(1r)2
(27)

Where “(no)x” is the fixed amount of amplitude being added by each pulse to the subsequently induced temperature fringe. This assumption was made to simplify the formulation leading to the sum of geometrical and arithmetical progressions [6

6. Laser induced dynamic gratings, edited by H.J Eichler, P. Gunter, and D.W. Pohl, Springer Verlag Series on Opt. Sciences, 50, 17 (1986).

]. The peak energy of the pulses increases gradually. Let us assume that the modelocking and Q-switching envelope is of Gaussian shape. The peak value of the amplitude may be given by

(no)x=(nop)k2πk=1ne[(tktoe)2ta]z1z2e[(zzop)2zb]dz
(28)

Where tk=tl→tn. Substitution of Eq. (28) into Eq. (27) leads to the following generalized expression.

(na)k=no(1rn)2π(1r)z1z2e(zy2zb)dz
+(nop)kr[1nr(n1)+(n1)r(n1)]2π(1r)2k=1ne(tx2ta)z1z2e(zy2zb)dz
(29)

where ta and tb are constant, tx=(tk-toe) and zy=(z1-z2)/2. The limits have been written in the above equation up to half of the envelope, however they can be extended up to the end of the envelope too. The shift in intensity profile of pump and DFDL lasers may be explained as shown in Fig. 2.

Fig. 2. Impact of pump laser pulse train profile on DFDL output profiles.

Case 3 2Lccτpandτq

This is the most simple case in which a dye cell is being pumped by mode-locked or repetitively pulsed laser. The typical pulse lengths are 20 to 30 ns and thermal relaxation time of the temperature gratings is of the order of 10 to 15 ns. The inter-pulse durations are more than τp or τq. This is the most general version of the accumulative model and can be applied to the q-switched lasers. The derivations are same as for mode-locked lasers. However, the value of r% would remain zero in this case. It will not cause any delay at all.

4. Experimental results

The second harmonic of a passively Q-switched and modelocked Nd:YAG laser was used to pump the solution of Rh6G in ethanol at concentration of 1.0×10-3 M. Experimental setup is shown in Fig. 3. A microdensitometer scanned streak record is shown in figure 4a showing both the pump laser streak record and the DFDL streak record. A half period delay was introduced deliberately in these results for clarity and it was subtracted for data processing.

The model can predict the mutual dependence of pump power and the percentage increase in amplitude of the temperature gratings. If the delay in the peak amplitude of the grating is equated with output power from the DFDL then the model shows a good correspondence between experiment and theory. The resultant delays were determined from the data by the computer program as well as being determined by visual inspection. Variations in the envelope shapes were found in different shots due to pump laser instability. The percentage increase in amplitude of the temperature gratings for a particular experimentally delay was estimated by creating a similar delay in the computer program. The computer program can create Q-switched and modelocked patterns with the option of different combinations of pulses and envelopes.

Fig.3. Experimental layout of Nd:YAG laser pumped DFDL.

Experimentally measured delays of 14.4±0.5 ns for 2Lc/c=4.43 ns, 12.8±0.5 ns for 2Lc/c=5.93 ns and 10.9±0.5 ns for 2Lc/c=8.31 ns correspond to computer calculated percentage increase of amplitudes of the temperature gratings of 74.2%, 67.5% and 57.6% respectively assuming Gaussian pulses and envelopes profiles. If the exact shape of the pulses and envelopes is known then the above values are 100% correct otherwise the accuracy depends upon the nearness of the actual envelopes and pulses to those assumed in the computer program.

5. Temperature grating time constant

The dynamically induced temperature gratings relax much slower as compared to other gratings such as population gratings and anisotropy gratings. If τq is thermal decay in liquids then it can be expressed by

n1=noexp((tτq))
(30)

where no is the maximum amplitude of the grating and for different pulses of the q-switched and modelocked Nd:YAG laser (assuming Gaussian profile) this varies as

n1=(no)mexp((tto)22)
(31)

Fig.4. Microdensitometer scanned streak record of Q-switched and modelocked Nd:YAG laser and corresponding DFDL output together with simulated output from the model.
n1=noexp(2Lccτq)
(32)
τq=2Lccln(r)
(33)

The plot of 2Lc/c versus r gives the value of τq as 15.8 ns. The same results can also be achieved in terms of the delay between the peaks of the envelopes as follows. The experimentally measured delay between the peaks of the envelopes was plotted against the percentage energy retained by the medium for similar delay in the computer program. The curve fitting program gave the following relation between r and τD.

r=0.2498x100.033τD
(34)

Combining these one gets a straightforward relation between thermal relaxation time constant and the experimentally measured delay between the peaks of the pump and DFDL.

τq=2Lcc(ln(0.2498x100.033τD))
(35)

The thermal relaxation time constant of Rh6G in ethanol at a concentration of 1mM was found to be 15.8 ns. Value of τq was also calculated using

τq=cpρKtK¯2
(36)

which was 16.88 ns. These values are very close to the tabulated values. Other researchers [11

11. A. Penzkofer and W. Falkenstein et al, Chem. Phys. Lett, 44, 82 (1976). [CrossRef]

13

13. P.Y. Key and R.G. Harrison, IEEE. J. Quant. Electron.QE-6, 645 (1970).

] previously have reported τq=12 ns for water and τq=20 ns for ethanol at some other concentration of Rh6G.

6. Conclusions

Population and phase gratings are produced by the spatially periodic distribution of light from a fringe pattern of a pump laser that is absorbed in a laser dye solution. This absorbed energy on thermalization produces a thermal grating. The temperature distribution in the thermal grating is sensitive to the interpulse excitation period. We have demonstrated an increase in the amplitude of the temperature grating for a Q-switched and mode locked pump laser. The theoretically calculated delay between the pump pulses and the DFDL output was found to be a function of the cavity length and envelope shapes but are not significantly altered by changes in the individual pulse shape. The microdensitometer records show that the DFDL pulse envelope tends to be peaked up one or two interpulse periods after the corresponding Nd:YAG pulse envelope peak. The overall tendency of the DFDL envelope is a slowly rising front and a fast decaying tail. This particular shape is formed by the gradual increase in the pump power in the first half due to the rising front of the pump laser and a proportionally lower increase in the amplitude of grating in the second half due to the decaying envelope of the pump laser. The use of the model enabled us to measure the thermal decay time of the gratings which was found to be 15.8 ns under the concentrations used in the experiments.

References

1.

V. Yu. Kurstak and S. S. Anufrick “Influence of thermal phase lattice on ultrashort pulses characteristics generated by DFDL”, LFNM, Kharkiv, Ukraine, 3–5 (2002).

2.

J. Liang, H. Sun, and Y. Hu et.al, “The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,” Opt. Commun. 216, 173 (2003). [CrossRef]

3.

A. A. Afanas’ev et al. “Effect of a thermal lattice on the generated line width of a dye laser with distributed feedback,” J. Appl. Spectr. 37, 899 (1982) [CrossRef]

4.

A. W. Broerman, D.C. Venerus, and J. D. Schiebler, “Evidence of the stress thermal rule in an elastomer subjected to simple elongation,” J. Chem. Phys. 11, 6955 (1999).

5.

R. Y. Choie, T.H. Barnes, and W.J. Sandle et al, “Observation of a thermal phase grating contribution to diffraction in erythrosine doped gelatin films,” Opt. Commun. 186, 43 (2000). [CrossRef]

6.

Laser induced dynamic gratings, edited by H.J Eichler, P. Gunter, and D.W. Pohl, Springer Verlag Series on Opt. Sciences, 50, 17 (1986).

7.

A. Bosh, M. Brodin, N. Orchair, S. Odulov, and S. Soskin, Soviet Physics, JEPT Lett.18, 397 (1973).

8.

H. J. Eichler, Ch. Hartig, and J. Knof, Phys Status Solidi (a)45, 433, 1978. [CrossRef]

9.

Zs. Bor, A. Muller, and B. Racz, “UV and blue ps pulse generation by a N-laser pumped DFDL,” Optic. Commun. 40, 294 (1982). [CrossRef]

10.

M. Fogiel, ‘Handbook of Mathematical Scientific and Engineering Formulas, Tables, Functions, Graphs and Transforms,’ TEA New York, 304, 7 (1986).

11.

A. Penzkofer and W. Falkenstein et al, Chem. Phys. Lett, 44, 82 (1976). [CrossRef]

12.

D.Y. Key, ‘The scattering of light from light induced structures in liquids,’ Ph.D. Thesis, London University, (1977).

13.

P.Y. Key and R.G. Harrison, IEEE. J. Quant. Electron.QE-6, 645 (1970).

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(140.2050) Lasers and laser optics : Dye lasers

ToC Category:
Research Papers

History
Original Manuscript: April 10, 2003
Revised Manuscript: June 12, 2003
Published: June 30, 2003

Citation
Nasrullah Khan, Tom Hall, and Norman Mariun, "Model of temperature grating relaxation times in distributed feedback dye lasers," Opt. Express 11, 1520-1530 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1520


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References

  1. V. Yu. Kurstak and S. S. Anufrick�?? Influence of thermal phase lattice on ultrashort pulses characteristics generated by DFDL�??, LFNM, Kharkiv, Ukraine, 3-5 (2002).
  2. J. Liang, H. Sun and Y. Hu et.al, �??The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,�?? Opt. Commun. 216, 173 (2003). [CrossRef]
  3. A. A. Afanas'ev et al."Effect of a thermal lattice on the generated line width of a dye laser with distributed feedback," J. Appl. Spectr. 37, 899 (1982) [CrossRef]
  4. A. W. Broerman, D.C. Venerus and J. D. Schiebler,�?? Evidence of the stress thermal rule in an elastomer subjected to simple elongation,�?? J. Chem. Phys. 11, 6955 (1999).
  5. R. Y. Choie, T.H. Barnes and W.J. Sandle et al,�?? Observation of a thermal phase grating contribution to diffraction in erythrosine doped gelatin films,�?? Opt. Commun. 186, 43 (2000). [CrossRef]
  6. Laser induced dynamic gratings, edited by H.J.Eichler, P. Gunter and D.W. Pohl, Springer Verlag Series on Opt. Sciences, 50, 17 (1986).
  7. A. Bosh, M. Brodin, N. Orchair, S. Odulov and S. Soskin, Soviet Physics, JEPT Lett. 18, 397 (1973).
  8. H. J. Eichler, Ch. Hartig, J. Knof, Phys Status Solidi (a) 45, 433, 1978. [CrossRef]
  9. Zs. Bor, A. Muller and B. Racz,�??UV and blue ps pulse generation by a N-laser pumped DFDL,�?? Opt. Commun. 40, 294 (1982). [CrossRef]
  10. M. Fogiel,�??Handbook of Mathematical Scientific and Engineering Formulas, Tables, Functions, Graphs and Transforms,�?? TEA New York, 304, 7 (1986).
  11. A. Penzkofer and W. Falkenstein et al, Chem. Phys. Lett., 44, 82 (1976). [CrossRef]
  12. D.Y. Key, �??The scattering of light from light induced structures in liquids,�?? Ph.D. Thesis, London University, (1977).
  13. P.Y. Key and R.G. Harrison, IEEE. J. Quant. Electron. QE-6, 645 (1970).

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