DEPARTMENT OF PMSICS

COLLEGE OF SCIENCES . )

OLD DOMINION UNIVERSITY / Y (f

NORFOLK, VIRGINIA 23529

Technical Report PTR-88-5

THEORETICAL STUDIES OF SOLAR-PUMPED LASERS

By

Wynford L. Harries, Principal Investigator

1<-CB-182929) IEECBETICAL S1UDJES OF £8*8-23971

SGIAF-SUHPED LA5EBS Progress fcepczt, 16 Jan.

- 15 JUB. 1986 (Cld rcn)iiiic£> Dciv.) 18 p

CSCL 20E Unclas

G3/36

f

Progress Report

For the period January 16, 1988 through June 15, 1988

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665

Under

Research Grant NSG-1568

Dr. Nelson W. Jalufka, Technical Monitor

SSD-High Energy Science Branch

Submitted by the

Old Doarinion University Research Foundation

P. 0. Box 6369

Norfolk, Virginia 23508

June 1988
THEORETICAL STUDIES OF SOLAR PUMPED LASERS

By

Wynford L. Harries*

SUMMARY

The investigation of stimulated emission causing transitions from the

Bn electronic state of sodium to the overlapping 2 J electronic state has

been continued. A new method of estimating the Franck-Condon factors has'

been developed which instead of fitting the molecular potential curves with

Morse functions, estimates the V(r) dependence by interpolation from given

potential curves. The results for the sum of the rates from one vibrational

level in the upper state to all the levels in the lower state show good

agreement with the previous method implying that "curve crossing" by

stimulated emission due to photons from the oven is an important mechanism

in sodium.

*Eminent Professor, Department of Physics, Old Dominion University, Norfolk,

Virginia 23529

J
1. INTRODUCTION AND PURPOSE

The experimental results at NASA Langley indicate that absorption of

the 488 nm Argon ion laser light in sodium occurred from the transition

X(v=0)> B(v=6) (path #1, Fig. 1). The increased fluorescence occurred at

wavelengths of 730 nm or greater, and the emission could only have come from

the transition A •»• X (path #4, Fig. 1). The dotted path #5 from B nu to

A £u is forbidden (u+u). However, the pathways B n -»-2 I , and subsequent

path 2 I •*• A £u are allowed. The question arises of the mechanism and rate

of transfer for these last two transitions, especially the transition for

"curve-crossing", B n -»• 2 J .

The effect of particle collisions on the transfer rates is first

considered. No buffer gas was used in the experiments with sodium.

At 450"C (723k) the total vapor pressure was estimated at about 1»31

torr, with the pressure of atomic Na about 1.24 torr, and of molecular Na

about 0.073 torr; thus Na./Na = 0.06. The calculated collision time of Na,

* -8 * Na? with an excited Na? was 4 x 10 s, and the collision time of excited Na?

* -7 with Na« was about 7 x 10 s. Both times are longer than the spontaneous

-9 emission time from the B level of 7 x 10 sec, to the ground X state and

therefore collisions can be neglected.

It is suggested that the transition between the two overlapping states

B n and 2 T , and the transition 2 £ +A n are due to stimulated emission

from radiation due to the oven at 450°C (723k). The energy differences

between the levels Vj = 6 in the B state, and v,, = 0 to 20 in the 2 Y state

correspond to

 a 0.3eV and require photons of wavelength * 4 pm. The Wien

displacement law, *maxT = 2898 (pm.K) indicates that the wavelength X max at
peak radiation intensity was indeed » 4ym at 723 k. The J" -»• A transition is

around 0.5 eV corresponding to radiation « 2y where there is still ample

energy.

The transitions B - 2 £, and 2 T - A would predominate only if the

stimulated emission rates were far higher than the spontaneous emission

rates B - X, and 2 I - X respectively. We shall first estimate the ratio R

of the stimulated emission rate W (B+2 I) with the spontaneous emission rate

(B^X).

2. TRANSITION RATES

As is well known, the rate of stimulated emission W = pB, where p is

the energy density of the black body radiation and B is the Einstein

coefficient. The value of p was obtained from Planck's radiation formula

and depended on the wavelength x , determined by the transition from vl

 V2

level v. to level v. and the absorbtion bandwidth. The latter was obtained

by assuming absorption occurred over of order 100 rotational lines, so AX

was 100 times the Doppler bandwidth of one line. Pressure broadening was

negligible under the conditions here.

The Einstein coefficient B is given by Ref. 1:

'v

where |p. | is the matrix element for the electron dipole moment for the

I,v1v2

transition B-»• 2^, from level v^ to level v2, EO is the dielectric constant

for free space, and-If is Planck's constant/211.

*

The matrix elements are of the form |u|= e / th r ^ dr where i|^, ^

are the wavefunctions of the two respective states, e is electronic charge,
and r the length of the dipole. Assuming the transition is instantaneous,

we take r as a constant and take it outside the integral. Then |y| can be

expressed as

= e r

where FI.V-^ is the Franck-Condon factor for the transition v^. In the

experiments the Argon ion laser pumped the molecules from the ground (state)

to the v, = 6 level in the B electronic state.

The rate of transitions from the v, = 6 to a level v« in the 2'£ state

is then

and the total rate of all transitions from v. = 6 to all the levels in the

2'£ state below v^ in energy (corresponding to an upper value of v2 = vp)

is then

n W ; Vl = 6 (4)

= 0 V1'V2

 L

The level corresponding to v in the 2 £ state must lie below the v = 6

level for the B state, for stimulated emission, and its value is

vn - 24.
3. PRELIMINARY COMPARISON OF STIMULATED TO SPONTANEOUS RATES

Next, we consider spontaneous emission, B + X with a rate coefficient A

given by (Ref. 1):

A =

si2

"2«V1'V3

3 h V x3

a

c_

; v1 = 6 (5)

with the same nomenclature as Eq. (1); |u2| is for the transition B -»• X and

X the wavelength corresponding to the transition between the vibrational d

quantum states, B(v.) to X(v_). However, the energy difference is about 2.6

eV, for V, = 6 and X = 0.5 urn, a value 10 times smaller than the peak i a

wavelength for black body radiation. The spread in x turns out to be small

a

or of order ±0.03 vm, as transitions are only appreciable for 0 < v < 4,

O

hence we neglect any spread in x due to different values of v and take its

a o

value as 0.5 ym.

Again, in the electronic transition, r is assumed constant, and is the

same r as in Eq. (3). The total rate of spontaneous emission is then

Ref. 1

A=ill_(er)2

 I 2 . (6)

3*Y. X3

a v, ' 1>3

o a 3

where F2,v,,v- is the Franck-Condon factor for the transition v., v3.

Experimentally, the spontaneous lifetime of the B state is known to 7ns, so

A = 1.4 x 108 s"1.

Equations (5) and (6) can be evaluated if r and the Franck-Condon
factors are known, but first it is instructivje to take the ratio

R = W/A = pB/A:

>

3

 ?

a IP (Xv,,v ? , T) • F^ v, v?

R = L_£

 l l 2 . (7)

4tf I F?.^ v3

Here, as'both transitions started from the B state, r cancels. The FranckCondon factors are finite for only specific values of x , . To get an v

l

 V2

order of magnitude for R, assume the contributions from the Franck-Conden

factors in the numerator and denominator cancel, and define a quantity R1 =

p (x , T) X AWT . The quantity R1 essentially reflects the contribution

V

1

V

2

 a

from the radiation density. Assuming x = 0.5 ym and constant, R' is

a

plotted as a continuous function of X , for different temperatures in

v

l

 V2

Fig. 2. It can be seen that for x , > 4 urn that R1 » 10; but if

vl V2

X , " 0.5y then R' < < 1, indicating that here R would be much less than

vl V2

1. Hence stimulated emission should predominate at around 5ym, and not

around 0.5 um.

4. CALCULATION OF TRANSFER RATES

We next confirm this conclusion by evaluating the actual stimulated and

spontaneous emission rates, based on estimations of the Franck-Condon factors. The basis of the method has been reported already (Ref. 2), but the

detailed calculations were done in two different ways, the difference being
in how the initial constants were obtained from the given potential curves

of Jeung (Ref. 3).

With the first method, Morse functions were fitted to the potential

curves, and the values of u , <o x etc., taken from Ref. 4 for the B'n

state and from Ref. 5 for the 2 ' V state except that the value of De was

made larger (0.7 eV) to fit the curves of Jeung. With the second method,

the potential curves were digitized using a computer interpolation technique, and all the required constants estimated from these numbers.

4.1 Evaluating Franck-Condon Factors Using Morse Functions

Morse functions were fitted to the curves of Jeung (Ref. 3) at three

points: the values were Te at the equilibrium distance of r^ the internuclear distance, the value of V at r. -»• «° which is Te + De, where De is the

dissociation energy, and the value of r. corresponding to the level halfway

between T and T + D . It can be seen the fit is approximate. The varia- e e e

tions from the curves shown in Fig. 1 may predicate that the energy levels

may be inaccurate by one or two vibrational levels, but nevertheless the

overall results in calculating W, the sum of contributions from v = 6 to all

the levels below it were relatively insensitive (± 10%) to changes of 50% in

D for example. The reason may be that an error in level v merely meant

that the next v + 1 level would give a compensatory contribution.

The wavefunctions were obtained from the one dimensional Schroedinger

equation with potential V represented by a Morse function (Ref. 5). First

asymptotic solutions were obtained for large values of r., the internuclear

distance. The values of r. were then reduced in small increments and

iterative solutions of the wave functions obtained by Euler's method. The
value of a) were taken from the literature but the anharmonicity constants

u x , were calculated from to and D , the dissociation energy read from the

e e e e 3

2

curves: coexe = u /4 Dg (Ref. 5). The second set of anharmonicity constants

to y were taken" as zero in all cases. Thus the calculations are approximate.

The wave functions were well-behaved and crossed the axis the correct number

of times up to the 50th level, although only 35 levels were used for each

electronic state, The maximum values of the Franck-Condon factors occurred

when the turn around points for the two electronic levels occurred at the

rl *

same value of r.. Checks on orthogonality showed that / ih ^ dr; r, =

8A° were always within or less than a few tenths of a percent. The Franckr

l

Condon factors / ip i|» dr (where i|> , i|> were normalized,) were then evaluated for all combinations of n, m from 0 to 35. The Franck-Condon factors

ranged from « 0 to « 0.2 and agreed (for the B -»• X transition) with Kusch

and Hessel (Ref. 7). The vibrational sum rule, (Ref. 5) that the sum of the

squares of the Franck-Condon factors summed over all values of the vibrational quantum numbers of both of the states should be equal to 1 showed

values approaching 1 from the lower side, as expected when the quantum numbers for both levels were truncated at 35. Plots of the Franck-Condon factors plotted on a v, V2plane, where v., v? are the vibrational quantum

numbers for both states, clearly showed a Condon parabola (Ref. 6).

4.2 Evaluation of Franck-Condon Factors Using Interpolation

In an attempt to get values which fitted the potential curves more

accurately, the second technique included digitizing eleven points on the

curves and entering the values into a computer. An interpolation sequence

then fitted a 10th order Lagrange polynomial to the curves. The fit could
be accomplished to approximately within the breadth of the line and thus V

as a function of r was known. Also digitized were Te, De and the value of

ri at the minimum of the curve. The values of o> , <u x and <o y were first

taken from Refs. 4 and 5, but the wavefunctions were not well behaved. Even

for v = 6 for the B nu curve the number of crossings of the wavefunction was

incorrect, being only 5. The functions were also unstable for values of r.

to the left of the intersection of V and the vibration energy corresponding

to v = 5.

Considerable improvement was obtained by not using the values of

to JD x and to y from Refs. 4 and 5, but instead estimating them from our

c c 6 c 6

digitized curves. The value of <o corresponds to the frequency of a harmonic oscillator, whose potential curve would coincide with the anharmonic

curves near the potential minimum. Hence if r is the radius for the potential minimum, then near r a parabolic relation would be obeyed.

-k (rrre)

2 (8)

where k is the spring constant. So a plot of + V - T vs (r-r ) should be

a straight line of slope ^ k/2 near r . Such plots showed constant slope

for a region ± 1A from r , hence yielding k. The value of <u , obtained

6 ' c

knowing the effective mass for Na?, showed differences of about 5% with

2

those in Ref. 4. The values of MX were then calculated from co x = to /4De c 6 c c C

and differed at worse by a factor of about 2 for the 2 J level, from Ref.

5, and as before u y was assumed to be zero. In this case the

wavefunctions crossed the zero line the correct nunber of times up to the

20th level, but the functions were still unstable to the left of the V
curves at higher value of v. As soon as instability set in the computer

clamped the value of the wavefunction to zero.

The Franck-Condon factors obtained from the two methods differed by

large factors when comparing the transitions from v.. =6 in the upper level

to individual levels in the lower state. For some values of v the Morse

curve results were larger, for others it was the reverse. The two sets of

Franck-Condon factors were then used to calculate the stimulated emission

rates, where again the answers were different for individual levels. However, when the rates were added for 21 lower levels, the differences averaged out and gave total transition rates, in good agreement as shown

below.

4.3 Stimulated Emission Rates

The value of W was obtained using the value of B from Eq. (1) where \i, was

taken as (er) F, with F, the value for B -»-2 Jn:

i, "»v '

2 2 F TT e r 1 v, ,v

12 , .2

3e h

The value of r was obtained from the known value of A, Eq. (5), where |v2l =

er F0, and X ? 0.5u; then r = 1.5A, a reasonable value, half the equilib- L a

rium internuclear distance.

A plot of W vs X , the wavelength of the oven black-body radiation. V1

V2

is shown in Fig. 3, for values of v, = 5, 6, 7, using the Morse curve numbers and shows a series of "lines" occurring near x = 5 ym with values »

12 -1

10 s . The addition of all the lines between 0 and 20 u gave a total

10
12 1 + rate of several times 10 for the stimulated emission rate B -»• J . The

9

interpolation method gave individual values of W, (v = 6) for different v?

in qualitative agreement with Fig. 3, but, with individual values which

showed differences of up to 5. However, when the rates for all the lines

were added the net rates calculated by the two methods agreed within 20

8

percent. The spontaneous emission rate from B -»• X is of order 10 , so the

ratio R again approaches 10 .

Stimulated emission does not occur from B -»• X. The 2 eV difference in

the electronic levels calls for wavelengths of 0.5 ym where the value of p

is about six orders of magnitude smaller at T = 450°C.

In a similar manner the stimulated and spontaneous emission rates were

calculated for the various pathways shown in Fig. 1 and are shown in Table

1. It is evident that the stimulated emission rate B -»• X far exceeds any

stimulated or spontaneous rate B-»• X. Likewise the stimulated emission rate

2 y •»• A exceeds the spontaneous rate and also both stimulated and

spontaneous 2 J + X. For the A + X transition the stimulated and

spontaneous rates are comparable.

5. CONCLUSIONS

The model we have assumed in the calculations is approximate only. It

may have inaccuracies in the value of r and also in the estimates of the

energy levels and hence the actual values of the wavefunctions and FranckCondon factors. Nevertheless, the insensitivity of our estimated total

rates to variations in parameters such as D , and the differences of 5 orders of magnitude between the estimated stimulated and spontaneous emission

rates (Table 1) suggest the pathways in Fig. 1 are valid. Moreover the

suggested mechanism of stimulated emission due to the oven black body

11
radiation seems valid for transfer from electronic energy states which

overlap, and for transfer in "curve crossing" in sodium.

ACKNOWLEDGEMENTS

The author gratefully acknowledges numerous stimulating discussions

with Dr. N. Jalufka of NASA/Langley Research Center, Hampton, Virginia.

N

12
REFERENCES

1. Svelto, 0. "Principles of Lasers" 2nd Ed. Plenum Press N.Y. p43, etc.

2. Harries, W. L. "Theoretical Studies of Solar Pumped Lasers. Technical

report PTR-81-9 for the period 15 Jan - 15 July 1981, under Research

Grant NSG 1568.

3. Jeung, C. J. Phys. B: At Mol Phys V. 16_ (1983) 4289-4297.

4. Huber, K. P. & Herzberg G. "Constants of Diatomic Molecules" Van

Nostrand 1979, p. 432.

5. Effantin, C., d'Incan J., Ross A. J., Barrow R. F. and Verges J., J.

Phys B: At Mol Phys V. J7 1515 (1984).

6. Herzberg, G. "Molecular Spectra and Molecular Structure I Spectra of

Diatomic Molecules, Van Nostrand Reinhold (N.Y.) 1950.

7. Kusch P. and Hessel, M. M. J. Chem. Phys. 68(6), 2591 15 Mar. 1978.

13
TABLE. 1

Transitions rates in Na2 with initial and final vibrational levels, _

average energy difference A(eV ) and average wavelength of transition X.

Initial Upper vb Final vb X Max Spont,

State Transition level v^ level v2 A(eV) (urn) W(s~*) A(s)~l

B B+ 2'I 6 0-20 0.3 4 8xl012 2xl05

B •»• A 6 0-35 Forbidden

B -»• X 6 0-35 2.5 0.5 - 7xl02 1.4xl08

2'Iq I + A 0-20 0-35 0.6 2 2xl013 l.SxlO6

I * X 0-20 0-35 2.0 0.6 - 1x105. 6xl07

A A* X 0-35 0-35 1.7 0.7 - 4xl07 3.7xl07

14
ENERGY

(eV)

5 8

Figure 1. Energy level diagram for Na~ according to Jeung (Ref. 3), with the

transition pathways shown. The dotted curves are best fitted

Morse functions.

15
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