H2 Lyman and Werner Bands Laser Theory

A. W. Ali and Paul C. Kepple

The laser power densities for the Lyman and Werner bands are calculated with a complete rotationalanalysis. For the Lyman band, P(3) and P(1) are favored and are stronger than Werner Q(1) and P(3)transitions even though the latter are favored in excitation. The transitions from Lyman levels to thelower ground state vibrational levels, where most of the strong Werner transitions terminate, reduce thenet inversion density for the Werner band.

L. Introduction

Recently Hodgson' and Waynant et al.' reportedlasing action in the vacuum uv from the H Lymanband. These experiments were stimulated by a feasi-bility study of Ali' and Ali and Kolb4 who calculatedlaser power densities for a traveling wave excitationsystem developed by Shipman.' Shipman's system,with its high current discharge and -3-nsec currentrise time, was recognized' as a potential short wave-length laser device. The feasibility calculations 4themselves were stimulated by the possibility of in-version7 in the resonance electronic states of H2 . How-ever, these calculations were based on a theory in whichvibrational-vibrational transitions were not resolved asto their rotational structures. In this paper a detailedrotational analysis is carried out for both Lyman andWerner band systems.

In Sc. II excitations of the Lyman and Wernerbands and the resulting transitions are discussed.Section III gives the basic equations describing theupper, the lower laser levels, and the laser power densityper rotational line. Rate coefficients used in thiscalculation are considered in Sec. IV while the resultsare given in Sec. V. The final section contains someremarks and conclusions.

II. Basic Considerations

The transitions in H we wish to consider are theLyman

(B1 2+ - X +)

and the Werner

The authors are with the U.S. Naval Research LaboratoryWashington D.C. 20390.

Received 24 January 1972.

(Cl7r - X 1+)

bands systems, which hereafter are referred to as B-Xand C-X transitions, respectively. At room tempera-ture the H2 molecules are mainly in the zeroth vibra-tional level of the ground electronic state (X, v = 0).This is due to the fact that the next vibrational level(v" = 1) is higher8 by 0.5 eV. Therefore, under a fastexcitation technique, the C and the B state vibrationallevels would be excited and would decay to the X-statevibrational levels (v" > 0), which are empty, resultingin inversion and possible laser action. However, inthis calculation consideration is given to the most pre-ferred C and B vibrational levels. The electron impactexcitation cross section9 aM for electronic state Mfrom the ground electronic state X, is

M JRZ ,,'2 = PO, jIR,,11,j2 (1)

where IR""12 is the square of the electronic transi-tion moment, PMo,V' is the band strength, andIJ t2 is the rotational line strength that can benormalized in such a way that the rotational sum

is valid. Assuming a mean electronic transition mo-ment Re the band strength is

PM.O,, = JR42qM.,o,,, (2)

where q,o,,,' is the Franck-Condon factor. Thus,one could use either calculated Franck-Condon factorsor experimental normalized band strengths Po,V/IRel12 to select the preferentially excited vibrationalstates. In this paper experimental normalized bandstrengths9 for the Lyman and the Werner bands wereused. From these band strengths one can see that forthe Werner band v' = 0 - 4 and for the Lyman bandv = 3 - 12 are preferred assuming a 50%0 valuerelative to the largest band strength within each band.The strong v' - v" transitions within each band can be

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2591

Then the population densities of the upper laser levelscan be written as

dNtr -- NN2XmqMo,,,RmJ',J" - AM,',tjNM,,'Jdit

- E OM ,V'~",J'J"[NMfV'J - (.M ,.IJ/aXV",J")NX,~",J"J ..VI"

X( >O)

,. IX(v=O,J= ,-,a)

Fig. 1. Lyman and Wener bands vibrational-rotational transi-tions that arise as the result of excitations from x(v" = 0, J" = 1,

-, a) state.

selected by their large transition probabilities. Transi-tions with Av > 7 X 107 sec-' were selected usingcalculationLs of Allison and Dalgarno.' 0

We considered electron impact excitations of the Cand the B states from the X(v" = 0) state. However,the rotational distribution with v" = 0 is importantand has to be considered. These can be calculated"by multiplying the Boltzmann factor by the rotationalstatistical weight (2J + 1) and the nuclear statisticalweight of (21 + 1). This last factor is 1 for para-and 3 for ortho-hydrogen. At room temperature therotational distributions in percent are: 13.3, 66.4,11.5, and 8.3 for J" = 0, 1, 2, and 3, respectively.Therefore, we considered excitations from the groundstate, (X, v" = 0, J" = 1) only. The transitions arisingwithin a v - v band for Lyman and Werner systems areshown in Fig. 1 following the selection rules." Lymanband is 2 - transition for which AJ = 4 1 must befulfilled. This implies that only B(V', J' = 0, 2) are ex-cited, which results in the following Lyman band transi-tions: B(v', J' = 0) -- X(v", J" = 1), B(v', J' = 2) -X(v", J" = 1), and B(v', J = 2) -* X(v", J" = 3).These transitions are termed P(1), R(1), and P(3),respectively. The Werner band is a 7r - transitionthat allows AJ = 0 in addition to the previous selectionrules. However, excitations from X(v = 0, J" = 1)lead only to C(v', J' = 1,2) since C(v', J' = 0) doesnot exist." Furthermore, because of A - type dou-bling, each rotational state actually consists of two levelswith different symmetries. The transitions arising inthe Werner band would be C(v', J' = 1) -X(v", J" =1), C(v', J' = 2) - X(v", J" = 1), and C(v', J' = 2)X(v", J" = 3). These transitions are termed Q(1),R(1), and P(3), respectively.

11. Basic Equations

Laser power density and its time history, for eachrotational line within the Lyman and the Werner bands,can be calculated in a manner similar to the theory"3"14

of the pulsed N2 laser. Let N,,,j denote the popula-tion density of the Jth rotational state of the vth vi-brational level belonging to the electronic state M.

where N2 is the population density of (X, v" = 0, '= 1), Ne is the electron density, XM is the total excita-tion rate coefficient for state M, RM,J',J" is the rota-tional line strength, AM,,',J' is the total transitionprobability of the state M, v', J' whose statisticalweight is gm V' j' and finally Om,v'v",,J'J" is thestimulated emission rate for a transition whose wave-length is XA,,V'V",J'J". Equation (3) was used forWerner (I==C) and Lyman (11M==B) bands with v' =0-4 and J' = 1,2 for the former and v' = 3-7 and J' =0,2 for the latter. The summation in Eq. (3) was from1 to 13. The population density of a lower laser levelcan be written as in Eq. (4), where thirteen vibrationallevels were considered.

= Z AAr,vi't",J'J"NS rv',J'dt M,v',J'

+ E 0Af,v'v"YJ'" [Nm,v'j"SKI,v', J'

- (qfl>',J'/gXvt",J")NXvt",J"] . ... (4)

For any given v' 0 v" transition one has three (3)lines in the Werner band and three (3) lines in theLyman band (see Fig. 1). Therefore, the laser powerdensity as it builds up from noise for either band can bewritten as

1 dP Mf,v'v"tJJ"

E1.VTIVJTJ dt' ''

+ OAf.v'v",J'J"[Nff,v',J' -(gMv''/gX,v",J") NX,V", J"]. . ., (5)

where a is a geometric factor. Equation (5) was usedto calculate laser power densities for P(1), P(3), andR(1) for Lyman and Q(1), R(1), and P(3) for Wernerbands.

These rate equations are coupled to the creation ofthe electrons, to their energies, and a circuit equation

dN6/dt = NeN2S, (6)

- (32NkT) = yR(t).I' - NeN2(XcE, + XBEBdt

+ XDED + XE4 + SEj), (7)

L(dI/dt) + R(t)-I + Rf-I =V, (8)

where S is the ionization rate coefficient, 3/2kT is theelectron energy, y is a conversion factor, I is the cur-rent, Xd is the electron impact dissociation rate co-efficient, X is the vibrational excitation rate coefficient,and EC etc. are the threshold energies for the correspond-ing process. R(t) is the gas resistance that is time de-pendent, Rf is a fixed resistance, and V, is the generatorvoltage.

2592 APPLIED OPTICS / Vol. 11, No. 11 / November 1972

C( ) i

I-~ itored and their power densities calculated for aninitial fill pressure of 30 Torr. The circuit parameters,

XjiEco L = 3.4nH, Rf = 3, V0 = 150 kV were furnished to usby Shipman" along with the dimensions of his traveling

xXBEt wave excitation system that was built for the H2 laserC-

7- and later used by Waynant et al.' The result of these

/CEC D o xxccalculations is shown in Figs. 3-7 where the transitionsW ~ / /// . are identified by their upper and lower vibrational

numbers and their wavelengths. The fractions of Ac-a I are omitted. These wavelengths were taken fromv) IC,-8-- I // Herzberg and Howe22 for the Lyman band. For the

Werner band data from Ref. were utilized. Somelinear extrapolation was used when data were not

-J I D available.>_ LLThese calculations show that Lyman band transitionsW 10-9 - are stronger than the Werner band transitions evenZ l ! \though the latter is favored in excitation. This is due

to the fact that transitions from B state vibrationalW l levels, to the lower vibrational levels of the ground-jX l electronic state where most of the strong Werner tran-

o --lo sitions terminate, reduce the net inversion density in theWerner bands. This results in lower laser powerdensities compared to the previously published re-sults4 for the Werner band, where the influence of theLyman band was not considered. The influence of the

10-lI L l l l l l Werner transitions on the Lyman band is small since4 6 8 10 12 1 1 8 20 22 24 most of the strong Lyman transitions terminate at

ELECTRON TEMPERATURE (eV)

Fig. 2. Electron energy loss rates to the most important processes 100.0in H2 as a function of electron temperature. ° (6 3)

IV. Rate Coefficients 1608( 5 (310)

The following rate coefficients, XB, Xc, XD, Xv, and 1596 7 1 1396 10)S. were mentioned in Sec. II. They are, of course,essential for the H2 laser calculations. These rate co-efficients were obtained by averaging the corresponding 10.0

cross sections with the electron velocity over a Max- -

wellian electron velocity distribution. A remark on E_this assumed distribution will be made in the lastsection. For the vibrational excitation Schulz's" mea- <sured cross section was used. For the B and the Cstates, Khare's' 6 calculated. cross sections were utilized. > .

For the dissociation and the ionization rate coefficients -1 1Zwe used measured cross sections of Corrigan 7 and of - 1

Tate and Smith, 8 respectively. Figure 2 shows theelectron energy loss rates 9 to these processes. Finally, 0the resistance of the gas, which is mainly due to electronmolecule collisions, is obtained from electron mobilitydata20 in H2. 0.1

V. Laser Power Density Calculations

To calculate the laser power densities in the Lymanand the Werner bands we selected transitions with lifetimes -_. (0.7 X 101) -1 sec whose upper levels are popu-lated preferentially. Accordingly, v' = 0 - 4 and 0.01 lllIv' = 3 - 7 were chosen as the upper laser levels for 1.0 1.5 2.0 2.5 3.0

the Werner and the Lyman bands transitions, re- TIME (nsec)spectively. For the lower laser levels v" = 1 - 13 Fig. 3. Laser power densities for Lyman P(3) transitions wherewere considered. With this setup a system of 202 the upper and the lower vibrational levels are indicated along withequations were solved where 150 laser lines were mon- the wavelengths.

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2593

100.0 ,,'I''''1''''I''''I ''' tion. The electron temperature is 4-5 eV, and the~ (6- 3) - electron density is (0.5-1) X 10"5 for the major parts of

1608 < (4 11) ~ the laser pulses, i.e., (1-2) nsec. The collision time for1608 (46--II) electrons to equilibrate is 4 = 1.7 X 10-2 T/Ne,(5-12) 16041610 \-/Y\ °(7-I13) where T is the electron temperature in degrees K.

1579This time is = 0.5 nsec, which is short compared with

10.0 ( 31591) the duration of the laser pulses (-2 nsec) or of the

plasma. This justifies the assumption of MaxwellianE1 velocity distribution for the electrons in the calcula-

tions of the rate coefficients.The gain in db/m, is 434.2,3, where ,3 = 1.33 X 10-1

X4An/AX. Here n is the net inversion density, andAX is the line width. The net inversion densities for

L 7._ the strong lines at the peak of the pulse are 3 X 101"U) 1 1 \ \cm-'. Thus for A = 3 X 107, X _ 1600 A, AX _ 0.01

A, the gain is -30, which is quite large indicating thatone does not need the presence of mirrors for amplifica-

3: 11 I D \\ \ \~ tion.

Finally within the framework of our choice, i.e., theselection of the upper vibrational levels, it is apparent

0.1 that this calculation predicts all the observed2" LymanP(3) and P(1) lines. A comparison of Figs. (3) and (4)with Fig. 6 of Ref. 23 shows this fact in addition to therelative strengths of the transitions. Furthermore,this calculation also predicts several strong Wernerband transitions (see Fig. 5) of which two lines, 1161

0.0,0.1 0.5 1.0 1.5 2.0 2.5

TIME (nsec)100.0 _ I I I

Fig. 4. Laser power densities for Lyman P(1) transitions areindicated by their upper and lower vibrational levels and wave-

lengths.

ground state vibrational levels with v" > 4. An un-coupled calculation, for Werner only, gives results 00 (2 126)similar to early calculations.4 Figures 3 and 4 show X 4the strong Lyman laser transitions of P(3) and P(1), - _ T 6(4)respectively, where the peak power densities are in E.3-7)kW/cm' and power half-widths of less than 1 nsec. (2_ byThese lines peak generally within the first two nsec of < 1176

the discharge as is the case for the Q(1) branches of the -

Werner band shown in Fig. 5 where peak power densi- > 1._ties are in W/cm. Figure 6 shows Werner P(3) transi-tions with peak power densities of -1/2 kWatt/cm'.These lines peak 2 nsec later than the Q(1) transitions.This delay is due to the fact that lifetimes of Q(1) are 206shorter than that of P(3). Some Lyman band tran-sitions that peak late in time are shown in Fig. 7. TheR(1) transitions in both bands are very weak and may 01 -not lase. Their weakness is due to their low inversiondensities and that their rotational line strengths aresmall compared to other transitions, which impliesthat their gains are very small. Therefore, for practicalpurposes one may drop all R(1) lines in calculationwithout any appreciable change in the power densities Al 1of the lasing lines. 0 1.0 2.0 3.0 4.0 5.0 6.0

TIME ( nsec)VI. Remarks and Conclusions

Fig. 5. Laser power densities for Werner Q(1) transitions areFigure S Aows the tifflc h1it~oi8 of t edltI'oll indicated by their upper and lower vibrational levels and wave-

density and the electron temperature for this calcula- lengths.

2594 APPLIED OPTICS / Vol. 11, No. 11 / November 1972

3.0 4.0 5.0 6.0 2.0 2.5 3.0 3.5TIME (nsec) TIME (n sec)

Fig. 6. Laser power densities for Werner P(3) transitions areindicated by their upper and lower vibrational levels and wave-lengths. These transitions arise late in time and are stronger

compared to Q(1) transitions.

A and 1230 A, have recently been observed' 8 in lasing.The improved average peak power density measure-ments26 per Lyman line is now -6 kW/cm'. Thisis an improvement over the previous measurements"by a factor of 20. We believe that further improve-ments in the system will result in larger power outputby at least a factor of 2. The uncertainty in this calcu-lation may only be in the'excitation cross sections forthe relevant lasing states, and these may be large byperhaps a factor of 2. A traveling wave calculationthat considers space dependence as well as time de-pendence should illuminate further the power outputfor such a system.

Note added in proof: While this paper was in pressHodgson and Dreyfus27 reported lasing in the Lymanband using the electron beam excitation technique.They indicated that stimulated emission is not presentin the Werner band. However, a short time later (2weeks) they presented a paper'8 to Phys. Rev. Lett.indicating the presence of the stimulated emission inthe Werner band. This event coincided with addi-tional observations of the stimulated emission in theWerner band by Waynant29 who used Shipman'sdevice.

Fig. 7. Some Lyman P(3) and P(1) laser power densities thatarise late in time. The P(3) transition is always larger than the

P(1) for a given v' - v" transition.

I-

za~

10.0

I-

z0.

crC-)~

0 1 2 3 4 5 6TIME (nsec)

Fig. 8. Time histories of the electron temperature and the elec-tron density.

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2595

References

1. R. T. Hodgson, Phys. Rev. Lett. 25, 494 (1970).2. R. W. Waynant, J. D. Shipman, Jr., R. C. Elton, and A. W.

Ali, Appl. Phys. Lett. 17, 383 (1970).3. A. W. Ali, "Vacuum uv Laser," Second International Con-

ference on Vacuum Ultraviolet and X-ray Spectroscopy (Uni-versity of Maryland, College Park, Maryland, 22-28 March1968); A. W. Ali "Vacuum UV Laser from the Lyman andthe Werner Bands of the H2 Molecule," The Catholic Uni-versity Dept. of Space Science and Applied Physics Report68-009 (1968) (unpublished).

4. A. W. Ali and A. C. Kolb, Appl. Phys. Lett. 13, 259 (1968).5. J. D. Shipman, Jr., Appl. Phys. Lett. 10, 3 (1967).6. M. A. Dugay and P. M. Rentzepis, Appl. Phys. Lett. 10,

350 (1967).7. P. A. Bazhulin, I. N. Knyazev, and G. G. Petrash, Sov. Phys.

JETP 21, 649 (1965).8. C. Rulon Jeppesen, Phys. Rev. 44, 165 (1933).9. J. Geiger and H. Schmoranzer, J. Mol. Spectrosc. 32, 39

(1969).10. A. C. Allison and A. Dalgarno, Atomic Data 1, 289 (1970).11. G. Herzberg, Molecular Spectra and Molecular Structure:

Spectra of Diatomic Molecules (Van Nostrand, New York,1950), Vol. 1.

12. J. H. Van Vleck, Phys. Rev. 33,484 (1929).13. E. T. Gerry, Appl. Phys. Lett. 7, 6 (1965).14. A. W. Ali, A. C. Kolb, and A. D. Anderson, Appl. Opt. 6, 2115

(1967).15. G. J. Schulz, Phys. Rev. 135, A988 (1964).16. S. P. Khare, Phys. Rev. 149, 33 (1966).17. S. J. B. Corrigan, J. Chem Phys. 43, 4381 (1965).18. J. T. Tate and P. T. Smith, Phys. Rev. 39, 270 (1932).19. A. W. Ali and A. D. Anderson, NRL Report 7282 (1971).20. S. C. Brown, Basic Data of Plasma Physics (Technology

Press, Cambridge, Mass., Wiley, New York, 1959).21. J. D. Shipman, Jr., NRL, private communication.22. G. Herzberg and L. L. Howe, Can. J. Phys. 37, 636 (1959).23. R. W. Waynant, J. D. Shipman, Jr., R. C. Elton, and A. W.

Ali, Proc. IEEE 59, 679 (1971).24. L. Spitzer, Jr., Physics of Fully Ionized Gases (Interscience,

New York, 1956).25. R. W. Waynant, 24th Annual Gaseous Electronic Conference,

Gainsville, Florida, Oct. 5-8 (1971).26. R. W. Waynant NRL, private communication.27. R. T. Hodgson and R. W. Dreyfus, Phys. Lett. 38A, 213

(1972).28. R. T. Hodgson and-R. W. Dreyfus, Phys. Rev. Lett. 28, 536

(1972).29. R. W. Waynant, Phys. Rev. Lett. 28, 533 (1972).

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2596 APPLIED OPTICS / Vol. 11, No. 11 / November 1972