The potential efficiency of using nonlinear up-conversion techniques for the high efficiency type lasers (CO,CO2, and chemical) is assessed. Results indicate that: the small pump photon energy necessitates the use of molecular media for conversion if resonance enhancement is to be used and that molecular systems present several problems. These difficulties include: their levels are complex; their transition probabilities are often unknown; and the oscillator strengths among vibrational levels in the ground electronic state of a molecule are much smaller than those among electronic states of an atom, thus limiting the magnitude of nonlinear interactions. It is shown that this problem can be eliminated by making use of vibronic transitions which, being primarily electronic transitions have much larger matrix elements and efficient conversion can be achieved with molecular systems

Billman, K. W.

She, C. Y.

English

NASA Technical Reports Server

IR PUMPED THIRD-HARMONIC GENERATION AND SUM-FREQUENCY
GENERATION IN DIATOMIC MOLECULES
C. Y. She
Colorado State University [,
c
N76-21513Kenneth W. Billman
NASA-Ames Research Center
It is well known that the efficient lasers we have, approximately 20 percent for C©2 around r
10.6 M and 60 percent for CO around 5 jz, have inherent disadvantages for certain types of power '
transmission. First, since beam divergence scales as X/D, we are confronted with either limited range
or excessively large optics (e.g., 30-m diameter collection mirror at geostationary orbit for
Earth-to-space transmission of 10.6-n wavelength) for reasonable transmission efficiency. Secondly,
the use of solid state converters is limited to 2/z and, for maximum efficiency, wavelengths below 1 M
are desirable. Even if the laser-driven engine is used for conversion, the need for high temperature
strength and transmission of the beam entrance window dictates the use of sapphire which has a
long wavelength cutoff at about 5.5 M- Thus the CO^ laser wavelength would certainly need .
up-conversion. . t
For these reasons, we began a study recently to assess the possibility and potential efficiency
of using nonlinear up-conversion techniques for the high efficiency type lasers (CO, C©2 and
chemical). This paper is a preliminary report on what we believe are some encouraging results.
Although the conversion efficiencies are not extremely high, it should be borne in mind that this
may be improved as we examine other candidate molecular gases. Also, especially for ground-based
laser type missions, one might be more tolerant of overall laser and up-converter efficiency,
especially if it reduces the need for many large onboard collection optics systems by area factors of
nine, say, or if this overall efficiency exceeds that of the best direct (no up-converter) laser
operating at the same wavelength.
Efficient up-conversion of visible laser radiation by third-harmonic and sum-frequency
techniques, using metallic vapors with added atomic vapor for phase-matching, has recently been an
active and successful area of research (refs. 1 through 4). Very high conversion efficiencies (ref. 5)
are attributable to the use of resonance enhancements, particularly a resonance at twice the pump
frequency (refs. 4 and 5). In principle, such conversion processes should also be possible for longer f
wavelength laser radiation. However, the smaller pump photon energy necessitates the use of
molecular media for conversion if resonance enhancement is to be used. Molecular systems present
certain difficulties: their levels are complex and transition probabilities often unknown. Also, the
oscillator strengths among vibrational levels in the ground electronic state of a molecule are either
zero or much smaller than those among electronic states of an atom, thus limiting the magnitude of
nonlinear interactions. However, as we will show here, this potential problem in the up-conversion
process can be eliminated by making use of virtual vibronic (vibration-electronic) transitions which,
being primarily electronic transitions have much larger matrix elements and efficient conversion can '"
be achieved with molecular systems. *-
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The energy diagram of a typical molecule is shown in figure 1. Both the ground and excited
electronic states are split into rotation-vibrational levels with each denoted by electronic,
vibrational, and rotational quantum numbers (e, v, and J). The incident infrared laser frequencies
are a>j and o>2 and the converted (or generated) frequency is 003. For many molecules, the
rotation-vibrational levels in the ground electronic state are well separated in comparison with their
linewidths; thus only one pair of such levels, with energy separation ftcoo, may be assumed to satisfy
the condition of two photon resonance, namely, CJQ ^  2coj. Only this pair of levels, denoted by| g > and | a > in figure 1, will make a significant contribution to the nonlinear interaction in
question.
Because there is no resonance requirement of the rotation-vibrational levels in the excited
electronic state, all those levels which are electric-dipole-allowed will participate in the nonlinear
process. The energy of all these levels in the excited electronic state may be approximated by their
average value M2 because £2 » 002 and 003. We therefore lump all these excited vibronic levels
into a single upper state | b >, and the molecule depicted in figure 1 may be represented by a
three-level system with energies M2 for |b >, hwo for | a >, and zero for | g >.
The basic difference between the molecular system considered here and the atomic system
considered by previous investigators (refs. 1 through 5) is that £1 is much greater than 003. Hence,
the only possible resonance enhancement effect which may be used is the two-photon resonance
namely, o>o = 2ooj. Because there is no resonance enhancement at 0)3 the system considered here
would have relatively lower conversion efficiency but, as will be seen, much wider bandwidth than
those considered by Harris and Bloom (ref. 6). The simplified expression for the relevant third-order
susceptibility for the three level system depicted in figure 1 can be found by standard methods (ref.
7) to be
- coo + 0093
 — (1)
4 h 3 ( G J W ) + i(5co/2)
where na^ is the electronic dipole matrix element between | a > and I b>, 8o?o is the linewidth of
level I a >, and all frequencies are expressed in circular measure. Since £1 » coj, 0^2. and 003,
the susceptibility of equation (1) is insensitive to o>2 (and 003), and the conversion efficiencies for
third harmonic and sum-frequency generation would be comparable. We therefore consider the
third harmonic generation only. By setting OJ2 = CJ1 and o)3 = 3cjj, the expression needed for
I x^ (3a>i) I2 may be easily obtained from equation (1).
The general expression for the ratio of third harmonic output to pump input, or conversion
efficiency, can be written as (refs. 7 through 9)
4 9 m I
,e = KTJ^N X (3toi)|
I
(2)
where the wave impedance is 77 = -^/n/e, N is the effective molecular concentration (to be explained
later), and the value of a is dependent upon whether the pump beam is unfocussed (plane wave) or
is a focussed Gaussian beam in the medium. For the former case, the cross-sectional areas for both
the fundamental and third harmonic are the same and hence ej = 1(003 )/I(wj), the intensity ratio,
and a = aT = 9cJ12I2(w1) F2L2. The coherence factor is F = sin(AkL/2)/(AkL/2),
82
Ak = 67r(n3 - nj )/Xj, where (n3 - nj) is the refractive index difference for 0)3 and u>j , and L is the
interaction length. For the focussed Gaussian case, the effective cross section of the fundamental is
three times that of the third harmonic because the pertinent induced nonlinear polarization is cubic
in the electric field (refs. 7 through 9). Then the conversion efficiency, written in terms of powers,
e- = P(cj3)/P(cj1) is found using a = a- = (3/4) (eoj/Xj)2 P2 (coj) | F' | 2, where F' is the phase
mismatch integral I(Ak, £, f) given by Ward and New (ref. 9).
In order to evaluate these efficiencies for a given molecular gas, the matrix elements of
individual rotation-vibrational levels between different electric states must be known to calculate
I X^(3wi) I • These transition probabilities have been reported (ref. 10) for the B 2U +- X1^-"*"
band system of molecular hydrogen. A slight complication arises because of the difference in
statistical weight of para (J = 1) and ortho (J = 0) states: at 300°K the ground state population
density of the J = 1 level is 65 percent while that of J = 0 is only 13 percent of the total molecular
density NQ. For this reason, we calculate the para-hydrogen conversion, that is, setting|g>= IX^g"1", 0, 1 >, |a> = |X1Sg+, 1, 1 >,N = 0.65N0. To evaluate the product of the four
matrix elements in equation (1), a sum over all the rotation-vibrational levels in the excited state
must be made. Standard application of the Born-Oppenheimer approximation | e, v, J > = | e, v > | > J,
the rotation sum-rule, and algebraic simplification gives the desired matrix product
M = (2JZ
where, for the case considered here, Ja = 1, v_ = 0, and va = 1. With the reported band oscillator
strengths fv'v" for B Su+ - X 2g of H2 (taole 4 of ref. 10), the linearly polarized dipole matrix
elements of equation (3) may be calculated through the relationship, fv>v" = 2fim | < e", v" | M I e',
v' > | /he , where m is the mass and e the charge of the electron. Using the averaged value of the
electronic state energy, fi = 103710 cm"1, we finally obtain M = 1.14 x 10"117 (C-m)4.
The value of the linewidth, 5o>o to use in equation (1) is complicated to a certain extent by a
pressure dependent quadrupole interaction in hydrogen which produces collision-narrowing (refs.
11 and 12). The full width at 300°K is calculated to be narrowed to less than 16 x 10~3 cm"1 for
pressures between 1-4 atm with a minimum of 12 x 10~3 cm"1 at 2 atm. For simplicity, we will
ignore this factor in calculated efficiency and use the Doppler full width of 36.5 x 10~3 cm"1,
together with an assumed pump frequency of coj = o;o/2 = 2077.6 cm"1 (at resonance) to obtain
IX ( 3 )(3co1) l2= 1.52 x 10"ll5m5V"2.
The conversion efficiencies may now be calculated. Without using special provisions for phase
matching, the maximum interaction length is just the coherence length LC = Xj/6(n3 - nj). Using
the standard equation (ref. 13) to calculate the index difference in hydrogen at 300°K, one obtains
Lc = 240 cm at 1 atm and LC = 60 cm at 4 atm. For the pressure range of 1-4 atm, NLC = 3.83 x
102^m~ , approximately pressure independent. Since the intensity conversion efficiency
ej«:(NLc)2, no enhancement is obtained by increasing the pressure. On the other hand, the power
conversion efficiency for the focussed Gaussian beam ep«xN , and there is a definite advantage in
using higher pressures. Of course, both cases must be more closely examined to include the pressure
narrowing effect, which cannot be done in this brief note. Using F = 2/ir and L = LC, the calculated
ej is 10 percent for I (wj)= 19.9 MW/cm2 and 30 percent for I(cjj)= 34 MW/cm2. For the
focussed case, we choose the confocal parameter b to be three times the interaction length, that is,
(b/L) = 3 and use the maximum | F' |2 which, according to Miles and Harris (ref. 8), is about 0.5.
83
l I l I H I I I I f i l l l i l l l l
The calculated e_ is then 10 percent for P(o>i) = 1.80 MW at 4 atm. Should the pressure narrowing
be included, only 780 kW is necessary to achieve the same efficiency. It should be noted that, for
the pressures and laser pulses of interest here, competing processes, such as two-photon absorption
(ref. 6) and gas breakdown (ref. 14) occur at intensities around 10^ MW/cm^ and therefore should
not present a problem.
It has been pointed out by Harris (ref. 15) that the microscopic processes for third harmonic \
generation with two-photon resonance enhancement and stimulated Stokes-Raman scattering are
similar, and that the calculated third harmonic conversion efficiency may be checked independently
by existing Raman gain data. We have derived the relationship between the intensity (plane wave)
conversion efficiency, ej, with L = Lc, and the normalized stimulated Raman gain coefficient g as
LcI(W l)g (4) I!
where cos = wg - wo is the Stokes frequency stimulated by a laser at frequency tog, and XR(^S) is
the associated Raman tensor. Using the measured gain (ref. 16) for H2 at 300° K and 1 atm, Lc = 240
cm, and the calculated ratio | x /XR I = 1-107, we find from equation (4) the pump in tensity neces-
sary for 10 percent conversion efficiency to be 39.2 MW/cm , a value in favorable agreement with
the value of 19.9 MW/cm^ calculated above.
The question of phase-matching is of interest. If the conversion medium is positively |j
dispersive, like H2> this can be accomplished by adding the appropriate mole fraction of a selected
negatively dispersive diatomic molecule. If heteronuclear molecules are selected as the interation
medium, and the v = 2 level is used for the two-photon level | a >, the medium could have negative
dispersion if the pump frequency cjj is smaller than, but very close to, the v = 1 level. For such a
negatively dispersive conversion medium, phase matching can be achieved by adding positively
dispersive diatomic molecules to the system. However, since neither oscillator strengths nor
refractive indices at infrared frequencies are known for most molecules, phase-matching may have
to be determined by experiment rather than a-priori quantitative calculation. f
k_
In summary, we have suggested a mechanism with which efficient third-harmonic and
sum-frequency generation, pumped by infrared lasers, should be practically achievable. We have
outlined the method for calculating the third-harmonic conversion efficiencies for unfocussed and
focussed Gaussian pump beams. A single relation for estimating the intensity conversion efficiency, ^
when the appropriate Stokes-Raman gain coefficient is known, has been given. Our major reason for
using H2 as our model molecular converter was the availability of the oscillator strengths needed for
the calculation. However, the required resonance pump radiation at coj = 2077.6 cm~* for r
hydrogen molecules, falls between the strong lines of the CO and chemical lasers, the nearest line
being a v' = 2 -* v" = 1, J' = 9 -»• J" = 10 transition in CO at 2077.1 cm~l. Other molecular systems,
which have either Rydberg transitions (ref. 17) or smaller Ji, should yield similar conversion
efficiencies. The purpose of this paper was to demonstrate that indeed efficient conversion in
molecules is possible and to motivate thereby the further study of other common molecules such as
HC1, N2, CO, and CN, which would undoubtedly result in suitable systems for experimentation.
For example, both CO and C^F molecules (ref. 18) would be candidates for up-converting CO^
laser radiation. Other possibilities, rather than such chance laser-molecule wavelength matches, are •»••-
available, however. With the recent advance in the development of high pressure infrared lasers, it
should be possible to tune the laser to achieve resonance. It must be noted, however, that if the
84
\
laser linewidth is larger than 5coo, the required power to achieve a given conversion efficiency is
proportionately increased. Alternately, the molecule can be brought to resonance with a nearby
laser line by Stark tuning with an external electric field (ref. 18).
Acknowledgements - Stimulating discussions with Professor S. E. Harris, who suggested the
use of virtual electronic transitions, and with Professor Y. R. Shen are gratefully acknowledged. f;
Discussions with C. Chackerian, J. O. Arnold, S. Bowen, and R. L. McKenzie have also been very c
helpful.
REFERENCES
1. Young, J. F.; Bjoiklund, G. C.; King, A. H.; Miles, R. B.; and Harris, S. E.: Third-Harmonic Generation in
Phase-Matched Rb Vapor. Phys. Rev. Lett. Vol. 27, no. 23, 6 Dec. 1971, p. 1551-1553.
2. Harris, S. E.; and Miles, R. B.: Proposed Third-Harmonic Generation in Phase Matched Metal Vapors. Appl.
Phys. Lett., Vol. 19, no. 10,15 Nov. 1971, p. 385-387.
3. Kung, A. H.; Young, J. F.; Bjorklund, G. C.; and Harris S. E.: Generation of Vacuum Ultraviolet Radiation in
Phase-Matched Cd Vapor. Phys. Rev. Lett. Vol. 29, no. 15,9 Oct. 1972, p. 985-988.
4. Hodgson, R. T.; Sorokin, P. P.; and Wynne, J. J.: Tunable Coherent Vacuum-Ultraviolet Generation in Atomic
Vapors. Phys. Rev. Lett. Vol. 32, no. 7,18 Feb. 1974, p. 343-346.
5. Bloom, D. M.; Yardley, James T.; Young, J. F.; and Harris, S. E.: Infrared Up-Conversion with Resonantly
Two-Photon Pumped Metal Vapors. Appl. Phys. Lett. Vol. 24, no. 9,1 May 1974, p. 427-428.
6. Harris, S. E.; and Bloom, D. M.: Resonantly Two-Photon Pumped Frequency Converter. Appl. Phys. Lett.,
Vol. 24, no. 15,1 Mar. 1974, p. 229-230.
7. Armstrong, J. A.; Bloembergen, N.; Ducuing, J.; and Pershan, P. S.: Interactions Between Light Waves in a
Nonlinear Dielectric. Phys. Rev., Vol. 127, no. 6,15 Sept. 1962, p. 1918-1939.
8. Miles, Richard B.; and Harris, Stephen E.: Optical Third-Harmonic Generation in Alkali Metal Vapors. IEEE J.
Quant. Elect., Vol. QE-9, no. 4, April 1973, p. 470-484.
9. Ward, J. F.; and New, G. H. C.: Optical Third Harmonic Generation in Gases by a Focused Laser Beam. Phys.
Rev. Vol. 185, no. 1, 5 Sept. 1969, p. 57-72.
10. Allison, A. C.; and Dalgarno, A.: Transition Probabilities for the B1!"1^ -X1Z+ Band System of H2 J.
Quant. Spec. Radiat. Transfer Vol. 9, no. 11, No. 1969, p. 1543-1551.
11. Lallemand, P.; Simova, P.; and Bret, G.: Pressure-Induced Line Shift and Collisional Narrowing in Hydrogen
Gas Determined by Stimulated Raman Emission. Phys. Rev. Lett., Vol. 17, no. 25. 19 Dec. 1966, p.
1239-1241.
12. James, Thomas C.: Calculation of Collision Narrowing of the Quadrupole Lines in Molecular Hydrogen. J. Opt.
Soc. Am., Vol. 59, no. 12, Dec. 1969, p. 1602-1606.
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1 1 1 1 M I I 1 I 1 E E I 1 1 I I I
13. Born Max; and Wolf, Emil: Principles of Optics; Electromagnetic Theory of Propagation, Interference and
Diffraction of Light. Pergamon Press, New York, 1959, p. 93.
14. Rockwood, Stephen D.: A Comparison of Laser Induced Gas Breakdown in Various Gases. IEEE J. Quan.
Elec., Vol. QE-10, no. 9, Sept. 1974, p. 732.
15. Harris, S. E.: private communications.
16. Hagenlocker, E. E.; and Rado, W. G.: Stimulated Brillouin and Raman Scattering in Gases. Appl. Phys. Lett,,
Vol. 7, no. 9, 1 Nov. 1965, p. 236-238.
17. Herzberg, G.: Spectra of Diatomic Molecules, van Nostrand Reinhold New York, 1950, p. 383.
18. Bischel, William K.; Kelly, Patrick J.; and Rhodes, Charles K.: Observation of Doppler-Free Two-Photon
Absorption in the i>3 bands of CH3F. Phys. Rev. Lett. Vol. 34, no. 6,10 Feb. 1975, p. 300-303.
le'.v'.J> T] I _ l«'.v'.J'>
f j i
i'itit
I «".v".J" >
1 -43
"0
o
Figure 1.- Energy levels of a typical diatomic molecule. Each electronic state is split into a series of rotation-
vibrational levels, with short horizontal lines indicating rotation sublevels. One possible path of virtual vibronic
transitions contributing to the product of four matrix elements in equation (1) is shown by the four vertical
dashed lines. For the up-conversion process of interest, the molecule may be represented by a three-level system
with the upper-state |b > forming a band containing all sub-levels in the excited electronic state. For the inter-
action considered (using the B1Zu+-X12g+ band system of H2), |e', v', J'>= IB1^"1", v', J'>,
|a> = |e", va, Ja> = IX1 Sg+, 1,1> and jg> = |e", vg, Jg> = (X1 £g+, 0, 1>.
86
• I I I I I I I I I I I I I I I I I I
DISCUSSION
Ernest Brock, Los Alamos Scientific Laboratory - With respect to the dispersion in the infrared in the hydrogen
molecule, there has been a recent report in JETP Letters, September-October, where some Soviet scientists use a
Raman pumping technique to pump hydrogen from v = 1 to v = 2, ground state. They observed a fringe shift which ['
they were able to interpret as an index change. Their experimental data is supported by a German calculation, which "
they reference.
Joe She: Thank you for calling it to our attention.
Dick Miles, Princeton University — You might consider focusing as a means of phase matching.
Joe She: Yes, for other molecules this may work, but you need an actively dispersive medium which is not the case
for hydrogen. F
\
I
87
I 1 I 1 I I I I I 1 1 1 1 I L 1 I I I


IR pumped third-harmonic generation and sum-frequency generation in diatomic molecules

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