SATURATION OF THE GAIN AND INDEX OF REFRACTION IN HF CW CHEMICAL LASERS

(1)

HAL Id: jpa-00220560

https://hal.archives-ouvertes.fr/jpa-00220560

Submitted on 1 Jan 1980

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

SATURATION OF THE GAIN AND INDEX OF

REFRACTION IN HF CW CHEMICAL LASERS

D. Bullock

To cite this version:

D. Bullock.

SATURATION OF THE GAIN AND INDEX OF REFRACTION IN HF

(2)

JOURNAL DE PHYSIQUE Colloque C9, supplément au n°ll, Tome 41, novembre 1980, page C9-37

Résumé.- Un simple modèle de laser chimique de type HF CW, comprenant trois niveaux oscillatoires différents, a été récemment construit. Les équations relatives au débit ont été résolues grâce à des transitions à double branche active P, qui relient les trois niveaux. Ces équations décrivent le développement des densités du groupement comme fonction de l'état des quanta oscillatoires et rotatifs, ainsi que de leur composante de vélocité le long de l'axe optique du résonateur laser. Un résonateur optique géométrique Fabry-Pérot est employé ici. En examinant le gain comme fonction de la fréquence, l'on peut rapidement noter dans le modèle la présence de brûlure de cavité, et d'aug-mentation de cascade. De plus, il y a lieu de souligner le phénomène de dispersion locale anomale, laquelle est due à l'absorption de résonance par les modes de résonateur longitudinaux.

La saturation du gain et de l'indice de réfraction est définie comme fonction de pression, de type de résonateur (onde stationnaire ou mobile), et de longueur de résonateur.

S A T U R A T I O N OF T H E GAIN A N D INDEX OF R E F R A C T I O N IN HF CW C H E M I C A L L A S E R S *

D.L. Bullock

TRW-DSSG, Eebondo Beach, U.S.A.

Abstract.- A simple three vibrational level model of an HF CW chemical laser has been constructed. Rate equations, which govern the vibrational and rotational state number densities as a function of their velocity component along the optical axis of the laser resonator, are solved in the presence of lasing on two P-branch transitions connecting the three levels. A Fabry-Perot geometrical optics resonator model is used. The presence of hole burning and cascade enhancement is readily observed in the model through examination of the gain as a function of frequency, as well as the phenomenon of local anomalous dispersion due to resonance absorption by the longitudinal cavity modes. The satura-tion of the gain and index of refracsatura-tion is reported as a funcsatura-tion of pressure, resonator type (stan-ding or traveling wave), and resonator length.

The operating pressure regime for HF CW chem- Rate equations which describe the develop-ical lasers is low enough (1-10 torr) so that inho- ment of the population densities, Nv > J( v ) , where v

mogeneous line broadening mechanisms are important is the molecular velocity component along the opti-factors in the determination of the gain saturation.1 cal axis of a Fabry-Perot (FP) resonator, are solved

In addition, these same mechanisms lead to index of in the presence of lasing on the two transitions. refraction effects which depend in a complicated way

on the degree of gain saturation. Knowledge of the gain and index of refraction saturation phenomena is important for determining their contributions to the optical extraction efficiency and optical quality of the gain medium. To investigate these effects, a simple three vibrational level model of an HF CW chemical laser has been constructed. Figure 1 shows the level diagram for this model. The levels model-ed are (V,0) = (2,J'-1), (1,0'), (O.J'+l), which are connected by the optical transitions Pp(J') and P,(J'+1). Here V is the vibrational quantum number and J is the rotational quantum number. The value for J' may be chosen from the range IsJ'si5.

Sponsor: Defense Advanced Research Projects Agency. Monitor: Air Force Weapons Laboratory.

FIGURE 1 LEVEL DIAGRAM FOR THREE VIBRATIONAL LEVEL MODEL

1H . Mirels, AIAA Journal 12, 478 (1979)

(3)

C9-38 JOURNAL D E P H Y S I Q U E

The FP resonator model may be of e i t h e r t h e t r a v e l - i n g wave ( r i n g resonator) o r standing wave ( l i n e a r resonator) type, and t h e spacing o f t h e l o n g i t u d i n a l modes ( f r e e s p e c t r a l range) i s determined by t h e 1 ength and t y p e o f resonator. The model o f t h e reso- n a t o r assumes gain equals l o s s f o r each l o n g i t u d i n a l mode amplitude. The l o s s i s determined by t h e r e - f l e c t i v i t y o f one o f t h e m i r r o r s o f t h e FPresonator.

The N V , J ( ~ ) a r e s u b s t i t u t e d i n t o i n t e g r a l s which determine t h e g a i n and index o f r e f r a c t i o n as a f u n c t i o n o f frequency. The presence o f h o l e burning and cascade enhancement i s r e a d i l y observed i n the model through examination o f t h e g a i n as a f u n c t i o n o f frequency. The phenomenon o f l o c a l anomalous d i s p e r s i o n i n the r e g i o n o f t h e l o n g i t u d i n a l mode a b s o r p t i o n resynances i s observed i n t h e index o f r e f r a c t i o n d i s t r i b u t i o n when displayed as a func- t i o n o f frequency. A s o l u t i o n o f s e l f - c o n s i s t e n t NV,j(v) and l o n g i t u d i n a l mode i n t e n s i t i e s i s d e t e r - mined by assuming i n i t i a l l y uniform mode i n t e n s i t i e s

f o r a l l l o n g i t u d i n a l modes w i t h i n f 1.5 Doppler widths o f t h e center frequency, and i t e r a t i n g t o a converged s o l u t i o n . The i t e r a t i o n process employs s o l u t i o n f o r t h e N V , J ( ~ ) , d e t e r m i n a t i o n o f t h e gain, a p p l i c a t i o n o f t h e g a i n and l o s s o f t h e r e - sonator t o t h e mode i n t e n s i t i e s , and s o l u t i o n again f o r t h e N V , J ( ~ ) . This process i s repeated u n t i l t h e g a i n d i s t r i b u t i o n and i n t e n s i t y d i s t r i b u t i o n s cease t o change.

The s a t u r a t i o n o f t h e g a i n and index o f r e - f r a c t i o n i s e x h i b i t e d i n a s e r i e s o f f i g u r e s as a f u n c t i o n o f pressure and resonator l e n g t h and type. These f i g u r e s show s o l u t i o n s t o t h e model f o r var- i o u s c o n d i t i o n s . F i g u r e 2 shows t y p i c a l g a i n d i s - t r i b u t i o n s f o r HF as a f u n c t i o n o f frequency i n t h e absence o f l a s i n g and a t a pressure o f 5 t o r r . The f l o w v e l o c i t y o f t h i s supersonic f l o w g a i n i s

5

1.6 X 10 cm/sec. The two t r a n s i t i o n s chosen a r e

-

d ! 1

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

NORflRLIZED fREOUENCY IN UNITS OF DOPPLER WIOTH

FIGURE 2

HF SMALL SIGNAL GAIN ON ~ ~ ( 6 ) AND P , ( 7 ) FOR PRESSURE OF 5 TORR, TEMPERATURE OF 4 0 0 ° ~ , AND FLOW VELOCITY OF 1.6 X 10' cm/sec.

'?

7

-1.5 -1.0 -0.5 0.0 0.5

T

1.0 1.5

NORMRLIZED FREOUENCY IN UNITS OF DOPPLER WIDTH

FIGURE 3

(4)

P2(6) and

P1(7).

The peak gain f o r ~ ~ ( 6 )

i s

%0.05/cm and f o r p1

( 7 )

i s %0.02/cm.

The t h r e s h o l d

gain f o r l a s i n g i s shown i n t h e f i g u r e f o r r e f e r e n c e

f o r a gain medium of length 152.4 cm i n a standing

wave resonator

w i t h

two m i r r o r s having 1.0 and 0.33

r e f l e c t i v i t y , r e s p e c t i v e l y . Figure 3 shows t h e in-

dex of r e f r a c t i o n a s a function of frequency f o r

t h e same c a s e .

Figure

4

shows t h e model s o l u t i o n s f o r t h e

t h r e e population d e n s i t i e s , i . e . , N2,5(v),

N~

, 6 ( ~ ) ,

and N 0 , 7 ( ~ ) , f o r t h e s i m p l e s t p o s s i b l e l a s i n g c a s e .

0,

I

5 -1.0 ' -0.5 I 0.0 0.5 1.0 1.5

NORMALIZE0 VELOCl TY IN UNITS OF THERMAL VELOCITY

FIGURE 4

HF SATURATED NUMBER DENSITIES FOR TRAVELING WAVE CONDITIONS LASING ON Pz(6) AND P l ( 7 ) . THE CAVITY LENGTH I S 1 . 0 0 METER AND THE MIR- ROR REFLECTIVITY I S 0.33. THE GAIN LENGTH I S 1

.524

METER AND THE GAIN MEDIUM CONDITIONS ARE THOSE OF FIGURE

2.

(THE INCONSISTENCY BE- TWEEN THE CAVITY AND GAIN LENGTHS I S IMMATE- RIAL, AS THE PURPOSE OF THIS CASE I S ONLY TO ILLUSTRATE THE EFFECT OF A SINGLE LONGITUDI- NAL MODE. )

d

/

1 +

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

NORMRLIZEO FREQUENCY IN UNITS OF OOPPLER WIDTH

FIGURE

5

HF LONGITUDINAL MODE INTENSITIES ON P z ( 6 ) AND P1(7) FOR CASE

OF

FIGURE 4

This c a s e ~ i s

t h a t of a i r a v e l i n g wave resonator

with a f r e e s p e c t r a l range which 5s of t h e o r d e r of

t h e Doppler width. Thus, on1

y

one longitudinal

mode i s a c t i v e f o r each t r a n s i t i o n . The t i c k marks

I

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

NORMALIZED FREOUENCY IN UNITS OF OOPPLER LlIDTH

FIGURE 6

(5)

C9-40 JOURNAL DE PHYSIQUE

i n t h e c e n t e r o f the f i g u r e i n d i c a t e t h e v e l o c i t i e s f o r molecules which a r e resonant w i t h the l o n g i t u - d i n a l modes. The s o l i d t i c k marks a r e f o r P2(6) and t h e d o t t e d t i c k marks f o r PI ( 7 ) . Local v a r i a - t i o n s i n t h e p o p u l a t i o n d e n s i t i e s a r e seen t o occur i n t h e neighborhoods o f t h e l o n g i t u d i n a l mode resonances. Figure 5 shows the l o n g i t u d i n a l mode i n t e n s i t y d i s t r i b u t i o n s f o r t h i s case. As mention- ed, o n l y one l o n g i t u d i n a l mode per t r a n s i t i o n i s a c t i v e . Figures 6 and 7 show t h e r e s u l t a n t g a i n and index o f r e f r a c t i o n d i s t r i b u t i o n s . Hole burn-

i n g t o t h r e s h o l d f o r t h e a c t i v e l o n g i t u d i n a l resonances i s seen i n Figure 6, w h i l e a l o c a l anomalous d i s p e r s i o n e f f e c t due t o t h e l o n g i t u d i n a l resonances i s seen i n Figure 7.

F i g u r e 8 shows t h e same case as F i g u r e 6, b u t w i t h t h e pressure reduced from 5.0 t o r r t o 2.5 t o r r .

V

NORMRLIZED FREOUENCY I N UNITS OF DOPPLER WIDTH FIGURE 7

HF SATURATED INDEX OF REFRACTION ON P,(6) AND P1C7) FOR CASE OF FIGURE 4

I

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 NORMRLIZED FREOUENCY I N UNITS OF DOPPLER WIDTH

FIGURE 8

HF SATURATED GAIN ON P2(.6] AND P1(7) FOR CASE OF FIGURE 4, EXCEPT THAT THE PRESSURE I S EQUAL TO 2.5 TORR

?

f

s,

I

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 I

NORMALIZED FREOUENCY I N UNITS O f DOPPLER WIDTH FIGURE 9

HF SATURATED INDEX OF REFRACTION ON P2C6)

(6)

The v a r i a t i o n s

i n

t h e neighborhood o f t h e l o n g i t u d i - n a l resonances a r e seen t o be more " l o c a l i z e d " as a r e s u l t o f t h e reduced pressure broadening. The associated i n t e n s i t y d i s t r i b u t i o n s ( n o t shown) as a r e s u l t o f t h e reduced pressure have dropped t o a v a l u e o f about 1/2 o f those o f F i g u r e 5. This i n - d i c a t e s t h a t , a t low enough pressures, pressure a f f e c t s t h e e f f i c i e n c y o f power e x t r a c t i o n from t h e g a i n medium. Figure 9 shows the sharper anomalous d i s p e r s i o n e f f e c t as a r e s u l t o f t h e lower pressure.

Figure 10 shows t h e same case as Figure 6, except t h a t t h e resonator has been lengthened t o 406.4 cm and converted t o a standing wave type. A double s e t o f t i c k marks denotes t h e resonances

f o r waves which now t r a v e l i n both d i r e c t i o n s i n t h e resonator. Since t h e r e s o n a t o r l o s s occurs i n t h e middle o f t h e round t r i p pass, t h e s a t u r a t e d g a i n i s s l i g h t l y below t h r e s h o l d f o r t h e pass t o t h e p o s i t i o n o f l o s s , and s l i g h t l y above t h r e s h o l d f o r t h e pass back, as i n d i c a t e d i n t h e f i g u r e . The n e t round t r i p g a i n i s , however, a t t h r e s h o l d . The associated i n t e n s i t i e s ( n o t shown) f o r t h e t r a n s i - t i o n s have increased by a f a c t o r o f approximately f o u r r e l a t i v e t o those o f Figure 5. T h i s i s t h e r e s u l t o f t h e increased g a i n e x t r a c t i o n e f f i c i e n c y provided by a d d i t i o n a l l o n g i t u d i n a l modes, showing t h a t resonator l e n g t h plays a r o l e i n e x t r a c t i o n e f - f i c i e n c y . F i g u r e 11 shows t h e corresponding index o f r e f r a c t i o n . The increased s a t u r a t i o n o f t h e i n - dex o f r e f r a c t i o n r e s u l t i n g from t h e increased e f - f i c i e n c y i s e v i d e n t by t h e r e d u c t i o n o f t h e s c a l e o f F i g u r e 11 r e l a t i v e t o Figure 7 by more than a fac- t o r o f three.

- _{Figures 12 and 13 show t h e e f f e c t s o f i n -} c r e a s i n g t h e g a i n and resonator lengths, as w e l l as decreasing t h e m i r r o r r e f l e c t i v i t y . More u n i f o r m s a t u r a t i o n o f both t h e g a i n and index o f r e f r a c t i o n i s evident. Figures 14 and 15 show t h e same case

-1.5 -1.0 -C.S 6 0 0.5 1.0 1.5

NORtlALIZEO FREOUENCY I N UNITS OF DOPPLER UIDTH

FIGURE 1 0

HF SATURATED GAIN ON P2(6) AND P1(7) FOR CASE OF FIGURE 4, EXCEPT THAT I T I S FOR A STANDING WAVE CONDITION AND A CAVITY LENGTH OF 4.064 METERS

hi I

'-1.5 -1.0 -0.5 0.0 0.5 1.0 _1.5I

NORnfitIZEO FREQUENCY I N UNITS OP DOPPLER W ~ T H FIGURE 1 1

(7)

C9-42 JOURNAL DE PHYSIQUE

-1.5 - 1 . 0 -0.5 d. o 0:s i. o 1'. 5

NORMALIZE0 FREOUENCY IN UNITS OF DOPPLER WIOTH

FIGURE 1 2

HF SATURATED GAIN ON ~ 2 ( 6 ) AND P1(7) FOR CASE OF FIGURE 10, EXCEPT THAT

THE CAVITY LENGTH I S 1 4 . 0 0 METERS

1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

NORMALIZED fRCI7UENCY IN UNITS OF DOPPLER WIDTH

FIGURE 13

H F SATURATED INDEX OF REFRACTION ON P Z ( 6 ) AND P i ( 7 ) FOR CASE OF FIGURE 1 2

r4

D

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

NORMRLIZED FREOUENCY IN UNITS OF DOPPLER WIDTH

FIGURE 1 4

HF SATURATED GAIN ON P,(6) AND PI ( 7 ) FOR CASE OF FIGURE 12, EXCEPT THAT THE PRESSURE I S EQUAL TO 2.50 TORR

I

-1.5 -1:o -0:s ' 0.0 0 . s I; 0 1:s

NORMALIZED FREOUENCY IN UNITS OF DOPPLER WIOTH

FIGURE 1 5

(8)

as Figures 12 and 13, but with pressure reduced from 5.00 t o r r t o 2.50 t o r r . Structure due t o the decreased pressure broadening i s evfdent.

The saturation of t h e gain and index of re-

fraction in HF

CW

chemical l a s e r s has been shown t o

be a function of pressure, resonator length and type, gain medium length, and resonator mirror re-

f l e c t i v i t y . The conclusions a r e supported by