Progress Report

to the

National Aeronautics and Space Administration (NASA)

Greenbelt, MD 20771

Theoretical Studies of Resonance Enhanced Stimulated Raman Scattering

(RESRS) of Frequency Doubled Alexandrite Laser Wavelengths in Cesium Vapor

NAG 5-526

from

Nabil M. Lawandy

Division of Engineering

Brown University

Providence, RI 02912

Report prepared by:

Nabil M. Lawandy

Associate Professor of Engineering

and Physics

Principal Investigator

Carl Cometta

Executive Officer

Division of Engineering

7 October 1986

(NASA-CB-179768) TEEGBETICAL STUDIES OF

BESONANCE ENHANCED STIHULATED .EAHAN

SCATTEHING (RESES) OF FREQUENCY DOUBLED

AIEXANDEITE LASEE WAVELENGTHS IK CESIUM

VAPOS Progress Eeport (Bro«n Uuiv.), ,; 16- p

N8 6-3 2717

Unclas

G3/36 4*1636
Progress Report/NASA -1- NAG 5-526

Introduction

This part of our work focused on understanding the effects of arbitrary transverse and

longitudinal relaxation rates on the susceptibilities of coherently driven three-level systems.

The approximation of a single relaxation rate often made in previous work is strongly invalidated by the variation in the spontaneous emission lifetime between various atomic level pairs

in systems such as cesium.

It is of great importance to the problem of nonlinear infrared generation to determine the

dependence both the real and imaginary susceptibility on relaxation rates. The imaginary susceptibility on the pump transition determines the absorption of pump photons and the imaginary susceptibility on the laser transition determines the spectral dependence of the gain.

This is of particular importance for pure Raman emission (i.e. absorption at linecenter of the

gain transition) as it determines the tunability characteristics we are aiming to predict. The

real susceptibility is important when cavities are used at the signal field as this will determine

the loaded resonance of the Raman oscillator. We will show that in some cases which result

from having different relaxation rates that mode splitting may result allowing more than one

frequency to have the same Raman wavelength, possibly resulting in a temporal instability.
Progress Report/NASA -2- NAG 5-526

Study of the Relaxation Rates in Coherently Driven Systems

The work on elucidating the rate dependence of the susceptibility is based on new solutions for the three-level density matrix driven by two classical fields of arbitrary strength.

The density matrix formalism we use is fully equivalent to the Schrodinger wave function but is more applicable to the statistical case, in which the wave function is not known

exactly. In this case the elements of the density matrix are taken to be ensemble averages of

possible configurations of the system. As a result every wave function can be expressed as a

unique density matrix, but not all density matrices can be expressed as wave functions,

reflecting the fact that the wave function contains only quantum uncertainties where as the

density matrix contains statistical as well as quantum uncertainties.

For a given wave function

|T> = £Cn|n> (1)

the density matrix is written as

Pij = crq (2)

where the bar denotes the ensemble average. The time dependence of the elements is given

by

Pij = •jkp.Hlij (3)

where H is the total Hamiltonian of the system used to describe the three level atoms.

The Hamiltonian in the dipole approximation is given by:

H = H0 - $3 (4)

where H0 is the unperturbed atomic Hamiltonian and -ft-li is the atom field Hamiltonian in the

dipole approximation. We will assume that the dipole matrix element u.12 is zero correspond-
Progress Report/NASA -3- NAG 5-526

ing to levels |1> and |2> being S-states. The pump transition connects the levels |1> and |2>

and the Raman emission is near the frequency of the |2>— »13> transition.

The transverse and parallel relaxation times per molecule level pair are given by i and T,

respectively. The levels are assumed to have equilibrium occupation ratios fy (ratio of 1th to j^

equilibrium populations). The energy level diagram with the associated rates are shown in

Figure 1.

The time evolution of the density matrix elements for the three level system are therefore

described by

Pll = -(P-3lPl3 ~ M-13P3l) ) - --telPll - P33)--(f2lPll ~ P22> (5)

n 12 13

P22 = -^(V-22P23 ~ U23P32)'£(t) ~ ~ tf32p22 ~ P33> ~ ~^-(P2Z ~

 f2lPll) (6)

P33 = 3lPl3 - ^13P3l) + (feP32 ~ M-32P23)(t) ~ ~(Pv ~ f32p22) - ~(P33 ~ ^SlPll) (7)

Pl3 = ^ [(P33 - Pll)Hl3 - ^23Pl2j'S(t) + i(0>31 + i^^PlS (8)

P32 = (P33-p22)^32-^3lPl2-(t)-i(C032-it:r)P32 (9)

Pl2 = ^13P32 - ^32Pl3-(t) + »(G>21 + ^3P\ 2 (10)

where (0^ = (E; - Ej)/h, E(t) = Epcos(copt + \j/p) + Escos(cost + \|/s) and p31, p23 and p12 are complex

conjugates of pi3, p32 and p12 respectively.

Steady State Solutions

To obtain the steady state solutions to the density matrix equation we first take out the

explicit oscillations of the off diagonal elements
Progress Report/NASA -4- NAG 5-526

r> -j ,,-,. De^ (11)

and then neglect all terms which oscillate faster than cop or cos in the off diagonal element

equations and all non DC terms in the diagonal element equations. This is the rotating wave

approximation which therefore disallows the possibility of Bloch-Seigert shifts in the resonances. If we then set all time derivatives equal to zero and define the following:

(12a,

(ilb)

Ls = ws - co32 + I/T! = 8S + i/Tj (13a)

Lp = cop - co31 - i/t2 = 5p - Ui2 (13b)

Lsp = cos - cop + co21 + i/x3 = 5S - 5p -I- i/T3 (13c)

and A32 = p33 - P22

 Ai3 = Pn ~ Pss *e

 equations become

LsX=ps*A32-p;D (14a)

LpA = -ppA13-psD (14b)

LspD = -ppX + p*A (14c)

and

0 = 2ImpsX - 4Impp*A + k^A^ - A,°3) - k2(A32 - A3°2) (15)

0 = 2Impp*A - 4ImpsX + k4(A32 - A3°2) - k3(AI3 - A,0

3) (16)

The constants kj, k2, k3, k4, A^ and A32 are defined in terms of the T and fjj by the following

relations:
Progress Report/NASA -5- NAG 5-526

.. _

k

 -

4f31

3 T,

.

~

2f32 2-2f31 2 + f21

T3

_ 1 2 - 2f32

K

3 - "T ~ 3

T

2

1 + 2f31 1 + 2f21

~

T

2

k =

2 + f21

T3

f21 + f31

We can now eliminate D from the first three equations and solve for A. and A

A

i3 ( A32

Lp Ls

An ! A32

[ L P L S _

I (17)

I (18)

where

^ _ . ,ly,

L IRI 2 TIRl^-l-TT T s 1 Ps 1 '-'p 1 Hp 1 ^ ^s^p^sp

In terms of two real quantities, RI and R2, we have that:

(5J

1

 ' 2

+ ^2)(5s2 + Tf2)(A - iB)

2 2 \^W

A = 5s(8p

2

 + T2-

2

)(5 1 + Tf2 - 1 PP 1 2) - 8p(52 + Tf2)(5p

2

 + <c2~

2

 - | (3S | 2) (21)

B = t3

 !(52

 + tf2X52 + T2-

2

) + ||

This then yields

-iPsi 2 Li , i

ImPs/V<

 |LS|

2

 f ' |

3P 1 V(8p2 + 12

2

) + | M V(S.2 + ^f2) , (22)

P I

2

 1 1 R I 2I R I2

P , IPp I IPsi . fT)\

_ OC-i Av> -• _ • ••« OtiAi^ (^-J)

Ls|

2

 'J |Ls|

2

|Lp|

2

and
Progress Report/NASA -6- NAG 5-526

-IPs

i13~ A

«2A32

where

- (5p

2

 - T2-

2

)R2

02 = (8ptf! - S/i^R, + (ifV + 6p6s)R2

a3 = 2Tf15sR1 + (S2 - Tf2)R2

Substituting these into the last two density matrix equations then gives

|Lp|

2

 |LS|

3

A

32

4IPPI

2

~~ I Ps I

k4A3

0

2-k3A1°3 =

2IP P I

2

r

-l

2a2

TLsi1

|2 IT ,2

I ILS I

+

Then by making the following substitutions and definitions

r,=

^ + T2

J

|2 f

a2

«l 2a9

>•+ k-,

(24)

(25a)

(25b)

(25c)

(27)

(28)

(29a)

(29b)

(29c)
Progress Report/NASA -7- NAG 5-526

4IPS

The equations become

2(5p

2

- k2A3°

2 = rtA13 - r2A

8

2

32

with the solutions

A13 =

A32 -

r4(k1A{)

3 - k2A3

°

2) + r2(k4A3

°

2 -

r,r4 - r2r3

i°3 - k3A3°2)

(29d)

(30a)

(30b)

(3 la)

(31b)

The polarization of the system can now be determined from the expectation value of the

dipole operators by:

<P> = Tr(np)

(32)

= 2Re(|i31p13) + 2Re(u32p23)

which, after substituting for p13 and p32 becomes

P = 2Re

2R

2Re „,, * 2hL

fill + J22

L

P

 LS

32 '(cos

t+Vs)

By comparing this equation with

the complex susceptibilities are found to be given by:

hLp

A13- IPS|

2R

A

13

LP

 + Ls

and

(33)

(34)

(35)
Progress Report/NASA -8- NAG 5-526

Xs = hLs*

A

13 A32

F*—"

p ^ 17s

(36)

In their complex forms

XP = Xs = Xs' + i

¥«. -I*1

"!' XpIPsl2

«i

(52 + T2-

2

:

A

32'

(5S

2

„ Itel'

Xs- M3-

YA 'p =P

 h(52

Xs' =

h(8s

2

 + Tf2)

132

I PP I

2

P Y3

+ A,

IPPl

2

a 2

IPs! 2Y2

IPPI

2Y2

(37)

(38)

(39)

(40)

where

Yi = (52 - 2T2-

1

8pR2

Y2 = (5PP5S

(41a)

(41b)

(41c)

These susceptibilities are the first results which allow for completely arbitrary relaxation

and dephasing rates as well as arbitrarily strong pump and signal fields.

The systematic study of the relaxation rate dependence was performed by setting two of

the pairs of rates (longitudinal, T and transverse T) equal and allowing the other pair of T; and

T; to have a different ratio. This amounts to eliminating the unity ratio of transverse to longitudinal rates between one pair of levels at one time. The results were astonishing and indicate

that the. single ratio limit is a pathological case of no great value for most atomic systems.

The next section will give results due to the variation of the ratio of longitudinal to

dephasing times. The curves will be labeled in terms of 8P and 5S; the pump and signal field
Progress Report/NASA -9- NAG 5-526

detunings in terms of T"1 units, and (5P and ps the pump and signal field strengths in terms of

saturation field values. For example, a P? = 1 indicates a value of pump intensity equal to the

saturation intensity based on a two-level model.
Progress Report/NASA -10-

Pump Susceptibilities %p'' %P'

NAG 5-526

8s = -5 pp = 2 ps = 2

0.01 -

-20 -15 -10 -5 0 5 10 15 20

detuning (units oft)

r i i i i i i i i i i i i i I i i i i I i

-20 -15 -10 -5 10 15 20

detuning (units of

-0.10
Progress Report/NASA -11- NAG 5-526

Pump Susceptibilities

5s = -5 = 2

CO

*=l, V.

^

: CL

X

-20 -15 -10 -5 10 15 20

detuning (units ofT)

o

W

. IX

X

0.10 -

0.05 -

0.00 ~

-0.05

-0.10^

I I I I I I I I I I I I

-20 -15 -10 -5 0 10 15 20

detuning (units of
Progress Report/NASA -12- NAG 5-526

Signal Field Susceptibilities (3 -> 2 transition): %s", xs';

o

W

N

=1,

^

X

0.08

0.06

0.04

0.02

0.00

I ' ' ' i I i i i i i i i i i, i i i i i i i i i i i i i i i i i i i i i i i i i [

- 2 0 -1 5 -1 0 - 5 0 5 1 0

detuning (units of T)

15 20

It is evident here that the case of equal rates Tj = t would predict pure Raman emission while

that of T! = lOt results in a sizable single photon gain contribution.

-0.06 -

-20 - 1 5 -1 0 -5 0 5

detuning (units of T)
Progress Report/NASA -13- NAG 5-526

Signal Susceptibilities

8p = 0 (linecenter pumping), pp = 2, ps = 0

0.12

_[ I I I I | I I I I [ I I I I | I I 1^1 | ly^l I I | I I I I | I I I I | I I I 1

T=T

-20 -15 -10 -50 5 10 15 20

detuning (units of T)

o

w

X

0.10

0.05 -

0.00 --

-0.05 -

-0.10

1 l I I I I I l I I I I

I i iVi i I i i i i I i i i i I i i i i I i i i i

-20 -15 -10 -5 0 5 10 15 20

detuning (units of T)
Progress Report/NASA -14- NAG 5-526

Mode Splitting and Relaxation Rates

It has been realized since the discovery of pulsations in inhomogeneously broadened Xe

lasers that complicated lineshapes (hole burning in the case of the Xe laser) could lead to conditions where a single cavity mode could be consistent with several frequencies. This is stating the fact that owing to complicated frequency dependent dispersion, several frequencies can

have the same wavelength. In our study of the signal susceptibility dependence on the T/T

ratio we discovered that the possibility of mode splitting due to nonunity ratios exists and is

likely to result in transient pulsations in the Cs system if a cavity were employed. As an

example we give the graphical solution of the Fabry-Perot Cavity resonance equation

k0 1 + y%s'(co) L = mre

for two relaxation rate ratios for the case of strong off-resonance pumping of a three-level

system (5P = -5, ps = 0, pp = -2):

0.04E

0° 0.03

*k 0.02

^ 0-01

0.00

-0.01

-0.02

-0.03

-0.04

x

-20 -15 -10 -5 0 5 10 15 20

detuning (units of T)
Progress Report/NASA -15- NAG 5-526

The intersection of the straight line with the x'(Ss) curve locates the frequencies where a cavity tuned to the center of the 1 3>- 1 2> transition would initially begin to oscillate. The curve

shows that these frequencies (labeled co, col5 0*2 and co3) can satisfy the oscillation condition

when T2 = 10T while only one can oscillate when T = T.

Conclusions

This phase of our work has clearly shown that all previous models for Raman emission

in coherently driven three-level systems are in most cases invalid for (1) predicting spectral

signatures and (2) predicting Raman oscillation frequencies. The various cases in the text

show that the ratio of the transverse (dephasing) to longitudinal (population) rates on any pair

of levels in the three-level system can strongly affect the susceptibilities. These situations are

in fact the rule rather than the exception when atomic species such as Cs, K or Na are concerned.

.In addition we have shown that when strong off resonant pumping such as would be

required for tunable IR generation, a larger dephasing rate results in strong modulation of the

real susceptibility. Using a graphical solution for the threshold resonance of a Fabry-Perot,

we have shown that such a Raman oscillator could begin oscillating on several frequencies.

This would most likely lead to temporally varying output even for smooth injected pulses.

The work we are currently pursuing involves the inclusion of the doublet effects present

in the Cs upper state (|3>) on the tf', and %' calculations. We will shortly produce predictions

of the time averaged IR output as a function of the power and frequency of the pump for

comparison of the experiments at Goddard Space Flight Center.