NBSIR 73- 312

NOTES ON INFRARED ABSORPTION EXPERIMENTS

IN A METHANE MOLECULAR BEAM

P. Kartaschoff

Stephen Jarvis, Jr.

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

May 1973

NBSIR 73-312

NOTES ON INFRARED ABSORPTION EXPERIMENTS

IN A METHANE MOLECULAR BEAM

P. Kartaschoff

Stephen Jarvis, Jr.

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

May 1973

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS. Richard W Roberts. Director
1
Contents

Page

1. INTRODUCTION 2

2. ABSORPTION IN SINGLE STANDING WAVE

OPTICAL BEAM 4

3. DISCUSSION OF MULTIPLE OPTICAL BEAM

EXCITATION 9

4. REFERENCES 15

APPENDIX A 16

APPENDIX B 22

iii
LIST OF FIGURES

Page

Figure 1. Geometry of molecular beam-optical wave

interaction 3

Figure 2. Phase p of individual trajectory 5

Figure 3. Approximate transition probability for individual

trajectory with Doppler shift iC^ 8

Figure 4. Schematic of three-field separated cavity

system 12

iv
NOTES ON INFRARED ABSORPTION EXPERIMENTS

IN A METHANE MOLECULAR BEAMP. Kartaschoff t Laboratoire Suisse de Recherch.es Horlogeres

Neuchatel, Switzerland

and

Stephen Jarvis, Jr.

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado USA

The problem of calculating the transition probability

of methane molecules in a molecular beam interacting with

an infrared (3. 39^) radiation beam is discussed. Contrary

to the usual microwave molecular beam experiments, firstorder Doppler frequency shifts cannot be neglected. This

makes the solution of the wave-equations more difficult.

Weak field approximations to the transition probability have

been calculated. Single optical beam experiments analogous

to the Rabi-type interaction result in a Doppler-broadened

absorption line with an estimated half-power width of a few

MHz. For separated multiple field experiments analogous

to the Ramsey-type interaction, no observable response is

predicted, the expected sharp resonance pattern being

smeared out by the random Doppler shifts due to the spread

of the molecular beam trajectories. Further investigations

are required in order to predict the resonance line shapes

for strong fields, i. e. , saturated absorption.

Key Words: Frequency standard; methane resonance;

moleciilar beam; Ramsey resonance;

saturated absorption; stabilized laser;

transition probability.

Contribution of the National Bureau of Standards, not subject to

copyright.

t P. Kartaschoff was a visiting scientist in the Time and Frequency

Division, Institute for Basic Standards, from October 1970 to

September 1971. He is now at Direction Generale des PTT,

Centre Technique, 3000 Bern 29, Switzerland.
1. INTRODUCTION

In 1970 H. Hellwig suggested a methane molecular beam

absorption experiment for the investigation of the photon recoil effect [1]

which might limit the accuracy of the 3. 39 -/im saturated absorption cell

frequency standard of R. Barger and J. Hall. This note summarizes

some first attempts to gain more understanding of the interactions

between the infrared radiation field and the molecular beam. We shall

see that one of the main problems is the first order Doppler shift which

is not negligible at these short wavelengths. The situation is very

different from that encountered in a microwave molecular beam

apparatus (e.g. , cesium). We first note some initial data and facts:

a) Dimensions of the interaction region

Optical beam diameter i. = 0. 1 cm

Separation for Ramsey interaction L = 5 cm

Molecular Beam Dimensions: Width ~ 1 cm

Height ~ 0.1 cm

Divergence of molecular trajectories < 0. 1 radian.

A compromise between divergence, which can be reduced by coUimation,

and useful flux has to be made. Similarly to the absorption cell, selection of trajectories is also possible by saturated absorption. However,

we then have more problems in understanding the recoil effect.

b) State selection

The source temperature can be situated between 78K and 300 K;

thus the ratio lies between 5 6 and 15 and the initial population of the

upper energy level of the transition can be neglected.

2
c) Popple r shift

We have

V where /3 = C ~ ' ^ being the velocity of the molecule and ^ « 1 the

angle

Figure 1. Geometry of molecular beam-optical wave interaction.

between the molecule trajectory and the wavefront in the plane of the

4 -1

trajectory. For methane, we have v ~ 3X 10 cms and thus with

c = 3X 10'''^ cm s''' i3 = 10^^. An absorption experiment using a traveling wave puts an unfulfillable requirement on the adjustment of

the optical and molecular beams with respect to each other, as

-3

^ = 10 radian would shift the absorption line by its entire width.

The shift can be reduced or compensated by using a standing

wave produced by a Fabry-Perot cavity. We then have an interaction

3
with two equal and opposite running waves. We thus expect no shift

but only a Doppler broadening due to the divergence of the beam,

assuming that we have indeed a pure standing wave. Actual resonators

will not be entirely perfect and this may limit the accuracy of the

experiment.

Z. ABSORPTION IN SINGLE STANDING WAVE OPTICAL BEAM

The following treatment is based on Ramsey's calculations of

the transition probability of a two-level system perturbed by a periodic

field [ 2, Section V. 3 ] . The form of the perturbation is different,

however, because we have to deal with two simultaneously applied

oscillating fields of equal amplitude but slightly different frequencies.

This case has not been treated in the earlier literature on microwave

molecular beam spectroscopy because there it was usually either

possible to neglect first order Doppler shifts [3] - [6] , or if

simultaneous perturbations by two frequencies were considered, one

was assumed to be very much weaker than the other. We therefore

could not see how the older work could be used. If we look at the

geometry as shown in figure 2, we have a perturbation by each of the

two running waves, which has to degenerate into the classical case

of Rabi for ^

= 0, i.e. for a molecular trajectory exactly parallel to

the plane in which the nodes of the field pattern lie.

4
/4

ELECTRICAL FIELD

STRENGTH

AXIS OF

OPTICAL

BEAM

Figure 2. Phase p of individual trajectory.

For the two -level system we have the wave function [2], [7]

lb (t) = C ij) + C ij) ^ P p q q

as a solution of the wave equation

Ml

B t i-h = (H + V) 0

Since we have two equal amplitude waves running in opposite directions,

the perturbation takes the following form

V

pq

= — (e + e )

V

qp

= - Qe + e ;
which, by adding an initial phase angle p (see fig. 2), can also be

written as :

iCOt

V = hbe cos(c<J_t + p)

pq

. V = hbe cos(oo^t + p)

qp D

where iC^ is the Doppler shift (angular frequency)

, - D o o c

With the assumed dimensions and velocity it is easy to see that ^^tDo

varies between the limits :

0 < CO^t < 40 D o ~

where t^ - — . The wave equations to be solved take the following

form; after transformation as in [2]:

d ioot . i — C (t) = CO C (t) + be cos(cOr^t + p)C (t) (la)

dt p P P D q

i-^ C (t) = be~^^^cos(CO t + p)C (t) + CO C (t) (lb)

dt q D ^ p q q

where

. H CO = w

P P

-h CO = W

q q

are the energies of the two levels p and q, so that

W - w = "h CO . q p o The above equations are to be solved with the initial conditions

6
C (0) = 1 C (0) = 0 . p q

(2)

It is easily verified that for OJ^ = 0 the case treated by Ramsey is

obtained and the solution in closed form is obtained for the transition

probability

P = Ic ^1 . p. q q

Unfortunately, a closed form solution in our case has not been

found.

We have obtained the following approximate solutions using

the approach of Appendix A.

The solution depends on the phase angle p, 0 < p < Ztt of the

field where the molecule enters the field region (see fig. 2) and is

to be averaged over (0, 27T).

For very small values of b t^ the solution is: (without higher

order terms)

. 2 2

P

X + 2 ^

" ^

2 ^0

) (

^0 \

(3)

with X = (60^ - CO) and t^ = — . This solution is valid for one

0 0 V

typical trajectory incidence angle and is to be averaged over the

possible values of = ——— ^, which again is not possible in

closed form but easily done on a computer.

It is nevertheless possible to discuss this result:

s in X 2 We have a superposition of terms of the ( ——— ) type. Finite (j^

yields the following picture for a single trajectory :

7
( CO - ) 0 D

(^0W + 0)^ ) D

Figure 3. Approximate transition probability for individual

, trajec tory with Doppler shift u> . 0

A finite spread in ^ and co^ respectively smears out the lateral

wiggles and we obtain a simple Doppler -broadened central peak at

0

with a half width of about 2 (rough estimate) i.e., for

max

0. 1 V

3 X 10

= 3. 3 X 10

00D

max

40

3. 3

6 7 1 X 10 = 1. 2 X 10 s"

= 3.8 MHz with 27^!^^^ = COj^

as an estimate for the observed linewidth. This is very broad indeed, so that we obtain a line-Q of only about

7 13

2 X 10 (y^ = 8. 8 X 10 Hz) .

As a conclusion, saturated absorption has to be attempted also in the

case of the molecular beam experiment. The possibility of a Ramsey

separated field excitation scheme has been discussed too. For the

moment, an attempt to solve the wave equation appears to be a rather

formidable task in the case of separated fields. To compute the

8
discussion of the single beam ("Rabi" -type) case, the following results

may be obtained:

a) Expressions valid for larger values of b t^ have been obtained, they

are asymptotic for b t^ —» , but

1) they are singular at the resonance X = 0

Z) they contain elliptic integrals.

b) At resonance X = 0 , the following expressions have been obtained:

^ = 0

CO t 4b . _D_0_

J^(x) being the Bessel-Function. This is again to be averaged over

the range of •

c) For a particle with zero Doppler shift we have

<|c '(t )|>

q 0

p

= 0, X = 0

The last expression allows, in principle, to estimate the required

power for saturated absorption.

3. DISCUSSION OF MULTIPLE OPTICAL BEAM EXCITATION

At this stage, we can only discuss the weak-field approximations

The result shown in the preceding section (fig. 3) is equivalent to the

superposition of the two independent solutions of the SchrodingerEquation, with one single, Doppler -shifted perturbation applied each

time. In other words, and this can easily be verified in [Z], we have

9
the sum of two shafted Rabi-type resonances, the shifts being + CO^ and

- ^j^) respectively.

It therefore seems reasonable to assume, without proof, that the

same should be true for Ramsey-type resonances produced by two or more separated fields.

We shall see, however, that for this weak-field assumption,

the average result, i.e., averaged over the spread of incidence angles

^ (see fig. 2) vanishes. In an experiment, we can predict that only

the Doppler -broadened (A ~ 3 MHz) Rabi-Pedestal will be observed.

The question if Ramsey-type "interference fringes" could be observed

by going into saturation remains still open.

In a first crude experiment performed in January 1970,

H. Hellwig and P. Kartaschoff have observed a weak saturation peak with

single beam excitation, but we were not able even to estimate the linewidth

The weakness of the signal and the bad signal to noise ratio was

believed to be due to lack of excitation power and mechanical laser

instabilities

.

For sake of completeness, we shall give below the results of

calculations of probability transition for the multiple field (2 and 3

field) cases, with single frequency perturbation.

A. Two-field case (Ramsey)

This is Ramsey's result (2). The perturbation applied is :

V =-hbe^^^ V -Hbe '^^^

(6)

pq qp

and the solution is :

P - <| C (7) P' q q

^ , 2^ .2 aT r XT aT ^ . XT . aTn^

= 4 sm 0 sm — [ cos —— cos —r— - cos 6 sm —— sm —r—

J

10
where

cos 9 = — sin 6 = - ^0 ^ . 2b

a

a =

[

( OJ^ - CO) + 4b ] 2 _L /I...2.1/2 K I ' A Y — —

X = co^ - CO T = —

0 V

For weak fields, i.e., bT «1 and near resonance i.e., X « b this

reduces to :

P « 4b^T^( + I cos XT) (8)

p, q 2 2

for one single frequency perturbation of amplitude b and angular

frequency CO .

If we now assume that the interaction with the two running waves

is analogous to the case treated in Section 2, we obtain for a given

angle of incidence ^ , i.e., for a given angular frequency Doppler shift,

CO^, the following results :

P., = ( + CO T) + p (XT - CO T)] . (9)

COj^ 2 pq D pq D

This probability has to be averaged over all possible angles ^, i.e.,

values of , corresponding to the divergence of the trajectories in

the molecular beam. The simplest, but crude example, is to assume a sharp collimation so that there is an uniform flux between two sharp

cutoff angles + • ^® then have :

11
<p> - D 0

T

D

(10)

From the beam geometry, we estimate the limit value

5i 40 .

CO V

D ^ c

If we do the integration for a close multiple of 277 , e.g.,

co't

D

1 27T

the integral vanishes exactly except for a constant term, and since

Co'^T is at least of that order of magnitude, slightly different values of

CO^T will produce only a very small term varying with X . This

means that due to the spread in angles of incidence, the Ramsey

resonance ("fringe pattern") is smeared out, and we cannot expect to

observe anything, at least for the assumed case of weak excitation.

B. Three-field case

There was some hope in early discussions that the application of

three successive separated oscillating fields might lead to an observable

resonance pattern. It was assumed that a majority of molecules would

be excited under a preferred set of phase relations. Unfortunately, the

Doppler shift was neglected then and these early assumptions are

wrong, at least for the present case of weak field excitations.

The 3 -field excitation geometry is assumed as follows:

MOLECULAR

BEAM

AXIS

STANDING WAVE PATTERNS

OF OPTICAL BEAMS

Figure 4. Schematic of three-field separated cavity system.

12
and the calculations of the transition probability is done using the same

notation and method as in Ramsey [2 ]

. The calculation is lengthy but

not difficult, an outline is given in Appendix B. We obtain the following

result

P = <|C (3r + 2T) r > (11)

p. q q

2 2 aT

= 4 sin 9 sin —— [ cos 9 sin aT sin XT

2 2 ar 2 aT

+ (cos 9 sm - cos ) cos XT

^ . 2 aT . 2 1 .2 + sm — sm 9 - — J

Near resonance and with weak perturbation as before, this reduces to

w b^T^(3 + 4 cos XT + 2 cos 2 XT) . (12)

p> q

This is again periodic in XT and the averaging over ^ or COj.^ respectively leads to the same smearing out of the resonance pattern

as in the 2 -field case.

At exact resonance X = 0 we find for all three cases"~the common

behavior

P = n^ b T b T« 1 p. q

where n is the number of interacting field regions.

* i. e., single-field (Rabi), 2-field, and 3 -field cases.

13
We would like to thank H. Hellwig and G. Kramer for their

valuable contributions through interesting discussions.

14
4. REFERENCES

Kolchenko, A. P. , Rautian, S. G. , Sokolovsky, R. I. Sov. Phys. , JETP, _28, No. 5, May 1969.

Ramsey, N. F. , Molecxolar Beams, Oxford, 1956.

Bloch, F. , Siegert, A. , Phys. Rev. 57, 522, 1940.

Stevenson, A. F. , Phys. Rev. 58. 1061, 1940.

Kolsky, H. G. , et. al. , Phys. Rev. 87, 395, 1952.

Ramsey, N. F. , Phys. Rev. iJOO, 1191, 1955.

Merzbacher, E. , Quantum Mechanics, J. Wiley, 197(

15
Appendix A

It does not appear possible to solve the wave equations (1), (2)

in closed form for arbitrary values of (cc, 00^, b), but some interesting

results can be obtained.

For convenience, we introduce some changes of variable and

dimensionless parameters. We take V to be the axial velocity of the

atom, and take Jl to be the width of the optical beam. Let

V

s = t —

so that s = 0 when the atom enters the perturbing field, and s = 1 when it emerges. Let also:

E = -— {00 - CO - CO) = 2V ' p q ' 2

a dimensionless measured of how near the field angular frequency CO

is to the atomic resonance ;

V

'

a field strength parameter ;

CO •'^ V D -i- G = = k^— = k^e ,

a Doppler parameter, where k is the optical wave number, and v^ is

the particle velocity parallel to the optical beam. Then putting

. ^ - l(CO + CO - CO) t C(t). . iCOt 2 p q = e e ^ ^ d(s)

P

16
— l( CO + 60 - 60) t C (t) = e'^ P ^ c(s) , q

we have the equations :

d = iEd + iF cos (Gs + p)c

c = -iEc + iF cos (Gs + p)d

subject to initial data

d(0) = 1 c(0) = 0 . We wish to find the transition probability

Ic^Dl = lc^'(-f )l .

Clearly the functions (c, d) are regular functions of the parameters E

and F.

Weak field approximation

The solutions can be expanded in power series in the field

strength parameter F :

d(s) = ^v^s)

k = 0 ^

OO c(s)= ^ F^'^'^cJ-)

where

k = 0 ^

( s ) = e

and the recursion equations are

17
3

( s) = i f dxe ^^^^ cos(Gx f p) d, (x) k J k

0

^ ^{s) - i dxe^"^^^ cos(Gx + p)Cj^(x)

0

The weakest field-dependent term is

f ^i[(E + G)s + p] g-i(Es - p)

^0(^^ - I j

. 2e/g

^i[(E - G)s - p] _^ -i(Es + p) 1

2 E - G !

Averaging |FCq(1) ] over the phase p leads to the result shown in

equation (3) in the text.

Expansions near resonance

The solutions can be expanded in power of E to describe the

solution in a neighborhood at the resonance E = 0 . For this, we put

oo k= 0

OO

k = 0 ^

and define

18
These new functions satisfy the system.

"A, - iF cos(Gs + p) lA, = i 0, k k k - 1

(t>, + iF cos(Gs -+ p)0, = i "A, ,

. k k k - 1

For k 0, the initial data are satisfied by the solutions

i— [ sin(Gs + p) - sinp]

>/'q( s ) = e F

-i— [ sin(Gs + p) - sinp]

s ) = e

The higher order functions could be obtained, with some pain, from

the recursions k>l

F

i — sin(Gs + p)

•A, ( s ) = i e k

. F

J dx 0 ( x)e

0 ^

-

i — sin(Gx + p) G

...

-i — sin(Gs + p) G

0, ( s ) = 1 e k

J

dx lAj^ _ ^(x)e

F

i — sin(Gx + p) G

k ~

Note that 0^s)=(-) (s) k k

19
Restricting ourselves to the resonance solution, E = 0

d = cos [ — (sin(Gs + p) - sinp) ]

c = sinf— (sin(Gs + p) - sinp) ] . O Lr

To this approximation

|C^^^)I = |c/(l)|

2 *

= sin [ F (sin(G + p) - sinp) ] where

^* _ F b_ - G ~ 00D

To average this result over the phase p, we note that we can write

|C^(^)| = ^{l-cos[ 2F sinp(cos G - 1) ]cos [2F cospsinG]

+ sin [ 2F sinp(cos G - 1) ] sin [ 2F cos p sin G ] } .

The third term in the bracket can be dropped, since it is antisymmetric about p = 0 . Using the expansions

cos (x sin p)

cos ( x cos p)

in p) 1

°o

OS p) J k = 0

^, J^, ( x) cos 2k

k 2k p

1

where = 1, = 2 (k > 1), and noting that

277 /

277 dp cos 2k cos 2 J& P P

(k, j^) > 0

20
we have, averaging over p.

k = - OO

where A = 2F (cos G - 1), B = 2F sin G . By Graf's Addition

Theorem

OO OO

k=-oo l = _c)o

. jJVa^ . b2

Thus,

<C ^ = - J.(2F"^(cosG- D) J (ZF^'sinG)

q V p 2 0 0

-2J^(4F"sin y)}

which leads to equation (4)

For a particle with no Doppler shift, CO^ - Oi

/ 2 i

<C ( -)>

q V p 2 ^ ^0^ V ' ^

as given in equation (5) . To derive even the linear dependence of | C | on E appears to

be a very difficult problem. Solutions for large F (saturating fields)

can be obtained. These are singular at E = 0, but if the E-series

could be determined to linear terms, its form for large F could be

matched to the large -F solutions to give a uniform solution for

saturating fields.

The solutions for large F will not be discussed in this report.

21
Appendix B

The problem is to solve the time -dependent Schrodinger

equation

in

^ = (H^ + V) ji at ^ 0 ' ^

where

0 (t) = C (t) J/) + C (t)j/)

p p q q

using the Rayleigh-Schrodinger Perturbation Method C^]- The initial

conditions are more general than in Section 2, i.e., we assume the

values

Cpi(t, ) , C (t. ) q 1

and look for the solutions at t = T . The perturbation is :

iCOt -iiCt V =-hbe V =-hbe

pq , qp

The general solution has been given by Ramsey [2] as follows ; we just

introduce the abbreviations

CO = w /n 00 = 00 /ii

p p q q

and assume these values to be constant throughout the interaction space,

(This simplifies somewhat the computations and it is valid for this case

since we can assume the Zeemann and Stark Shifts to be negligible. ) The solutions are :

22
C (t, + T) = {[icos 0sin-^ + cos-^lC (tj pi Z Z-'pl

+ [ i sin 0 sin e^^^^ 1 C (t, ) | Z q 1 J

1 T/Z CO - CO - C0„)

X e

C (t + T) ={[isin9sin^e"^^Mc (t J

q i ^ Z -"pi

r. .aT aT-,_,.-i

+ - 1 cos 0 sm—r— + cos —— C (t, ) r Z Z

J q 1

-if(C0+ COp + COq)

X e

where sin 9, cos9, a, are the same as defined in Section 3 A. The

computation for three successive separated oscillating fields follows

closely that of Ramsey [Z, pp. 1Z7-128].; we used the same notation in

order to allow an easy comparison. Before proceeding further, let us

note the special case of b = 0 , which is used for the regions between

the oscillating fields :

°C (t^ + T) = C (tje P

pi pi

-iCO T

""C (t^ + T) = C (t.)e

"1 q 1 q 1

To avoid further complication, we restrict ourselves to the case of

equal b in all three field regions of length ^ , separated by regions of

length L, where b = 0 (see fig. 4).

The molecule enters the first region at t^ = 0 and leaves it at

1 - T , Ehiring that time T, b is constant (not exactly true, but simpler)

Furthermore, C (0) = 1, C (0) = 0. We then have, after the first

P q

Z3
field region

3. T 3, 7* C {T) = ( i COS 9 sin—r— + cos —r- ) P 2 ^ iI{a)_co -co ) Xe ^ P ^

-i— ( C0+ +00 ) C (T) = ( isinesin^) X e ^' P ^

q 2

These results are the initial conditions for the next step : b = 0 for

tinn.e T . We obtain

C (T + T) = ( i cos 0 sin + cos

^) p 2 2 + i ( ^- ^ - ) - T 1 X e ^

p q P

'

a T C (r + T) = ( i sinf5 sin -— ) q 2

J" Xe '

In the second field region, another perturbation b is applied for time

Tjwith the above C (T + T) , C (T +T) as initial conditions. We

P q

therefore apply the general solutions by setting therein: t^ = 7 + T

and T = T and obtain

C (2r+ T) ={ [isinesin

^ + cos 1 C ( T + T)

D 2 Z v>

+ 1 sin9 sm — e C (T + i) J- - 2 q

+ i-:r { 00 - 00 - 00 ) Xe ^ P ^

24
C (2t + T) = [ [ i sin e sin

^ e "^^ ^

'^^] C (t + T)

q ^ P

+ [ - i cos e sin — + cos -y ] C (r + T) }

7 ^2-i — (CO + CO + 60 ) P q X e Up to this point, the results are copied out of reference [2]. To go

beyond, we apply again b = 0 for time T and obtain:

-ico T

C (2t + 2T) = C (2t + T) e ^

P P

-ico T

C (2t + 2T) = C (2t + T) e ^

q q

Finally we need only:

C (3t + 2T)= { [ i sine sin

^ e '^^^^'^ + C (2t + 2T)

q ^ P

+ [ - i cos 0 sin

^ + cos

^ ] C {2t + 2T) } 2 2 q

T

- — (CO + CO + 00 ) 2 p q X e f -±

in order to obtain:

3^ = < I C (37 + 2T) I ^

> P ' q

' p, q

where the ( )just denote that this is an expectation value. There is no

need for a further averaging over initial phases of entry into the first

field, since Ramsey's solution is independent of the initial phase. (This

is no longer true for the case treated in Appendix A. ) For the detailed computation we introduce the following

substitutions:

CO
Then:

a = 1 sm 9 sm —

jS = icosQsin —

ar

y = cos — .

T i[-(c«j-aj -a;)-coT]

C (2t + T) = { (jS + y) e P P

P

o -i[T(C0 + W + ) + 60 T - 00(7 + T ) ] } c c p q q

1— ( CO - a; - CO ) 2 p q

(CO - co - C0)-60T - co(T + T) ] C (27 + T)= [a O + y) e P

^

P

q

(60 + CO + CO ) + CO T ] }

+-(y-i3)«e ^ p q q

-1 — ( CO + CO + CO ) 2 p q

We want to obtain

C. (37 + 2T)= [ae -i^(2r + 2T)^

(27 + 2T)

+ (y-i3)C (27 + 2T)}

q

-1 — ( 60 + CO + CO ) xe ^ P ^ .

This expression can be written in the following form:

26
a, a,

C (3r + 2T ) = A e + e + A e + A e q L d 6 1

where

= a + y)^

3 A^ =

A^ = a{y - /3 ) A^ = a{y - iS)^ .

And by introducing X = CO - OJ - CO, CO - CO = C0_ X = co_ - c q p q P ^ ^ we have

4 = - ^(2T + 3t)(C0 + C0 +co)+XT

^2 = ^3 = - i

+ 3t) (CO + cOp + co^) = - m

- 1

(2T + 3t) (CO + cOp co^) - XT

and thus:

C (3t + 2T) = (A e^^'^ + A^+A +A e"^^'^)e"^^,

q

'

' 1 2 3 4

Of this last expression we need only the terms in the

- i u

parenthesis, since the factor e drops out in calculating the

modulus of this complex quantity, and we obtain, by using

. , ^ 1 , iXT -iXT

I , iXT ^ -iXT

cosX T = — (e + e )

27
the final solution:

..2.2arr . P -4 sxn B sin —r— 1 cos 0 sm aT sm A. T

p, q 2

, 2^.2aT 2a r, + (cos 9 sm — - cos — ) cos AT

I . 2 a r . 2 1 i 2 + sm — sm 9 - -r /

having re -substituted for a, and "Y .

28
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 73-312

/. Gov t Accession No.

3. Recipient's AccessicAi No.

4. TITLE AND SUBTITLE

XNNTz-^-f D L oC cO

r\-n Ull ±J.ll. TttF r*X d^ Xf^r\ AV^QnT o \J ± ni" L/lf ion ± WXl T«^"V"r*<=^ 1 i LJ viX XXXTnXdlLtrf c;O inXll Cta XVX f^i" v:^ UXlCLll Vi3 n ^ Molecular Beam

5. Publication Date May 1973

6. Performing Organization Code

7. AUTHOR(S)

P. Kartaschoff and S. Jarvis, Jr.

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, CO 80302

10. Project/Task/ Work Unit No. 2730244

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as #9.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The problem of calculating the transition probability of methane molecules in a molecular beam interacting with an infrared (3. 39 jU) radiation beam is discussed.

Contrary to the usual microwave molecular beam experiments, first-order Doppler

frequency shifts cannot be neglected. This makes the solution of the wave -equations

more difficult. Weak field approximations to the transition probability have been

calculated. Single optical beam experiments analogous to the Rabi-type interaction

result in a Doppler-broadened absorption line with an estimated half-power width of a few MHz. For separated multiple field experiments analogous to the Ramsey-type

interaction, no observable response is predicted, the expected sharp resonance

pattern being smeared out by the random Doppler shifts due to the spread of the

molecular beam trajectories. Further investigations are required in order to predict

the resonance line shapes for strong fields, i.e. , saturated absorption.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Frequency standard; methane resonance; molecular beam; Ramsey resonance;

saturated absorption; stabilized laser; transition probability.

18. AVAILABILITY STATEMENT

[x] UNLIMITED.

I I FOR OFFiaAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

22. Price

USCOMM-DC 66244.P7I

USCOMM • EW.