Some coherent transients at millimetre wavelengths

Journal of Molecular Structure, 97 (1983) 203-214 Elsevier Scientific Publishing Company, Amsterdam

SOME COHERENT

TRANSIENTS

203 -Printed

AT MILLIMETRE

in The Netherlands

WAVELENGTHS

Bruno MACKE Laboratoire

de Spectroscopic

Universite

de Lille

Hertzienne,

Associe

I, 59655 VILLENEUVE

O'ASCQ

au C,N.R,S.,

Cedex - (FRANCE)

ABSTRACT Millimetre wavelengths'are shown to provide a favourable time scale for the observation of coherent transients, Basic phenomena (optical nutation, photon echoes) are examined with a special attention paid to a realistic description of the electromagnetic field, Experiments involving an exchange of electric polarization between two molecular species and the propagation of a non resonant step through an optically thick gas are also described.

INTRODUCTION Coherent

transients

are observed

action

between

on-off

in a time long compared

a dilute

when the resonant

gas sample

and a coherent

to the period

to the times characterizing

the evolution

to wavelengths

than the sample

corresponding Coherent

much smaller phenomena

transients

cal (especially

of the former, length

have been extensively domains

studied

(for recent

measurements

etc...

of novel techniques,

and the development

point of view

and the gas relaxation ximations

tizers

namely

transients

here the

resonance,

reviews

and opti-

see e.g.

relaxation

times,

the pulse Fourier

have been examined

trans-

for some kind microwave

roughly

line profile

encountered

is easily

from a quan-

as Z-level quantum

by two relaxation

transitions,

appro-

Millimetre

as the 3rd power of the frequency) role. On another

in some infrared

saturated

systems

times. TRese

ones owing to their much larger absor-

effect which may play an interesting

(= 1 us) remains

refers

in the microwave

comprehensive

are considered

to centimetre

(evolving

trary to the situation whole

magnetic

of dipole moments,

is simply described

are well justified

ption coefficient

coherent

: molecules

lines have been preferred

their Doppler

"Optical"

(Ref. 1,2,3).

In our group from Lille, tum optics

field is switched

in order to distinguish

in nuclear

Ref. 1,2). They originated

form spectroscopy

"optical"

inter-

of the latter but short compared

from their analoques

infrared)

or near1.y resonant

experiments

and the Doppler

much below the time resolution

and to

hand, con-

(Ref. l), the

characteristic

time

of conventional

transient

digi-

us to reexamine

the basic pheno-

(= 10 ns).

The choice

0022-2860/83/$03.00

of millimetre

wavelengths

@ 1983 Elsevier Science

allowed

Publishers

B.V.

204 mena of optical spatial

nutation

and frequency

concerning

spectrum

the exchange

and the propagation

and photon echoes

by taking

of the field,

of electric

to undertake

polarization

of a non resonant

into account

stepwise

original

between field

the actual experiments

two molecular

through

species

an optically

thick

gas.

OPTICAL

NUTATION

We consider

a dilute

gas submitted

to a C.W. field.

frequency,

1-1the transition

matrix

element,

relaxation

time, v o (wvo/c)

the most probable

and E (u) the C.W. field amplitude fully

characterized

c1 = w - w. which reaction

by the Rabi flopping

ximation),

a quadratic

time dependent larization

detector

signal

in the w-rotating

frequency

switched

placed

(Doppler

shift)

interaction

is small

is

Stark field.

If the

(thin sample

appro-

at the cell termination

(more exactly

(polarization)

WI = /uE/hl and the detuning

by a suitable

to the imaginary

frame

velocity

The gas-field

on the C.W. field

proportional

We denote w. the line

the population

molecular

(frequency).

can be conveniently

of the gas polarization

TI(T2)

delivers

then a

part of the complex

to its projection

gas po-

P(t) on the

C.W. field mode).

In this section we are interested initially

at equilibrium

unpolarized, the linear its value driven

far from resonance

is put in exact limit

resonance

corresponding

to the steady

processes

transient

is expected

(fully saturated

line). This phenomenon

with the well

passing

that these Rabi oscillations domain

known spin nutation

(wl >> kvo) and the optical The actually

observable

to the spatial nutation where

when

signal

depart

distribution

strongly

of the C.W. field. and experimentally

the field distribution

is much better

made for a gaussian

effect

is negligible.

P(m) = 0 nutation

studied

result

in the

experiments.

saturated

then to P(t) Q sin w1 t.

is multivalued

To illustrate

defined

in

Let US note in

resonance

is easily

a

(a: TI T2>>1):

from this very simple

beam resonator

(T,_, T2 + -) leads to an analytical the Doppler

reduces

is only

are responsible

in double

the Rabi frequency

transition,

was theoretically

The calculation limit

nutation

limit WI with

resonance.

line profile

up to

On the contrary

called optical

in magnetic

splitting

the whole

signals

even for an undegenerate

saturation

in the time domain

of the Autler-Townes wavelengths,

but this growth

effect).

is currently

t = 0. In

grows monotonously

then at the Rabi frequency

analogy

At millimetre

Doppler

in the strong

1/T2) and thus

the C.W. field.at

state absorption

(collisions,

P(t) oscillates

when the gas,

(CX >> w I, kvo, I/T1,

(a = 0) with

the polarization

frequency

observed

(WI ' T I T 2 << l), the gas polarization

by incoherent

true coherent

in the transients

law since, according

this point, optical

(Ref. 4) in a resonator

than in a travelling in the infinite

(P(t)Q[l-

Due to the C.W. field

wave.

saturation

Jo(wlt)l

/ wit)

inhomogeneity,

the

205

Rabi oscillations polarization

are obviously

P(t) remains

strongly

damped

in the absorptive

and, moreover,

the transient

domain

(Fig, 1). A moderate

41-r

6’lT

Doppler

~ItUW

Fig. 1 : Optical Nutation in the infinite saturation limit : theoretical shape in a gaussian beam resonator. The polarization, projected on the fundamental gaussian mode and normalized to unit, is plotted as a function of the nutation angle wit, WI being the maximum Rabi frequency.

broadening

(kv, = WI/~)

does not affect

responsible

of an extra damping

lar motion,

especially

of a hybrid resonator (perpendicularly results

across

to the Stark plates.

The above mentioned damping

exp

motions

are negligible

geneity

affects

a molecular

echo

(kvo << WI),

only the macroscopic

to a dephasing

polarization

character.

echo.

during

with the theory.

the experimental

the collisional

When the molecular that the field inhomo-

but not the polarization

It is then possible

by switching

the molecular

shape of the nutation motions

is

molecular

to "reverse

signal free of inhomogeneous observation

of

damping

of the different

a short time AT (Fig. 3) with ,a detuning

When the molecular

parallely

The "inhomogeneous"

In the first experimental

and the antisymetrical

were made by means

hide generally

of the Rabi oscillations

(Ref. 5), this was achieved

The location

between

let us note however

and to obtain a macroscopic

to the molecu-

(Fig. 2).

(- t/ZTl - t/2T2) of the Rabi oscillations.

a nutation

resonance

Experiments

The agreement

effects

(wit G 2n) but is

related

field distribution

is very good

"inhomogeneous"

and has not an irreversible

the history" called

calculation

wave.

(guided)

subset with a given Rabi frequency.

thus related subsets

the standing

with a gaussian

and a numerical

the first Rabi period

of the Rabi oscillations

effects,

of a nutation transition

off

ct such as aLT=

(2n+l)%

echo are in accordance

are taken into account,

the

CH3 F J=O-J=l pressure

O.BmTorr

Fig. 2 : Optical Nutation : experimental study at X = 6 mm in a hybrid resonator (inserted). Geometric data : 1 = 1 m, b = 0.2 m, radius of curvature of the cylindrical mirror R = 5 m. The crosses are experimental and the solid line is derived from a numerical calculation taking account of the actual mode distribution of the Doppler effect (kv,/wl = 0.16), of the residual collisions and of the resonator rise time (2 180 ns). The difference is given by the lower curve.

J=O-

J=l

P=O.riBmTorr

Nutation

0

) I

Nutation

+s

5

1

Fig. 3 : Nutation Echo : observation at X = 6 mm in a travelling wave. The inhomogeneous damping of the Rabi oscillations is reversed at.the time T= 2.5 1_1sby switching the molecular transition off resonance during AT such as (w - wo) AT = 7~r. The nutation is then observed around the time 2T + AT as predicted theoretically. "history

reversing"

of a travelling date

is inferior

the travelling

cannot

be perfect

but we showed

(Ref. 5) that,

in the case

wave,

the nutation echo is unaffected in so far as the echo to (T/W,) l/2 where T is a mean molecular transit time across

beam. Since the time characterizing

the inhomogeneous

damping

207 as ~/WI, the latter condition

evolves incident

C.W. fields

been actually switching

PHOTON

determined

instead

by a nutation

of the Stark

always

rate

described

fulfilled

for large

(l/Z TI t l/2 T2) has using a source frequency

here

(Ref. 6).

ECHOES

reaches

its first maximum,

nance. The gas being uncoupled then freely

portional

which

and, as soon as the

precession.

in the w-rotating

effect

frame

The detected

wavelengths

between

region

: the whole Doppler profile is easily saturated and Doppler

damping

rates

sinu-

the incident wo.

of the optical

the infrared

collisional

(Ref. 2). At millimetre

pro-

pre-

(Ref. 1) but prevails

mediate

case

signal,

is simply a damped

of an extra-damping

at centimetre

evolves

time T2, a phenomenon

by the gas at its eigenfrequency

is responsible

is negligible

far from reso-

its polarization

w. with a damping

(w-w,) which may be seen as a beating

and the field reemitted

The Doppler

experiment

with the C.W. field,

decay or optical

to the polarization

soid of frequency C.W. field

nutation

let us put again the molecular

at the new eigenfrequency

known as free induction

cession

damping

echo technique

switching

Let us come back to the optical signal

is obviously

and the collisional

wavelengths,

in

we are in the inter(WI >> kv,) but the

are of the same order of magnitude

(kv, Q 1/T2).

It is then interesting

by submitting

the gas to an inhomogeneous

to artificially

sion and to use a photon echo technique

speed up the extra-damping

Stark field during

the optical

(Ref. 7) as illustrated

preces-

on Fig. 4. The

Fig. 4 : Photon Echo : observation at h = 3 mm (CH3F line J, /KM\ = 1,l -f 2,l. Pressure = 0.6 m Torr). The first optical precession and the photon echo appear as beats of frequency 5.5 MHz, the mean molecular frequency shift induced by the Stark pulses. Their short duration is related to the Stark field inhomo0 geneity.

208

sequence

is the following

("~r/2 pulse"), nutation

(duration

dephasing

: a first nutation stopped at its first maximum

a first precession AT) stopped

("n pulse"),

inhomogeneously

at its first

a second

optical

tions

are again

echo.

Let us note that the photon

polarization to a forced In fact, motion

the echo obtained

effects

2T- 71 sequence

shown

(Ref. 7) and it allowed

are uncommon

motion

to the photon of the

echo related

This difficulty

by the molecular

prevents

a perfect

is overcome

by using the

us to obtain

precise

wavelengths.

measurements

Obviously

of the wall

0

of T2 (Ref. 7)

the experimental

collisions

results

and of the molecular

in Ref. 8,9,10,11.

Y

Y

rephasing

Fig. 5. The second echo is free of motional

to take account

as indicated

affected

Stark field which

at millimetre

have to be corrected transverse

leading

polariza-

to the free evolution from the nutation

Fig. 4 is strongly

the inhomogeneous

of the microscopic- polarizations.

which

the inhomogeneous

polarization.

through

~/2-T-n-

distinguished

T), a second

The microscopic

the TI pulse,

echo is related

be carefully

(duration

zero and reversing

precession.

in phase at a time T after

and must

damped

‘IFS 10

5

20

15

Fig. 5 : n/Z-TT- 2T- 71 photon echo sequence at?, = 3 mm (CH3F line J, /KM] = 1,l -f 2,l. Pressure = 0.65 m Torr). In the limit case presented here, the first echo, expected at t = 10 I_IS,is fully smeared out by motional dephasing whereas the second one, unaffected by the molecular free flight, is still present.

The long sequence shown

equivalent

in conventional

time resolved rized

durations

that the used sources

speaking,

frequency

by these photon

echo experiments

may give quite different

(Ref. 12). Whereas

of its field

the knowledge

of its r.m.s.

spectroscopy

spectroscopy

by the width

requires

required

have

must have a very low phase noise and that sources

power

spectrum

of its frequency deviation

a$ and its correlation

results

the latter

function

or, at least,

time l/q. Roughly

of the source frequency

noise being usually

in

is fairly well characte-

for the former,

autocorrelation

cr+ and q may be seen as mean values

of their rate. The amplitude

a source

negligible

jumps

and

and the condition

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6ulaq Irn >> ?D

210 attenuated

in the same manner

as exp { - (~ir/Z)Ij~d? t/q } for the same time t. 2 2 that for very large q (such as'qt >> 06 t ) this atte-

Let us note however, nuation

becomes

modulation

again negligible.

index

frequency

d8, a situation

EXCHANGE

Stark-switched sible exchange

BETWEEN

transients

of electric

2 MOLECULAR

is related

exchange

17) but it differs

by its more

of the A(B) molecules

pair a- a' (b- b') during efficient

separated

exchanges

(secular

The experiment

width much below the r.m.s.

to the Dicke narrowing

between

two level pairs

strictly

evolves

of a pos-

(a- a') and

A and B (Ref. 15,16). This problem

in the so-called resonant

"resonant

character.

at the eigenfrequency

times of about T2, polarization

are usually

(Ref. 14).

SPECIES

species

if 2~r /vb- vaj 5 l/T2 whatever

polarization

to a very small FM noise

as a good tool for the detection

polarization

molecular

to the energy

spectrum

related

appear

(b- b') or two different

zation

corresponds

(0$/q) and to a source

deviation

POLARIZATION

This

the exchange

neglected

when

collisions" Since

va(ub)

the polariof the level

exchanges

process.

(Ref.

can be only

For this reason

the eigenfrequencies

are well

approximation). was made on CH3F in para

(A) and ortho

(B) states

(Fig. 7).

1 Polarization

A

I B

Frequency

Detuning

va-vb

(MHz)

I

Fig. 7 : Polarization exchange between two molecular species. The relevant (a') for the A (para) molecule states of CH3F are J, KM = 1, -1 (a) = 2, -1 (b) = 2,0 (b') for the B (ortho) molecule. CH3F pressure = and J,K = 1,O 6.8 m Torr (T2 = 1.3 us). Duration of the polarization exchange phase = 1 ps. The polarization gained by the B-molecules is plotted as a function of the during the exchange phase. The Gars are experimental frequency mismatch L, -v and the solid curve ?s theoretical.

Using

a suitable

Stark

sequence,

we actually

tion from the A to the B molecules. achieved

: 1) a polarization

observed

The following

is selectively

an exchange

operations

of polariza-

are SuCCesSiVely

put on the A molecule,

2) the

211 frequencies

va and v

field frequency, detected.

are nearly equalized during a time Q T2 far from the c.w. b 3) the polarization gained by the B-molecules is selectively

The phenomenon

be explained

either

has the expected

by collisional

B-molecules

of the field resulting

A-molecules.

The two explanations

signals

of opposite

former,

s P2 for the latter).

the polarization

processes

(Fig. 7). It may

(Ref. 15) or by a trapping precession

lead to the same theoretical

effect

collisions

for L varying

generally

(Q R for the

from 0.5 m to 2.5 m

prevails,

having an upper

by the

of the laws but to

cell length dependence

Experiments

trapping

exchange

behaviour

from the optical

sign with different

show that the radiation

resonant

the cross section

of

limit of s 2000 i2

(Ref. 16).

PROPAGATION

OF A NON RESONANT

The radiation related

trapping

effects

to the gas optical

ment

to a problem

We consider through

for such studies

encountered

an off-resonance

a dilute

contains

ponse and another

one (transient)

step is then accompanied

absorbed

and rebuilt

: Stimulated

The experiment

during

22 = 31.2

to

now an experi-

w propagating

w. (Fig. 8). The induced

at the eigenfrequency by a resonant

foreward

having

gas polarization to a forced

w. (inelastic).

pulse which

when the sample

res-

The

is continuously is optically

thick.

scattering.

(Ref. 18) was made on the J= 14 + 15 line of OCS at X= 1.6 mm

in a L = 17m long X-band cell in the collisional

and we present

field of frequency

the propagation

resonance

are clearly

coefficients,

: one at frequency w (elastic) corresponding

incident

Fig. 8

section

approximation

in laser spectroscopy.

stepwise

gas of eigenfrequency

two terms

in the previous

the thin sample

Due to their large absorption

lines are favourable

related

discussed

thickness,

be left for their calculation. millimetre

STEP

(Fig. 9). The resonance absorption coefficient -1 leading to an optical thickness

limit is = 1.84 m

(135 dB). The detuning

II

c1 = w - w. is such as the-gas

(a >> w 1, kv,, l/T2 and Z/n 2 Ts << 1) and moderately

dispersive

is transparent (Z/aT2 < l/Z)

212

Square Modulated Rise Time z 7ns

Bias Mixer

Fig. 9 : Experimental arrangement. The stepwise 180 GHz signal is delivered a frequency multiplier the bias of which is square modulated. The resonant pulse beats with the main incident field on the mixer and this beating is treated by a multichannel averager.

in steady

state at the incident

the resonant observed

pulse

beats with

frequency

the main

alone by substracting

modulated

features

being clearly

experiments, by adding ment with relative first

related

especially

solutions amplitude

experiments

cal thickness,

of the linearized

at "retarded"

TR = T2/Z

levels

independently

of the source

achieved

se propagation Promising nal diameter developped coefficient made

sample

involving

(13 dB). This (self induced

possibilities

is also required

Z is adjusted in good agree-

(Ref. 19)

: the

is PZ/aT2,and

its

then constant

The situation

in the TEOl mode.

would

be diffe-

but this requires, much

below the one pre-

for more general

solitary

by oversized

(from 2 to 5 dB per km, between

at 100 GHz

Different

time. For large opti-

remains

non linear effects

ago for telecommunications

of a 11 m long waveguide

has been reached

supperradiant

pulse envelop

transparency,

50- 60 mm) excited

some years

thickness

equations

a cell attenuation

are offered

is

times 01= 3.67 TR, 02= 12.3 TR,

excitation.

highly

problem,

pulse

effects.

for rise time effects

is the so-called

an efficient

rent at power

sently

Bloch-Maxwell

of the beat corrected

the area of the resonant

and this prevents

thickness

the optical

without

time T2, these

gas (see Fig. 1 of Ref. 18), give results

zeros are observed

03= 25.8 TR where

optical

where

c1 is

the unit step detected

much below the relaxation to sample

detector,

This beat of frequency

of Fig. 10 show that the resonant

and shortened

an inactive

field.

in the averager

gas in the cell. The recordings strongly

w. On the output quadratic

by

waves

studies

of pul-

etc...).

circular

waveguide

These waveguides

(inter-

extensively

have a very low absorption

40 and 180 GHz). With a resonator

and two grid mirrors

! We are preparing

a quality

factor

of 1.8 x lo6

two 90 m long cells of expected

213 attenuation

below 0.5 dB. Independently

cells are obviously

iilil\A

interesting

T,,‘Z=133ns;

of pulse propagation

for more conventional

experiments,

such

spectroscopy,

Z/aT,-_,O.lO

TI/Z=420ns:Z/aT=0.04 0x5.3

time 0

t

0.5

p-3)

2

1.5

e

Fig. 10 : Experimental and theoretical study of the propagation of a non resonant step through an optically thick gas .,The step obtained without gas in the cell being substracted, the output beat is plotted vs the "retarded time", i.e. the real time minus the transit time R/c. The recordings (on the left) give for different gas pressures an experimental evidence of the resonant pulse shortening and modulation. As other experiments not reported here, they show that the lobes of the pulse envelop have a duration proportional to the superradiant time Tz/Z. The theoretical curves (on the right) drawn for clT2= 60 TT illustrates the continuous passage from an optically thin sample (Z/aT = 0) with a beat damped as exp(- t/T?) to an optically thick gas with the s ape experimentally observed.

2

ACKNOWLEDGEMENTS The main results F. Rohart butions

described

in this paper were obtained

in the course of his thesis

of H, Deve and J. Legrand

lar waveguide

components

Telecommunications.

has

in collaboration

(Ref. 11) and with B. Segard,

are also acknowledged.

The millimetre

circu-

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214

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