Optical hole burning in ruby

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Physical Review B, 21, 3, pp. 926-936, 1980-02-01

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Optical hole burning in ruby

Jessop, P. E.; Muramoto, T.; Szabo, A.

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PHYSICAL REVIEW B VOLUME 21,NUMBER 3 1FEBRUARY 1980

Optical hole

burning in ruby

P.

E.

Jessop,

T.

Muramoto,

"

and A.Szabo

DivisioIt ofElectriea/ Engineerittg, NatioIIal Research Courteil ofCaItada, Ottawa, Cattada

K13

OR8

(Received 23July 1979)

We report experimental and theoretical studies ofoptical hole burning in the inhomogeneous

R&line ofruby at low temperatures. Both Stark-shifting and pump-probe techniques using

narrow-band single-frequency lasers were employed. Inaddition to the observation ofnarrow holes, we have found that a large (up to 70%)decrease in absorption occurs outside the hole and that the relative size ofthis decrease isconstant over the entire inhomogeneous line. This effect isascribed to fast resonant cross-spin relaxation in the ground-state levels which drives all

spins within the optically pumped volume to acommon spin temperature. A theoretical model

isformulated which describes the power-broadened hole shapes aswell as the off-resonance de-crease inthe absorption coefficient. Fair agreement with experiment isobtained for the case of amagnetic field applied along the ~ axis; however, sizable deviations from the theory are seen

for zero field. We conclude that further studies are needed to elucidate the nature of the Al superhyperfine-broadening mechanism. An upper limit of1200nm isdeduced for the size of macroscopically broadened regions in ruby.

I.

INTRODUCTION

There has been increasing interest recently in

optical-hole-burning and fluorescence-line-narrowing

(FLN)

studies

of

inhomogeneously broadened lines

in solids. The resolution

of

these laser techniques is

limited ultimately by the homogeneous rather than the inhornogeneous widths. Therefore, high-resolution Zeeman, ' Stark,

'

and strain-broadening spectroscopy become possible. Such studies have re-cently allo~ed measurements

of

small hyperfine split-tings

of

optical transitions

of

ions in crystals' and have given information on line-broadening mechan-isms.

"

These experiments compliment similar stud-ies using coherence techniques.

'

Also

of

interest are various phenomena in the area

of

hole burning.

These include short-time phenomena, such as the

cross-relaxation effects described in this and earlier work, as well as long-lived effects, for example pho-tochemical hole burning in organic solids

'

and color centers.

"

Higher-resolution spectroscopy is

possible with hole burning than with FLN because in the latter technique resolution islimited by the

in-terferometer. With hole burning, resolution is

limit-ed only by the laser stability. Finally, there is

in-creasing interest in the possibility

of

using hole

burn-ing for a high capacity, frequency-selective optical memory. '

In this paper we extend earlier studies

of

ground-state cross-relaxation effects on optical hole burning

of

the R~ line in ruby. Previously a sideband-spectroscopy technique was used. The frequency coverage

of

the optical probe used to study the hole

was limited to

+30

MHz about the hole. Here we

describe two additional techniques, one

of

which has

allowed us to make probe observations over the

en-tire inhomogeneous [

—

2

6Hz

FWHM (full width at half maximum)] R~ line at

693.

4 nm when a hole is

burnt into it by anarrow-band (

(

2MHz jitter peak to peak) single-mode, cw ruby laser. In addition to the hole, a large (up to 70%) decrease

of

absorption

coefficient is seen over the etitire inhomogeneous line, unequivocally indicating the presence

of

spatial transport processes. In this paper we present a de-tailed theoretical and experimental study

of

this surprising effect.

InSec. IIwe describe experiments using a Stark-switching technique to measure the hole shape. Ad-ditional hole-burning studies are reported in

Sec.

III in which two separate lasers were used, one to satu-rate and one to probe the hole. In

Sec.

V, a physical

model and theoretical description

of

hole burning is

presented. In Secs. VIand

VII,

a detailed compari-son between theory and experiment is presented along with a discussion

of

implications for macro-scopic versus micromacro-scopic inhomogeneous broadening

in ruby.

II. STARK STUDIES A. Introduction

When an electric field isapplied along the caxis

of

ruby, a linear "pseudo"-Stark splitting' isobserved for the Cr +R~and other lines. There are two kinds

of

sites occupied by the Cr'+ ions in A1203which are related to each other by inversion symmetry. The R~

line associated with each site shifts linearly with field,

in opposite directions, to produce the pseudo-Stark-splitting. In this section we describe experiments

FREQUENC Y

(a)

E-0

SOX CAR

H

C AXIS

REcORD

ABSORPTION

Ei

setu for Stark-shifting

FIG. 2. Schematic of.apparatus seup experiments.

(c)

LASER

, S k- hifting technique for measuring hole a e. {a){top)shows the hole inzero field. n

shape. a top s

'

ld ulses shift the hole in-{c)successively larger electric fie pu

creasing 1yaboveve aand below the laser frequency.

which use the Stark effect to measure p

'

o tical hole shapes in the Ri line

of

ruby.

Fi

.

l.

The basic l'dea o

f

the method' is shown in Fig. Fi

.

1 a) we show a hole burnt into an

inhomo-f

t' n electric field is pulsed on, in Figs. 1 an

1(c),

the hole splits alld shlf'ts to hlg

e

.

Thus the hole can be scanned through

eq e

is.

s e

the laser frequency either point by poin as ' and eariier wor or

an l' k or by the continuous tuning.

In the present experiments,

t.

e ho e

by monitoring the change in transmitted light when the field was pulsed on.

the crystal. M

.

ost experiments were done at a

tem-—

5

K.

A transparent conducting

coa-perature

of-t' _{oated to give a transmission o}

86%at

6900

A, was applied to two surfaces o e

'd _{the electrodes. The electric}

ruby sample to provi e

a air

of

coaxial

field signal was applied through a pair

of

coaxia

transmission lines with the output line being

ter-minated in 50 ohms.

Tuning

of

the laser in the vicinity

of

the

4~,

(+-,

E(+

—

₎

_{transitions was achieved by}

t erature control

of

the laser ruby. The laser beam was circularly polanzed and o y

tempera u

4A&

——

)

E(

——

)

transition was studied. Usually

t e aser ' ' in

—

100

MHz

of

the laser frequency was fixed

within-the peak

of

this line.

s littin using FLN is

A measurement

of

the Stark splitting using is Fi

.

3.

Since only a pseudo splitting"

oc-d in order to ob-curs, the electric field must be pulse in o

serve the Stark-s-

h'f

i te_d

_FLN.

_We used square-wave modulation at 10kHzz, a rate sufficiently high to

of

minimize

E

relaxation effects. As expected, each o

59

MHz

8.

Stark switching

—

experimental Thee expex erimental setup is shown '

inFi.

'g.

_2.

A

uced by a cw ruby single- re

-f

quency laser beam pro uce y

f

ed into the ruby sample along the c axis.

laser is ocuse i

re a lied along the

Th

e

eec

1 tric and magnetic fields are app ie a

in this paper,

.

In all the experiments reported in h p p

a Czochralski-grown ruby samp

e,

nufac-t.

% Crl03 (as specified by the manu

ac-turer, Union Carbide) was used. The samp

e

em-d b aLakeshore silicon diode

erature was measure y

h d t the copper structure holding thermometer attac

e

o

FIG.3. Measurement ofStark splilittingin ruby at 5K

us-1' _{narrowing. The Stark lines appear}

ing fluorescence ine narr

1'

928 P.E.JESSOP,T.MURAMOTO, AND A.SZABO 21

the two Stark lines has half the intensity

of

the un-shifted line, The observed' shift

of

the lines is

0.

110+0.

008 MHz/Vcm

'.

The slight asymmetry

of

the Stark-line positions in Fig. 3about the unshift-ed line arises from drift

of

the laser frequency and

Fabry-Perot interferometer spacing during the scan.

Several scans were averaged to give the quoted Stark

coefficient. Our value is consistent with an earlier result,

"

0.

114

MHz/V cm ' obtained by a nonlaser techique.

The beam-collection geometry is shown in Fig.

4.

This geometry was devised to allow an accurate mea-surement

of

the beam intensity and to restrict

obser-vations to the central 10'/o

of

the beam to avoid edge

effects. The intensity in the center

of

the beam in

the sample is

d mPp

AL Fd

where Pp is the total beam power, Fd isthe fraction

of

power transmitted through the aperture diameter

d, and Lis the distance between the aperture and

fo-cal plane. We chose d sufficiently small that the in-tensity was approximately constant in the aperture.

A reed-switched transmission-line pulser supplied a

rectangular voltage pulse, 400 nsec in width and with a height which was variable up to 100volts. A PAR

162 boxcar with a 10-nsec gate averaged the signal at

a 80-Hz repetition rate. This gave a duty factor

of

3

x 10

'

during the pulse for an average probe po~er

far below the value

of

0.

1 W/cm' for which saturation

effects begin to appear. The signal level

correspond-ing to a transparent sample was measured by heating the sample to 120 Kwhich moves the R~ line 43

6Hz

out

of

resonance with the laser. The

unsaturat-ed absorption coefficient o.

„was

measured by

record-ing the decrease in the transmission

of

a low-power

laser beam as the s'ample was cooled from 120K back to 5

K.

A value

of

ot„=11,

0+0.

4 cm ' was

determined from the observed attenuation

of 5.

2in

our

1.

5-mm-thick sample. Data showing the fre-quency dependence

of

the hole shapes, obtained by

scanning the electric field pulse amplitude, are shown

in Fig. 5 for various laser intensities in the sample.

0.03'/o RUBY I5mm THICK

I750 Oe I( TO c AXIS 5 K 12— IO-O u B u LJJ 0 O 0 ₄ I— o. 0 In 0

UNSATURATED ABSORPTION COEFFICIENT

+/Cm2

FREQUENCY (MHz)

90

FIG. 5. Experimental hole shapes in ruby using the Stark-shifting technique.

Measurements were made near the end

of

the pulse to avoid the small power-broadened free-induction-decay signal seen near the leading edge

of

the pulse. Two features to note in these results are: (i)as the intensity increases, the hole deepens and becomes in-creasingly power broadened, and (ii) a large decrease

in the absorption coefficient is seen up to

-90

MHz

away from the hole.

III. PUMP-PROBE STUDIES A. Introduction

B. Experimental

The aim

of

these studies was to extend the hole-burning observations to cover the entire inhomo-geneous line shape. A disadvantage

of

the pump-probe method was that the spectral resolution was

about an order

of

magnitude less than in the Stark studies because

of

the relative drifts

of

the two lasers. In the Stark as well as iri the earlier sideband tech-niques, this relative frequency drift was absent since the pump and probe beams were derived from the

same source. However, since the main purpose

of

the experiment was to study the off-resonance

sat-uration behavior where high resolution was not im-portant, this deficiency was not serious.

8EAM

IN

SAMPLE AT

FOCUS

APERTURE

FIG. 4. Laser-beam collection geometry used to observe only the central part ofthe beam in Stark experiments.

J

The pump-probe experimental setup is shown in Fig. 6, The saturating pump beam was thermally tuned and fixed near the center

of

the inhomogene-ous line, while acounter-propagating, weak,

nonsat-urating probe beam was tuned to measure the inho-mogeneous line shape. The probe-beam diameter

was adjusted to be about three times smaller than the

pump beam to minimize edge effects arising from the Gaussian beam shape and the slight misalignment between the two beams required to prevent feedback and consequent frequency instability

of

the lasers.

/lazily ~Xg PUMP PROBF .A"xg 'vb '~ &''~"_„PROB H C E Q PUMP PUMP b%%

"""&

h&/PROBE .'_Fg /g 0 O Luo C9 p 0' I —I.O -0.5 ~fl H=0 II ~ I I 1 II ~ ~ I 05 FREQUENCY (GHz) I I.O p 'p

~p

p

—

p

—

p-~H—.~0. SPEC TRUM ANALYZ ER PROBE SIGNAL

FIG.6. Experimental setup for two-beam studies ofhole burning.

FIG. 7. Normalized hole-burning data using the pump-probe technique. Note that the off-resonance change in ab-sorption isindependent of'_frequency difference from the pump beam.

A2(

——

1

)

E

(

——

) Zeeman transition. The

fre-.2 2

quency difference between the pump and probe was

measured by heterodyning the beams on a fast diode

detector and observing the beat frequency on a

spec-trum analyzer.

C. Pump-probe results

vs frequency difference from the pump beam. We see here two striking effects. First the fractional

off-resonance change in

n(v),

F, outside the hole is

in-dependent

of

frequency and second

F

is much larger

in zero magnetic field. These two effects are

qualita-tively explained by the cross-relaxation mechanism discussed later.

The unsaturated absorption profile

of

our sample

was measured by blocking the pump beam and tuning the probe beam across the inhomogeneous line. The

absorption coefficient was calculated from the usual

expression

u(v)

=

—

—

l ln

P(v)

r _Pp(v)

where Po isthe incident power and tis the sample

thickness. Automatic compensation for various

win-dow and lens losses was obtained by measuring Po

with the sample temperature detuned asdescribed earlier. The linewidths observed were in the range

1.

5 to

1.

8

6Hz

depending on position in the sample.

These values are typical for the low-strain,

laser-quality Czochralski-grown rubies we have studied. When the pump was unblocked, the probe power transmitted by the crystal increased. In addition to the optical hole burnt at the pump-laser frequency, there was an overall decrease in absorption across the entire inhomogeneous line. Checks were made to ensure that the pump-induced signals were not due to pump leakage or scattering, by blocking the probe.

Pump leakage into the probe beam was typically less than 10%

of

the probe power and was corrected for in

the data analysis. The quoted pump intensity was not as acurately measured as in the Stark experiments and was estimated by calculating the focal-spot

diam-eter from the diffraction-limit formula.

In Fig. 7we show normalized results in which the

percentage change in absorption coefficient isplotted

D. Temperature dependence I,O H 0Oe B W/cme 0.5— u p p pp p p p p p p 0.05 I I O.I O.I5 (TEMPERATURE) ' (K') I O.20 0.25

FIG.8. Measured (0)and theoretical (

—

) temperature dependence ofthe fractional off-resonance change

F

in ab-sorption coefficient at zero field,

As a further test

of

the cross-relaxation model, we

studied the temperature dependence

of

F.

The two-beam technique was used with the probe beam set

—

600MHz from the pump frequency. The sample temperature was controlled by adjustment

of

the helium-gas _temperature flowing into the sample chamber using an electronically controlled heater.

Figure 8shows the dependence over the temperature range 5

—

40

K.

F

isseen todecrease steadily as the temperature is increased.

930 p.

F.

JESSOP,T.MURAMOTO AND A SZABO 21

IV. SUBSIDIARY MEASUREMENTS

SysTEM RESOLUTIPN

+

7~3MHz

O.PI%RUB~

4.2K

'

on

of

theory and experiment,

know-For comparison o

es and

homogene-of

the various relaxation times an

p

f

h homogeneous linewidth or e

dependence o t e

nc-4A

(+

—

₎

Et+

—, ran

' '

A2

—

—

₂ ( —,

)

t sition and laser-freque

y-jitter measurements.

9.3MHz

A. Spin-lattice time E-Ap('/2)

E-

Ag (/p)

'

ths

—

field dependence

B. Homogeneous linewid

ments' in ruby have yielded

t edetailed dependence

of

the homogeneous

ine-nts

of

echo li etimes exis

measurements

'dth

measure-we resent homogeneous linewi

Here we presen

0

—

200Oe using the Stark ments in the field range

Measurements were

i ue for hole observation.

d o

bod

made at low laser poower to avoi pow The results are shown in Fig.

9.

At zero ie

l50 & _IOO 0.05%RUBY 4.2 K E('~a)

~

A~('&a) so LIJ I 200 I00

MAGNETIC FIELD (Oe)

ndence ofthe hole linewidth FIG. 9. Magnetic field dependence o

4A

(+

—

_{) Etransition in ruby.} ofthe A2

nown to be very sensitive to S in-lattice times are known o

'

s.' In particular

Fe

an thepresence

of

impurities.

f

ma nitude

'+

'

ruby can produce orders o g '

'+

_l t' _{n time by resonant and}

of

the Cr re axa io

reduction o

To

categorize our sam-nonresonan nt cross relaxation. ocae

al technique. The

measured Ti by an optica ec

pie, we mea

two-beam methododwas used with the probe beam

dis-beam. Pulsing

-600

MHz from the pump beam.

placed-and monitoring the recovery

of

tion coefficient by the probe yie e

2

30K

fr

ob d tern erature range

temperature depe ndence

of

i was o

ne- honon relaxation process.

at li uid-helium tempera-serveed valuev

of

215msec at liqui - eiu

le"

ture is typical for a pure samp

e.

- _{arrowing spectra of 0.01%}

FIG.10. Fluorescence line-narrowing ruby in afield of400Oe.

width (half the observed FWHM homogeneous linewid

eswith '

th is 65

+

10 MHz. This value agrees wit

FLN observations made with t is sam th

tth

ho

t Stark data.

'

As an interesting aside, we see t a

'dth

of

the F. (

+

—,

)

A2 and

F. (

+

—

)4A2 transitions become t ee same ats zero

E

+

2 r

hoton-echo studies,

"

field. As is well known from p

a ma netic field isapplied, these transition linewidths rapidly diverge. is is

FLN results in

F'

ig. 10 for dilute ruby in w ic a

f

—

35 MHz is observed for the resolved width o

—

z

E

(+

—

) Aq transition and an unreso ve wi

+

—

)4A_{2. For an}

ap-(

5 MHz isobserved for

E

+

—,

field

of

—

—

1 kOe, recent photon-echo studies'

ahomogeneous

of

the present sample indicate a

linewidth

of

160kHZ for F.

(+

—,

)'A,

.

C. L'aser jitter

ntal arameter which arises An important experimen a p

n-

_of

_{theory and experiment is t e}

p

r fre uency jitter; both its magnitu e an

j

'itter was ac ieveh' d by using a new laser con i

t frequency jitter arising

Th

f

t

all elimina es

- ower fluctuations. isc

(0003'/o)

d very low concentration ruby

.

o a

use ave

tin effects and a rto suppress pump-heating e

cr stal (0 05%) s ltfter.

more concentrated y

were operated in series wi

-l rbeam transmitted throug e

gon-ion pump- aser ea

um the amplifier.

op

p

Measurements were made with a

n econfocal Fabry-Perot with a

spectral-range con o

F

500P)

The

peak-to--.

rwas

1.

5

—

2MHz giving an rms width o

—

_0.

₇ MHz

—

a value similar to a

in heterodyne methods. The jitter fre-quenency rate was in arange up

to—

mari1y by cryostat mechanica1 vibrations and pointing fluctuations

of

the pump argon-ion laser. Unlike

jet-stream dye lasers, which exhibit frequency fluctua-tion rates up to

—

1 MHz, such high rates were total-ly absent in our system since a solid block

of

ruby constitutes the 1aser cavity.

V. THEORY A. Zero laser-frequency jitter

3A

PUMP RE'DUCED

ABSORPTION

.We have previously described how cross relaxation

in the ground-state levels

of

ruby can produce an

off-resonance decrease in optical-absorption

coeffi-cient when a hole is produced in the R~ line by a monochromatic laser beam. Also, in FLN experi-ments in

0.

03%ruby, the fluorescence

inhomogene-ous background was measured to be less than 1%

of

the line-narrowed peak. Therefore we concluded that there is no optical spectral diffusion. This conclusion

on the lack

of

optical energy transfer between

reso-nant and nonresoreso-nant ions isfurther supported by

re-cent'9 grating experiments in ruby which show no

spatial transfer beyond

30

—

50nm from optically

ex-cited ions. We contend that the spatial energy transfer, which must exist to account for our results,

occurs in the ground-state levels.

Our earlier three-level model isextended here to

the more realistic six-level system shown in Fig.

11.

The six-level model is appropriate for

E

A2

transi-tions in an applied magnetic field. In order to estab-lish some basic rate-equation approximations, we as-sume initially that the ions may be divided into two groups, those resonant with the laser and those

non-resonant. Using these model equations, we then

for-mulate a theory in which a continuous frequency dis-tribution is treated.

In Fig.

11,

level 2 isthe A2(

—

2

)

level which is

pumped to level 3A

[E(

—

₂

)].

The usual ratio

of

o spontaneous rates from the

E

levels is taken, and the much slower m spontaneous rates are neglected. The

populations

of

the resonant ions will be denoted by

n,

and the nonresonant ones by N .

For

level 1A

of

the resonant ions we have

dt 3Pn

~'~

—

_+3yn3A

+P(n2

+ nlc+

n]s)

,[nt'q(Nt'

+N)'s+N&c)

—

Nt'g(n2

+

nt'a+

nic)

~

(2)

P

is aground-state spin-lattice relaxation rate which

isassumed to be the same between all levels so that the spin-lattice time Tt

=

(4P)

.

Actually, according to relaxation theories for ruby in which aquadratic spin-phonon Hamiltonian

of

the

form"

Hsp

=

S D S produces relaxation, the latter assumption is not

jus-tified.

For

example, direct relaxation between the

+

—,levels isforbidden as in fact has recently

'

been

verified at very high magnetic fields

(50

—

130 kOe).

However, sizeable deviations from the theory have been

seen"

at lower fields such as used here. Lack-ing further information, we have assumed for simpli-city equal spin-lattice rates rates between all levels.

In Eq.

(2),

cois a resonant cross-relaxation rate which

for

0.

03%ruby

is24"

—

106sec '

»P

—

1sec ' (at

4.

2

K)

and also

»

y

=

(Sr)

' where

r

is the F.

fluorescence lifetime

of

4

X10

sec. Writing similar

equations for n~'~ and n~'~, we obtain for

I I I I n)

—

—

—

n)„+n)~+n)~

I

ct

= —

Pnt'

+3Pn2 +4yn3

—,

N' (n~'N2

—

n2Nt') 1C lA

RESONANT IONS NON-RESONANT .IONS

where N' is the total population

of

off-resonance

ions, N~' =N~'&

_+N~'&+NIc,

and n3

=n3~

+n3g.

Be-cause

of

the fast Orbach relaxation in the

'E

levels at the temperatures in our experiments, we take

n3~

=

n3~.

For

N2 we have FIG.11. Six-level model used for ruby in amagnetic

field. The 2 3Atransition is optically pumped, and cou-pling between the resonant and nonresonant ions occurs via cross relaxation in the four ground-state levels. _Pisthe spin-lattice rate assumed equal between all levels (only two levels are shown coupled for simplicity), eoisthe

cross-relaxation rate, and (5y) ' isthe fluorescence lifetime. 4yn3

=,

(nt'N2

—

_n2Nt') N'

(5)

I

=

pNt'

—

_3pN2

—,

(nt'Nt'

—

n2Nt' )

.

(4) dt N' ~I ~ /

932 P.

E.

JESSOP,T.MURAMOTO, AND A.SZABO 21 We can neglect terms involving Pin Eq.

(3)

because

the spin-lattice relaxation in the ground state

of

the resonant Cr'+ ions is very much slower than the cross

relaxation to the nonresonant ions. Combining Eqs. (4) and (S)gives

4yn3

=

pNI'

—

3pN3

We might remark that the results obtained by using

the above approximation along with the spin-tem-perature approximation [Eq.

(14)]

are identical to those obtained when the rate equations are exactly solved for a three-level system and due approxima-tions made because ol

»

_y,

_P.

We now consider acontinuous frequency depen-dence for the populations. Define the total popula-tion in level i by

f

eo

n,

—

=

~

n;(v')

dv',

the same spin temperature, i.

e.

,

111(V

) =/n3(V )

(14)

~here

for all v', and

f

is aconstant to be evaluated. The

thermal-equilibrium value

of

f

is

f

=3

in the high-temperature approximation used throughout. Equa-tion

(14)

assumes a limiting size 1'_{or macroscopic}

re-gions in which ions have a constant optical frequen-cy, as will be discussed further in

Sec. VII.

Combin-ing Eqs.

(12)—(14)

gives

Npgi

(v

vo)

(1S)

2+

_(f

+1)

[1+10y/o.

(v'

—

vL)P] We write pNog;

(v'

—

vo)

1+4[(v'

—

vl.

)/8]'

where v' is the optical frequency

of

the ion for the transition from level 2 to 3A. The absorption

coeffi-cient for this transition is

n(v)

=J

[n3(v')

—

n34(v')]a(v

—

v')dv'

0 where and

.

=

_{[(1+f)/S+ (3+f)]-I}

(17)

(18)

G'(V V

)

=

(Togs(V V)

and g~ isa normalized Lorentzian

homogeneous-line-shape function. Rate equations for population packets at v' may be written

dn34

(v')

₌

rr(v'

—

_vt

)

P[n3(v')

—

n34

(v

) ] dt

—

5_yn34( v')

+

R [n3s( v')

—

n34( v') ]

Here

S

=

IrpP/Syvrwp and wp equals the

homogene-ous linewidth (FWHM). 5is seen to be the

power-broadened linewidth

of

the population in 3A. From

these sets

of

equations, the frequency-dependent ab-sorption, Eq.

(8),

may be evaluated

4u„(v)

p(3

+f)

8(

w„+

5)

nv="

1

——

19 1

+f

₍wp

+8)'+4(v

—

vl

)'

and

(10)

dn„(v')

R[1134

(P

)

n313(v')]

—

5yn313(v') dt

where in the integration, we assume w;

[the.

inhomo-geneous linewidth (FWHM)]

»

wh sothat

g;(v'

—

vp) may be taken outside the integral.

n„(v)

is the unsaturated absorption coefficient.

~,

(v)

= —

„(~oNo)g;(v

1

—

vo)

.

where

8

is the Orbach relaxation rate, I'is the pho-ton flux in photons/cm sec, and vL is the laser

fre-quency. Adding Eqs.

(10)

and

(ll)

and setting

n;=0

gives

o(v'

—

vl

)Pnl(v')

=

[10y+

Ir(v'

—

vL)P]n34(v )

(12)

Population conservation requires

Equation

(19)

describes the shape

of

the power-broadened hole (width

=

wp

+

5)

as well as the

off-resonance change in

n(v)

due to cross relaxation.

Our remaining task isto calculate the spin-tem-perature factor

f.

We now assume that Eq.

(6)

derived for asquare line shape also applies for a Lorenztian line shape so that

ill

(

v

)

+

(

nt)v+ 113(v )

=

cVpg;(v'

—

vp),

(13)

4yn3 gy1134

=

p(nl

—

3n3)

(21)

where No is the total population, g; isa normalized Gaussian inhomogeneous-line-shape function, and vo

is the center frequency

of

the inhomogeneous line.

Because

of

fast cross relaxation, we assume that ali

spins within the optically pumped volume come to n)

+

n2

—

No (22)

remembering now that n sare total populations. In

writing Eq.

(21),

we assume that the hole width is

much smaller than the inhomogeneous width.

Combining Eqs. (22)and

(14)

with Eq.

(21)

gives

(f

-3)

87n3g

=p&p

( )

Integration

of

Eq.

(16)

over v' _gives

1

n3g

=

₂'rr_pSNpg;(VL

—

_vp)

(23)

(24)

—

3 2

[&(3+,f)

+(I

+,f)

j

=(&&)',

(

f+

I)

where

(25)

JC

=8(~ln2)'~'

~

P

w(. (26)

Equations

(17), (18), (23),

and (24) may be

com-bined to give an expression from which

f

may be

ob-tained in terms

of

known parameters. We take the

~aser frequency vL

=so.

We have

w„w„+J.

The hole width Wj, in Eq.

(19)

becomes 1/2

W=(

+J)1+

f

+I

(I

+

f)

(31)

which in the limit

of

low power

(S

.

0)

becomes

Wj,

=2(

wp

+ J)

as might bephysically expected. The case

of

slow jitter is more difficult to treat and

we proceed intuitively. Looking at the line shape for

the n3~ population in Eq.

(16),

we replace

(32)

5~5+

_J;

i.

e.

, the Lorentzian jitter width simply adds to the power broadened width 5. The next question is, how

is ~ modified by the jitter?

To

first order, we expect eto be independent

of

jitter. In other words as the laser frequency moves about, the peak

of

the n34

population remains at the no-jitter value. This is what we mean by "slow"jitter. These considerations

change Eq. (25) for

f

to the equation

a„(v)

—

n(v)

n„(v)

(27)

An experimental parameter, in which we have been particularly interested (see Fig. 7) is the frac-tional change in off-resonance absorption defined by

—

₃ 2

[S(3+.

f)

+(I

+ f)

j

(f+

I)

J

(1+

f)

wp

&(3+f)

+(1+f)

(33)

where the observation frequency v is sufficiently far

from vL to be outside the hole. From Eq.

(19)

(28)

Thus,

oncefis

numerical, ly evaluated from Eq.

(25)t

experimental quantities such as

F

and the observed hole width W~

=

~~+5

may be calculated.

B.

Effect

of

laser-frequency jitter

i/2

Wp

=2J+

wp

1+

+1

(1+

f)

which again approaches the proper limit

2(

wp

+ J)

in

the limit

of

low power.

(34)

VI. COMPARISON OF THEORY AND EXPERIMENT

AND DISCUSSION

Since the probe laser also exhibits ajitter

J,

the width

of

the observed hole from Eq.

(19)

becomes

We calculate the effects

of

laserjitter in the limit

of

two extremes, (i) the jitter rate is much faster than the system time constants which determine the time development

of

the hole, and (ii) thejitter rate is

much slower than these time constants.

For

simplici-ty, a time-averaged Lorenztian shape for the

jitter-broadened laser line shape is assumed.

Consider first fastjitter. In this model we write for

the spectral distribution

of P

in Eqs.

(10)

and

(11)

Comparison

of

theory with experiment for the

off-resonance fractional change in absorption F,the hole

P(v")

=Ppg,

(v" —

vL) where

(29)

/I

2J

g/(v vL)

=

[J'+4(

"—

)'j

(30)

I

Here

J

is the laser-jitter linewidth

(FWHM).

On

in-tegration over

v"

in Eqs.

(10)

and

(11),

this simply

results in aconvolution

of

J

with ~q sothat all the earlier analysis may include jitter by replacing

0 O.I / / / / l I I.O iNTENSITY (W/ cm~₎ IO0

FIG. 12. Comparison oftheory ( )and experiment

(-0-)for the fractional change

F

in off-resonance absorption coefficient for zero field.

934 P.E.JESSOP,T.MURAMOTO, AND A.SZABO 21 CU 0 1750Oe 5K Zo 0 Oy N R II u T O hJ_g Z LLt O Z 1750Oe 5K 0 O.I I I.O I0.0 INTENSITY (W/cm ) O O.I I I.O INTENSITY (W/ cm~) I 10.0 IOO

FIG. 13. Comparison oftheory and experiment (0)for the fractional change

F

in off-resonance absorption coeffi-cient vs pump intensity for an applied magnetic field along the caxis. Theoretical curves (1)no laser

jitter,

(2)

slow laser jitter ofwidth 0.7 MHz FWHM

—

—

—,(3)fast laser jitter ofwidth 0.7MHz FWHM

—

—

.

FIG.14. Comparison oftheory and experiment (0)for the holewidth (FWHM) vs pump intensity for an applied magnetic field along the c axis. Theoretical curves (1/ no laser

jitter,

(2)slow laser jitter ofwidth 0.7MHz FWHM

—

—

—,(3)fast laserjitter ofwidth 0.7MHz FWHM

width, and the temperature dependence

of

F

isgiven

in Figs. 12

—

14and in Fig. 8. These are discussed in

the following paragraphs.

A. Zero-field studies—

temperature dependence

The two parameters in the theory which are tem-perature dependent are T~ and the homogeneous linewidth. The direct process and Raman-induced phonon linewidths were calculated from the theory

of

McCumber and Sturge

'

using the photon-echo-derived" direct process coefficient

P'=g.

l cm ' and the Raman coefficient

e'

=544

cm

'.

The phonon-induced linewidth was then added to the homogene-ous linewidth

of

65 MHz measured at 5

K.

The cal-culated variation

of

F

with temperature is shown in Fig. 8along with experimental data. The observed decrease in

F

with increasing temperature is

qualita-tively predicted by the theory; however, the

theoreti-cal curve lies well above the observed points. A similar deviation isalso observed in Fig. 12where the power dependence

of

F

is shown for zero mag-netic field.

The

F

values in an applied field (Fig.

13)

are

small-er than in zero field (Fig.

12).

This is qualitatively explained by the theory as arising from the larger homogeneous linewidth in zero field.

One assumption

of

our theory which could explain part

of

the discrepancy at low powers (small satura-tion) issimply that our sample is not optically thin. However ifwe assume a worst-case situation in which the smallest measured intensity isattenuated by a factor

of

5 (corresponding to the unsaturated

at-tenuation), the theoretical

F

value still exceeds the experimental value. Another possibility isthat the six-level model is no longer appropriate at zero field.

%e

believe, however, that a major source

of

the discrepancy isrelated to the question

of

the nature

of

the line broadening produced by the superhyperfine interaction between the A1 and Cr spins. As dis-cussed by Grischkowsky and Hartmann,

'

this broadening arises from the time-dependent magnetic field at the Cr site produced by the flipping Al spins

of

the host lattice. This produces a modulation

m(t)

of

the Cr energy-level separation, which has a

minimum value when a magnetic field isapplied along the c axis

(8=0').

As qualitatively confirmed

in the FLN results

of

Fig. 12,this broadening will be

larger for

Az(+

₂)

E

transitions than for

4A,

( +

—,)

E

transitions because

of

the different

splitting rates Sv/5H. When 8

AO', m(t)

rapidly

in-creases because the Al precession axes no longer

ap-proximately coincide with the Cr precession axes.

This results in an increase

of

the projection

of

the time-dependent part

of

the Al spin vector on the Cr spin vector, hence increasing

m(t)

This feature

.

is

responsible for the increase in linewidth at zero field. An unknown factor is the nature

of

the time depen-dence

of m(t)

which is determined by the complex Cr-Al-Al spin interactions. As discussed by Shelby etal

3for

the syste.m Pr +:LaF3,

m(t)

may exhibit a

slow component, since Al spins near the Cr ions will

be detuned from distant Al spins, and hence their flip-flop rate will be much slower than those

of

the distant spins. This idea that superhyperfine

broaden-ing has an approximately static inhomogeneous

com-ponent along with a homogeneous component is sup-ported by recent hole-burning observations' in

Pr'+:LaF3which gave a frequency-domain linewidth

of

200kHz compared to the time-domain width

of

20

kHz inferred from coherent transient studies. Itis

tempting to model the 65-MHz width

of

ruby in zero field as arising from a smaller homogeneous width

plus an "atomic jitter" width. Unfortunately accurate

echo data do not exist for ruby at zero field to test

this idea. Assuming slow jitter, this kind

of

model could bring theory closer to the data by varying ~I,

and

J,

where

J

is now identified as the atomic jitter. B. Magnetic-field studies

Comparison

of

theory and experiment for the two

laser-jitter models are shown in Figs. 13and

14.

Fair

agreement between experiment and theory is

ob-tained assuming that fast jitter occurs at low power levels and slow jitter applies at high power levels.

This is consistent with the idea that the hole-formation time isexpected to shorten with increasing power.

Regarding atomic-jitter, a similar problem exists as

in

Sec.

VI A in that no high-resolution data exist for

the width one would see in the frequency domain.

The 160-kHz

(

T2

—

2

psec)

width inferred' forour

ruby from echo studies ispresumably produced by

resonant cross relaxation" since T2

—

co '

—

1p.sec. It might be mentioned that

calculations"'

of

Al

superhyperfine broadening in a magnetic field predict

a width

of

-0.

5 MHz using an approach developed

by Wenzel

'

for EPR broadening. However, this theory does not consider the time dependence

of

the

Al spin flips. The width is calculated taking random static Al spin configurations around the Cr ion. Since

photon-echo measurements have yielded' linewidths

as narrow as 25 kHz in dilute ruby at high magnetic field, it seems clear that the linewidth calculated from the static approach is in fact inhomogeneous.

VII. FURTHER COMMENTS AND CONCLUSIONS

Despite the lack

of

agreement between theory and

experiment for some

of

our observations, we believe

that cross relaxation is the major mechanism which

produces the large change in off-resonance absorp-tion coefficient. This isfurther supported by

obser-vation

of

small increases in

F

for magnetic field values at which the ground-state energy-level spac-ings are harmonically related.

(See

note added in

proof.

)

As is well known from EPR studies,

addition-al cross relaxation occurs' under such conditions.

We might emphasize that in the field

of

spin ther-modynamics, the present studies represent a rather unique situation in which spins labeled by a particular optical frequency are initially driven by optical

pump-ing to a temperature far different from the non-resonant spins. Our significant observation that all spins within the pumped volume come to the same temperature has interesting implications regarding the spatial distribution

of

ions having the same optical frequency or in other words on the question

of

ma-croscopic versus microscopic inhomogeneous

broadening in ruby. By macroscopic broadening we

mean the existence

of

regions in which all ions have

the same optical frequency as opposed to microscopic

for which the frequency varies randomly from site to

site. This question is

of

importance in recent specula-tions

of

the possible occurrence

of

an Anderson transition in ruby. An approximate upper limit on the size

of

macroscopic regions may be calculated.

The number

of

Cr spin flip flops which occur before

thermalization occurs by a phonon interaction is AT~

=2

&

10'.

Assuming a random-walk model and a step distance l

—

2.

6

x

10

'

cm for

0.

03%ruby cal-culated from

Lyo's"

formula

I3=0.

17/No, we obtain a diffusion distance

of

(2.

6

x

10

')

(2 x 10

)

'i'

-1200

nm

which represents the maximum size

of

macroscopic regions. Also, since this distance is much less than the diameter

of

the pumped filament

(

—

3x 10

'

cm),

significant thermalization

of

the pumped spins by the unpumped spins outside the fi-lament isnot expected. _{As the sample temperature}

increases, the diffusion distance decreases because

of

the decrease in T~. It is possible that the

anomolous-ly low

F

values observed at higher temperature may

arise because the diffusion distance becomes less than the macroscopic region dimension. However, the generally low

F

values seen atzero field do not permit a firm conclusion.

In summary, an earlier hypothesis that ground-state cross relaxation occurs over only part

of

the R~

inhomogeneously broadened line in ruby is incorrect.

Using a simple three-level model and a square hole shape, this earlier hypothesis seemed to be necessary tocorrelate theory with the large observed decrease in absorption coefficient outside the hole. However, the present studies show that resonable agreement between a more sophisticated theory and experiment isobtained for cross relaxation over the entire inho-mogeneous line. The latter observation places an upper limit

of

1200nm on the size

of

macroscopically broadened regions in ruby. Further studies are need-ed to elucidate the nature

of

the R~ line broadening mechanisms in ruby due to Al superhyperfine in-teractions to explain some aspects

of

the observed

optical saturation behavior. '

Note added in proof. Asimilar observation using a

different technique has also been reported by A.

C.

Boccara; Colloq. Int. CNRS

255,

299

(1977).

ACKNOWLEDGMENTS

We thank

L.

E.

Erickson for commenting on the manuscript and M. Kroll and

E.

L.

Dimock for

P.

E.

JESSOP,T.MURAMOTO, AND A.SZABO *On_{leave from Faculty ofEducation, Shiga University,}

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