Photon echoes of terbium in lithium yttrium fluoride
by Paula Louise Fisher Darejeh
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Physics
Montana State University
© Copyright by Paula Louise Fisher Darejeh (1983)
Abstract:
The photon echo experiment is a method of measuring the homogeneous dephasing time (T2) from
which the homogeneous linewidth of an optical transition can be calculated. In this thesis, the theories
of linewidths of transitions and of the development of a photon echo are presented. Also discussed is
the experimental setup by which the first photon echoes in the rare-earth terbium were observed and
measured. The decays of echo intensity with laser pulse separation at the externally applied fields of 45
KG and 25 KG were plotted and from these plots a homogeneous dephasing time of HO nanoseconds
was calculated. Data at 15 KG were taken, but because of the non-exponential behavior of the decay, Tj
was not calculated. At 0 KG, no echo was observed, implying a decrease of the dephasing time below
15 KG. PBIOTOT ECfflOSS OF TEfflfflZfflH SK
LITfflIUH TTTfflIUH FLUOSIBB
by
Paul a Loui se Fisher Darejeh
A thesis su bmitted in partial fulfillment
of the requirements for the degree
of
Master
of Science
in
Physics
MO N T AN A STATE U N I V ER S I T Y
Bozeman, Montana
December 1983
main lib.
M372
ii
cop. A
AP PR OV A L
of
a thesis
Paula Louise
submitted by
Fisher Darejeh
T h i s t h e s i s has b e e n r e a d by ea c h m e m b e r of the th es i s
c o m m i t t e e and has been found to be satisfactory regarding
content, English usage, format, citations, bibliographic
style, and c o n s i s t e n c y , a nd is r e a d y for s u b m i s s i o n to
the College of Graduate Studies.
7 W 2 g , / ? g 3
Date
Chairperson,GraduateCommittee
Ap pro ve d for
the Major
Department
Dat e
App ro ved for
-
Date
the College
of Graduate
2raduate Dean
Studies
iii
S T A T E H B N Y OF P B S H I S S ION TO U S E
I
In p r e s e n t i n g
this
thesis
in p a r t i a l
fulfillment
of
the r e q u i r e m e n t s for a m a s t e r ' s
d e g r e e at M o n t a n a S ta te
University,
library
available
a g r ee
that
to bo r r ow er s
quotations
from
permission,
source
I
under
rules
the
make
Library.
that
accurate
it
Brief
special
acknowledgement
of
is m a d e .
for
extensive
r e p r o d u c t i o n of this
professor,
Libraries,
of
or
financial
or
when
the
copying
in
t h e s i s m a y be
his
absence,
in the
material
use
of
g a i n s h a ll
opinion
is
the
for
rajuiLa ctfyujno -/JloA
S i g n a t u r e /I
j 8,
quotation
by
the
scholarly
material
in
the
of
proposed
purposes.
this
or
my major
Director
not be a l l o w e d w i t h o u t
AJoMji J o
from
g r a n t e d by
of eithe r,
p e r m i s s ion.
Da t
of
shall
this thesis are allo wa ble w i tho ut
provided
Permission
use
the
thesis
Any
for
my written
iv
T AB LE OF CO OT E O TS
Page
L I S T OF F I G U R E S ........................
AB STR ACT
1.
.................
v
.........................................vii
INTRODUCTION
.........................................
I
2. HO M OG EN EOU S AND I NH OM OGE NE OUS L IN EWI DT HS . . . . .
3.
PHOTON
ECHO
4
........................
12
F o r m a t i o n of the E c h o .............................12
Damping of the E c h o ................... ............22
Di r ec ti o n of Echo P r o pa g a t io n
. . . . . . . . . .
23
Laser Intensities Re qu ire d for n/2 and n Pulses.
24
4. TERBI UM ION ST RUCTURE AND T R A NS I T I ON S
5. E X P E R I ME NT A L SETUP
............
.....................
6. COMPONENTS OF SETUP
28
. . . . . . . . ..............
31
Lasers . . . . . .
. . . . . . . . . . .
. . . . 3 1
Lenses and Waists
. . . . . . . . . . . . . . .
Po la ri z at i on Di scr im i n a ti o n Technique
. . . . . .
Spatial Filt eri ng
...........
.
Pockels C e l l ..............
Elect ro nic and Computer Control of Experiment
.
7. A DAY
( OR TWO
) IN THE LAB
8. DATA AND CON CLU SI ONS
REFERE NC ES
CITED
. . . . . . . . . . .
34
37
40
41
43
46
......................
......................
26
60
.
70
A P P E N D I C E S ................... ............................ 7 3
Ap pendix A - Crystal An is ot r o p y
.................
App en dix B — Da ta — A cq u is it io n and Control
Progr am for Photon Echo Experi me nt
.
74
79
V
LIST ©F FISliMS
Page
1.
The p re ce s si on
of <p>
about x
........................ 19
2.
The p r e ce ss i on
of <p>
about z
......................
3.
The
pr ece ss i on
of <p>
about x
........................ 20
4.
The
pr ece ss i on
of <p>
about z
...............
5.
Setup for pho ton echo experiment
6.
Block diagram of laser
7.
Ni tr oge n-1 a se r-pumpe d dye laser
8.
Po l ar iz at i on di scr im i n a ti o n
9.
Pockels
10.
Delayed trigger generator
cell
trigger
drift
20
21
.
30
compensator
circuit
. 32
..................... 33
............
39
........................
42
..........................
43
technique
circuit
11. Flowchart of data — a cq u is it io n and control
pr og r am for photon echo experiment . ............... 45
12.
13.
The two Nj- Iaser pumped dye lasers with
monit or ing p ho to di ode s and Fabry-P er ot e talon
. . 47
The
. . 48
14. The
optics
b etw ee n the lasers
slicing of
15.
A diagrammatic
16.
Pulse
17. Pulse
18.
the beam by
and the crystal
the razor blade
cross- se ct io n
of the
dewar
. . . .
50
. . . .
52
se paration = 35 n s e c ............................ 61
se paration = 65 nsec
.................
61
Semi— logarithmic plots of ec ho.intensity
vs. pulse se paration at 25 KG
'..................... 63
vi
ILZST ©F FZffiWSES — Cont inped
Page
19.
20.
Se m i- lo gar it hmi c plots of echo intensity
vs. pulse separation at 45 K G ............ ..
64
Phot on echo signal with p o la r i z at i o n
di sc ri mi n at io n technique ..............................
66
Echo
22.
intensity vs. pulse separation at 45 KG . . . 67
/
The p o l a r i za ti o n rotation by a half-wave plate . . 76
23.
intensity vs.
pulse
se paration at 25 KG . . . 6 6
21.
Echo
24. The Babinet -So l iel compensator
........................ 77
vii
ABSTRACT
T he p h o t o n e c h o e x p e r i m e n t is a m e t h o d of m e a s u r i n g
the h o m o g e n e o u s
dephasing time
(Tg) f r o m w h i c h
the
h o m o g e n e o u s l i n e w i d t h of an o p t i c a l t r a n s i t i o n c a n be
calculated. In this thesis, the theories of li ne wi d t h s of
t r a n s i t i o n s a n d of the d e v e l o p m e n t of a p h o t o n e c h o are
p r e s e n t e d . A l s o d i s c u s s e d is the e x p e r i m e n t a l s e t u p by
w h i c h the first photon echoes in the rare-earth terbium
were obs er ved and measured. The decays of echo intensity
w i t h l a s e r p u l s e s e p a r a t i o n at the e x t e r n a l l y a p p l i e d
f i e l d s of 45 KG and 25 KG w e r e p l o t t e d and f r o m these
plots a
h o m o g e n e o u s d e p h a s i n g t i m e of H O n a n o s e c o n d s
was calculated. Data at 15 KG were taken, but because of
the n o n - e x p o n e n t i a l b e h a v i o r of the decay, Tg w a s not
c a l c u l a t e d . At 0 KG, no e c h o w a s o b s e r v e d , i m p l y i n g a
decrease of the dephasing time b el ow 15 KG.
I
CHAPTEB I
INTBOHU CTIOH
The
first
observation
of
the
phenomenon
known
as
p h o t o n e c h o e s w a s r e p o r t e d by I . D. Ab e11 a , N. A. K u r n it,
a nd
S . R.
That
Hartmann
group
crystal
at
Columbia
performed
has
been
an
made
experiment
to
emit
i n t e n s e b u r s t of r a d i a t i o n ,
echo
...w .
In
predictions
direction
of
echo
Dicke's
theory
found to
decay
between
excited
hyperfine
exponential
of
separation,
same
group
derived
states"^].
a
function
pulses
one
of
the
interactions,
the
th e
a photon
were
would
inversely proportional
behavior
hy
short,
propagation
with
to T 2• the h o mo g e n e o u s
decay
a
ruby
polarization,
energy
echo
then
be
a
l e ve l s
separation
decay
time
of
superimposed
about
the
a function
hom og ene ou s
time
of the
the
echo
if split
upon
with
to the s p l i t t i n g . Thus,
as
echo was
for examp le
modulation
intensity
in fo rm a ti on
of
a
and
utilizing
The
dephasing
transit io n was nearly degenerate^-*,
by
a
ec h o
excitation
If
which
intensity,
e xpo ne nti al ly as
state.
1964^*^.
w h i c h w e w i l l ca l l
of "super-radiant
the
pro porti onal
echo
"in
in
spontaneously
analysis^]
later
of
University
the
period
f r o m the
of
p u ls e
li newidth
2
(which
is
inversely
transition
involved
ec h o
and
about
energy
by
isolated
the
can
ion.
The
several
(1)
reports
on
significance
of
the
information
the
non­
photon
echo
in o r g a n i c
^ ^.
observation
in the r a r e - e a r t h
this
photon
of
doped crystals
first
the
usually
linewidths
of
ec h o
of
The
in g a s e s
the
of p h o t o n e c h o e s
of
splittings
applications
reported
T2 )
extracted.
extract
in rare-ea rt h
thesis
measurement
be
inhomogeneous
been
m o l e c u l e s a n d
This
could
N um e r o u s
have
to
hyperfine
thus
broad
atoms.
experiment
the
levels
experiment
masked
proportional
experiment
and
terbium
is
due
to
factors:
The
linewidths
previously
been
of
studied
terbium
in
an y
transitions
detai l.
In
have
not
particular,
m e c h a n i s m s of ho mo ge n eo us broade ni ng are only speculative
at
this
time.
(2) This
splittings
ex per im e nt
or other
u n r e so Iv a b Ie by
could
directly
structure
other
of energy
w h i ch
experiment.
homogeneous
mechanisms
provide
In
the
and
of
the
second
thesis
basis
are
for
chapter,
inhomogeneous
each
any
hyperfine
levels on a scale
experiments.
C h a p t e r s 2 through 4 of this
theories
show
are devoted to the
un de rstanding
the
theories
broadening
presented.
Th e
third
and
this
of
the
chapter
3
contains
the m a t h e m a t i c a l
f o ll ow ed by a less
echo
arises.
echo
are
The
also
formal
laser
ion's
crystal
environment
complete
the
of the
setup
setup
conclusion
data,
drawn
the
about
lack
In
and
structure
(or
of
tra ns it io n
in terbium.
and
of how
the
fourth
chapter,
allowed transitions
in detai l.
the
in the
6 describes
each
Chapter 7 describes
run. The e i g h t h c h a p t e r
the
an
Chapter S describes
chapter
analysis
the
required to produce
are p r e s e n t e d .
experimental
relevant
discussion
intensities
structure
experimental
component
physical
calculated.
terbium
the
p r e d i c t i o n of p h o t o n e c ho e s ,
of
that
homogeneous
structure)
of
data,
a
contains
an d
the
linewidth
and
the
P^
4
CMA P T S S 2
H CH O0 S N E O W S
In an en se m bl e
each atom
i m © E © ® B M B © M S L I ME U I D TO S
of two-level
atoms,
the v avefu n c t ion of
can be w r i t t e n as a s up erp os iti on
?(r,t)
where
AMD
= c 1 (t)n1 (r)
+ C2 (I)U2 (T)
= ^ cn un
n
Eg,
I
cn (t) = <un (r) IY(.r,t)> and H u n = E n u n . Ic ^ I^ or
Ic2 I2 is the probability that the atom is in the lower
state or upper state,
Fo r
the
ensemble,
respectively.
the
general
density
operator
is
defined as
P = I P j l?j><?j I
JLflj— 2
j
where
P
is t h a t
f r a c t i o n of
atoms
w h i c h h a s the state
J
vector 7 j• Using
P = J Pj I I
equation I for
.V,
we
get
Iun X u m I = I Pnml*n><*ml
n m
where Pno has been defined as
Pnm =
I
= V
cn
Eg. 3
5
T h n s , for
an e n s e m b l e
of
t w o — level
atoms,
the
density
m a t r i x is
f
c / J ’o / C j )
I pj
P
j
For
/
t he
)
Eg . 4
V
Ic2 (J)I2
c 2 (j>cl0<j)
ensemble
of
two-level
P u
atoms,
is
the
p r o b a b i l i t y of f i n d i n g an a t o m of the e n s e m b l e in the i
state.
If N is the d e n s i t y
the
average
the
two
F or
density
of
the
pf a t o m s ,
popula ti on
t h e n N ( p ^ ^ - p 2 2 ) is
difference
between
levels.
ah e l e c t r i c
dipole
interaction Hamiltonian,
gaB(t), w i t h pjj = pj2 = ® an d pjj = pjl =
average
^
V =
e ensemble
<|i> = tr(pji) = p(pjj + P21^* But s i nc e pjj = P21°
<ji>
Ba.
P(P21
The m a c r o s c o p i c p ol ar i za t i o n
P = N
Taking
5
is given by
<p> o
the
respect to time,
partial
derivative
of
equation
2
with
we obtain
dp
at
J P j t l V j X T j I + I V j X T j I]
Eo . 6
6
1
o
i
From Schrodinger's equation,!?>=-
Taking
the
hermitian
conjugate
—
H |?>
an d
Ba.
noting
that
H
7
is
he rm it i a n
I H|^>]t
[!?>]* = [-
&
.
i
< Y I= + - <?|E
h '
Using
equations
9p
--
Eq. 8
7 and 8 in eq uation 6, we get
i
=-
- J Pj[ n|?j><?j I + iVjXVj Ih ]
Theref ore,
9p
—
1r
=—
i
— I H, p J
at
■
Eq. 9
a
if H is the s a m e for all. I?>.
The H a m i l t o n i a n
describing
the
internal
e n e r g i e s of
t h e . a t o m s an d the i n t e r a c t i o n b e t w e e n the a t o m s and the
electric
field
same
all
wh ich
fo r
a to m s ,
describes
= E n u n ). V
is
is. H = H q + V. Here,
the
the
H 0 is
the
internal
dipole
H is t a k e n to be the
unperturbed
energies
interaction
of
the
Hamiltonian
atoms
Hamiltonian
(®0un
(V =
7
— E(t)ga). The
matrix
r epr es en tat io ns
of
the
operators
H 0,
p, and p are
f Ei
fPu
H0 =
,
\°
E2j
Thus, equation
P 12)
Z0
P=I
„
,
21
and
P22J
p =
\
0J
9 for the t i m e e v o l u t i o n of the d e n s i t y
m a tr ix yields the fo l l ow in g fo rm ula s
8 Pzi
i
= " - [ H,p
at
J 21
h
i
= " " [ (H0P)2I + (Vp)2I - (PH0)2I " (PV)21 ]
a
= --- [ E 2 P 21 - Eftlpp 11 - E 1 P 21 + E( t )Pp 22 ]
a
3 p 21
----- -
-
i&)0 p 2 1 + i P E f t l f p 1 1 -
P-2 2 ) / a
Eq .1 0
at
where
M q = (E2 -
Using
the
E 1 ) / h.
normalization
c o n d i t i o n P 11 + P 22 = I and
using the same approach given above,
a
—
at
we
find
6
(P l I -
P2 2 I =ZipEftlfp2 1 -
P 21
)
Ba.11
8
In
the
absence
of
an electric
field,
equation 10
has
the
solution
p 21 = P21
At
e
itoO t
E g . 12
this
point
the
atoms
has
to
included.
would
expect
be
effect
P 21
th at
of r a n d o m
When
would
app roach zero as the relative
to r a n d o m
random
inelastic
effects.
Thus
is t u r n e d
decrease
phase
collisions
for
B.(f)
dephasing
of
the
off,
we
and
eventually
coherence
is lost due
with
the ensemble,
phonons
an d o t h e r
equation 12
should
become
P21
where
p21
Tg
®
is
("itoOt
the
dephasing
m o d i f i e d to include
9p21
---
tZT2 )
time.
Equation
10
is
then
the dephasing and becom es
= - Im q P2I + ipE (t ) (P u - P22)ZA " P2I/T2
Eq «13
81
In a s i m i l a r m a n n e r ,
the
to
re la xat io n
its
constant
of
the
equilibrium
e q u a t i o n 11 is m o d i f i e d to i n c l u d e
po pu l a t io n
value
difference
-r p 2 2 )0
N(p^^
with
- p2 2 )
a
time
r.
8
— (pll"P22)
8t
2 i p E (t )
^
(pl l " P 2 2 ) " ( pl l " p22)o
= -------- (p2 1 " p21 ) " ------ =
----- 1
-- " T -----h
v
Ea .14
I
For
an e l e c t r i c
defining
the
cr1 2 (t)
we
obtain
field
of
the
form
E(t)
= E cos <ot and
slowly varying variables
from
Eq .15
= P i 2 <t )
= O2 I
equations
13
an d
14
the
optical
Bloch
e q u a tions
i(iE(t)
- i(M-(Oq )* 21 + —
---(P11- P 2 2 )
*21
2h
T0
ifiE(t)
— <P11-P22)
at
&
Eo .1 6
- _
(P l l " P 2 2 )“ (Pll“ P 2 2 )i
’(* 21_ *21
)
Ba .17
where
terms
Equations
which
vary
as
e + 2 i<ot
have
been
dropped,
16 and 17 yield
0T2 (Pll” P22)o
E q ■1 8-
Im * 21
I + Cm-(O0 )2T2 2 + 4Q 2T2 t
I + Cw-(O0 )2T2 2
Eo .19
(P H - P 2 2 ) - (P h - P22 ^
I + C(O-W0 )2T22 + 4fi2T2
Combining
p o l a r iz at i on
equations
5,
6,
is found to be
P = HCcr21 e " i6>t + O 2J e iwt)
and
15,
th e
macroscopic
10
w h i c h can be r e w r i t t e n as
P = 2|i (Re [<?2 ^ (t ) ] costot + Im [»21 (4 ) i sintot)
Th e m a c r o s c o p i c
atomic
p o l a r i z a t i o n is r e l a t e d to the c o m p l e x
s us ce pt ibi li ty X by
P = Re [80X E 0 e
So the
found
Eg .20
]= E 0 (6 0Re [X] co stot + 80 Im [X] s ineit)
i m a g i n a r y part
from
equations
of the
18, 20,
complex
and 21
Eg .21
susceptibility
is
to be
P2T 2A N 0 _____________ I_____________
Im X
e0h
where
The
the
I + (to-to0 )^T2 ^ + 4 Q ^ T 2 t
pr ece ss i on
population
equation 19
frequency
Q is
difference
per
defined by 0 — pE0 /2h.
unit
volume
from
is
I + ((D-U0 )2T 2 2
AN = AN
I + (to—u 0 )2 2 2 + 4 Iii2T 2T
Thus ,
Im X ~ AN
wh ich has
the
I + 4iT ( f - f ^2™
T 2
a full w i d th
absorption,
which
at half m a x i m u m Af = (nTj)” 1 . Thus
is
proportional
to
Im
X,
has
a
11
full-width
at half m a x i m u m
of (nTj)
the h o m o g e n e o u s l y b ro ade ne d
of a t o m i c
fli ps ,
collisions,
this
case
have
the
the
coherence
the
occurs.
different
this
are
co nsidered
homogeneous
in
linewidth
conventional
the
photon
directly
measuring
linewidth
can be
and
broadening
a
spread
the a t o m s
of
each
in
due
atom.
larger
where
the
to the
Usually
than
the
feature
in
W h e n inhomo ge neo us broade ni ng
is
sp ec troscopy
lineuidth.
echo
T2
etc. In
in ho mogeneous
much
ab sorption
used to study h o m o g e n e o u s
3.,
le v e l s,
so it is the d o m i n a n t
the absorp tio n spectrum.
chapter
s u c h as s p i n
case
of
is
to the loss
E2- E 1 . In the
e n vi r on me nt
frequency
called
in di stinguishable
reflects
transition energies
spread
present,
energy
broadening
crystalline
effects
to o t h e r
distinguishable,
This
individual
to r a n d o m
tr ansition
are
I i n e w i d t h . is due
transitions
atoms
same
ato ms
due
This width,
will
experiment
from
calculated.
As
which
is
the
cannot
be
shown
in
a method
of
be
homogeneous
12
CHAPTBE
s
pm©TOT ECHO
F o rm at ion of the Echo
Without
an
externally
applied
H a m i l t o n i a n for an n— atom en se mbl e
H
H.
radiation
field,
the
can be w r i t t e n a s ^ ^
H j iHj3
H q represents
the
translational
e n e r g y of the e n s e m b l e
a nd the i n t e r a c t i o n e n e r g i e s b e t w e e n at om s . It w i l l not
be
discu ssed
ERj g
is the
eigenvalues
of
here.
internal
+gE. We
energy
can write
of
the
the j — —
internal
atom
and has
energy portion
the en sem bl e e ig en fu nct ion as
V =
where
atom
j=l
2
3
4
5
6
• ‘
( +
+
—
+
—
—
***)•
a + or being
sign
in the
)
j— — p l a c e
refers
in the excited or ground state,
to
the
j— —
respectively.
Rjg is an o p e r a t o r a n a l o g o u s to one of the P a u l i sp i n
operators.
It o p e r a t e s on the p l u s
j— — p l a c e a n d has
or m i n u s
the e i g e n v a l u e +1 / 2
s i g n in the
or - 1 / 2
depending
13
on w h e t h e r
the
atom
Dicke
introduces
three
Pauli
is in the
t h r ee
spin
Rj
e x c i t e d or g r o u n d
operators
operators
and
analogous
having
the
state.
to
the
following
propertie s :
j
j
• • )
*jl(
=
I
-(
2
• )
• •
i
. . . + . • • ) = + -( ' * . + .. . • )
Rj2(
B a . 22
2
I
r
j3(
. . . + . • • ) = + -( * * • '+ e e • )
2
The H a m i l t o n i a n de scribing the interac ti on b e t w e e n the
atoms and the
electric
field
is
j
S in ce
|ij
is
an
od d
operator,
elements
Hj = M x R j 1 + yRj 2 )
whe re
p is a constant.
it ha s
only
off-diagonal
14
Thus,
the
energies
atoms
of
to tal H a m i l t o n i a n d e s c r i b i n g
the
atoms
and
the
the i n t e r n a l
interaction
between
the
and the electric field is
Uuo0R j 3 ~{2 ji(EsR j 1 + Ey Rj 2 ))
H = J
Eo .23
j
This
form
of
the H a m i l t o n i a n
a I ^ , so that an electric
E s = E cos tot ,
w o u l d cause
was
chosen
by
Abel la,
et
field of the form
Ey = E sin tot
t r a n s i t i o n s b e t w e e n the e x c i t e d and g r o u n d
st at es, w h e r e a s the o p p o s i t e p o l a r i z a t i o n w o u l d not. As
that g r o u p state d,
general
If,
this c h oi c e ,
though arbitrary,
g iv es
results.
following
define
the
example
a pseudo-electric
electric
of
Abella
et aI,
dipole
moment
and
we
now
a pseudo­
field by
p = {2 n J (Rj1 X + Rj 2 y + Rj 3 z )
Eq .24
5 = E s X + Ey y -
Eq .25
and
respectively,
simp ly as
H = - P ’ 5
then
(Eto0 / V2 (i ) z
the H a m i l t o n i a n
can be
expressed
very
15
The fo r mu la for the time
d<p>
i
dt
h
dependence
of any operator
is
H , p
Since
p
zero,
and w e are left w i t h the f o ll ow ing
We
is a .constant
of
the motion,
the second term
is
formula:
can solve individually for each component
--- -—
dt
----< [ H . p J >
h
Equations
23,
24,
commutators
for the
25,
, etc.
and
26,
Eq .2 6
together
with
the
R j 's , i. e .,
[Rj o ' R j ' p ] = iRj r 8 j j ' 8aPr
where
e a p y = + I d e p e n d i n g on w h e t h e r
odd p e r m u t a t i o n of the i n t e g e r s ,
more
indices
are repeated,
will
apy
is an e v e n or
and 8 a py = 0 w h e n t w o or
be used to find d <p>/dt.
16
[h ,p x] = H P x - P s H
= [lrha,0 R j3 -f 2p(ExRjl+EyRj 2 )]]o[f2^
i
Ry i ]
j'
[Ithm0 R j3 - V 2 p ( E 2 R jl+E y R j 2 )]]
j'
[ H <P x I = ^ 2 ^ 1
j
Ih(I)0 ( R j 3 R j fl- R j 11R j 3 )
jj'
— "/2 p.Ex (Rj ^Rj P1-Rj P1 Rj 2 )
- /2 pE y (Rj2 R j p i - R j , i R j 2 )]
[H, p x ]
=/2pJ [ Ihio0Rj 2 + IZpEy R j 3 ]
j
= / 2 p ( - i | zpy + ISyP 2 )
-— —
at
=
- ( [ h , P x ] ) =/2p-< [-iS zPy + i S y P z ]>
h
a
/2p
=—
d<p >
--- L _
dt
I < P y > ? z “ <Pz>&y 1
VIp
_
= _
[ <p > s § ]2
17
The p r o c e d u r e s u s e d to o b t a i n d < p y >/dt and d < p z > / dt are
identical
results,
to
we
d<p >
the
procedure
tfn
This
field
s
£.
It
is
describes
the
applied
allows
rotating
electric
d<p>
8<P>
dt
---dt
equation
of
motion
transverse
us
to vi e w
about
used
a
to
we make
the system
from
a
a
at fr equency <0, the frequency
field.
+ 1» x <p>
to = toz. In the
rotating
p processing
the f o r m a t i o n of a pho to n echo. First,
of reference
the
a vector
basic
t r a n s f o r m a t i o n w hi ch
(Ex)
the
_
<?>
equation
describe
where
Combining
h
dt
of
above.
obtain
----
frame
used
Ba.27
rotating
frame
of
reference,
field E(xcostot + y s i noit) is
the
stationary
so eq uat io n 27 b ec om e s
3 <P>
----dt
_
= Y <p > 2
_
(-«0+ to) _
[Ex + -------- Y
w h e r e y .= -/2 Ji / h. If we call
z]
the q u a n t i t y (-to0 + <i>)/y the
18
effective
field
€ z e f£>
describes
the v e c t o r
whichever
is dominant.
It
is p o s s i b l e
a
slightly
different
equation
simply
to i n c l u d e
effect
of c r y s t a l l i n e
replacing M q with <i>0j - that is,
its ow n rate
different
the
of preces si on
transition
energy
co rr esponding
ht»0
due
to
to
its
environment.
8 <P >
-----at
T he
this
<p> p r o c e s s i n g a b o u t Es or £ z e ffZ»
field i n ho mo g en ei t ie s by
each atom has
then
_
= Y <P > X
sequence
_
[Ex + ? zef f i z ]
of e v e n t s
leading
Eq.28
to the p h o t o n e c h o
is
as f o i l o w s :
1) I n i t i a l l y ,
the s y s t e m
no exter nally applied
is in e q u i l i b r i u m ,
electric
field,
and <p>
th er e is
is directed
a l o n g the z axis.
2) At
t=0,
an e l e c t r i c
field
E
>>
£ z e ££
is
applied
and equation 15 becom es
9<p>
-----81
Thus,
_
= y<p>
2 Ex
<p> processes
y E T = jt/2,
where
T
about
is
the x axis w i t h per io d 1/yE.
the
amount
of
time
the
If
electric
19
field
is applied,
y axis
then after
the pulse,
<p>
is along
the
( f i g u r e I).
z
Figure
I. The
3)
Without
governed by
d<P>
:----
and
<p>
(f i g u r e
electric
pr ec es s i o n of <p> about
externally
applied
field
the equation
= r<p>
1 S ze f f j 1
pr ocesses
2).
an
x.
Since
about
each
dipole m o m e n t s
the z axis wi t h period
atom 's
quickly
period
is
YSzeffj
different,
the
get out of phase wi t h one
a n o t h e r and the m a c r o s c o p i c p o l a r i z a t i o n e x p o n e n t i a l l y
dec
<p>
is
20
z
Figure
2. The
4)
< p > to
time,
again
the
At
of <p> about
t = T , another
process
about
the
pulse
z axis
z.
is
applied,
(figure
3). This
y E T = n, so that e a c h < p j > is r e f l e c t e d a c r o s s
x axis,
axis
pr ec es s i o n
back
to <pj>
to the
x-y
plane.
was 6 j before
second pulse,
the
angle
the
If
the
angle
s e c o n d pul se ,
from
3. The
is n-@j.
pr ec es s i o n of <p> about
the y
t h e n a f te r
x
Figure
the
x
causing
21
5)
After
moments
again
Y 5 zeffj
along
the
second
process
(figure
4).
processes
through
f r o m the y - a x i s
about
A t t =
the y axis again.
pulse
the
2t ,
In the
is
z
the
time
removed,
axis
the
with
period
<Pj>'s w i l l
interval
x,
dipole
all
be
each Cpj >
an a n g l e G j . Thus at t = 2x,
the a n gl e
is (Jt-Gj ) + G j = jt.
x
Figure 4.
The
visual
involves
The
p rec es si on
example
imagining
ga t es
are
o pen ed ,
t he ir
respective
that at t= r , all
of
this
horses
and
at
the
s pe ed s,
the horses
of <p> about
sequence
a racetrack.
horses
getting
circle
out
z.
of
ev e nt s
At
t=0,
the
track
of phase.
turn around and start
the
at
Suppose
running
in the o t h e r d i r e c t i o n . At t = 2 r , the h o r s e s are all b a c k
at
the
was,
gate
the
neck-to-neck
less
distance
because
the
it had covered.
slower
the
horse
22
Damp ins of the Phot on E o ho
The dephasing
of
the dipole m o m e n t s
due
to each atom's
■'
unique
That
trans iti on
is,
after
frequencies
frequency
a
time
is
8 t,
a
d e te r mi ni sti c
two
atoms
differing by S m q will
with
•
*
process.
resonant
have a phase difference
8 M q S t . It is the d e t e r m i n i s t i c n a t u r e of this d e p h a s i n g
which
allows rephasing,
If,
however,
t h e re
dephasing
mechanism
amplitude
will
moment
be
and thus
is a l s o
such
as
greatly
the photon echo.
a stochastic,
random
collisions,
diminished.
is d a m p e d at a rate Tg w h i l e
rephasing along w i t h
all
the other
echo.
the
echo
Consequently,
factor
of
initial
jt/2
expC^r/Tg)
pulse
and
if
the
Each
echo.
is
the
atom's
echo
dipole
it is d e p h a s i n g and
dipoles to produce
amplitude
2t
or r a n d o m ,
the
the
is s m a l l e r by
time
between
a
the
23
Direct ion of Echo Propagat ion
A b e l l a t21
echo
in
found
the
the
r a d ia ti on
di rection k due
intensity
of
to excita ti on
the photon
pulses
with
w a v e v e c t o r s k 1 and k^ to be
I
,
= — N Iq
4
I(k)
where
the
Iq
is the
direction
sample.
The
between
Thus,
beam.
ra diation
k
a to m s .
intensity
of
a single
and N is the n u m b e r of
squaring
if the
first,
I e x p [i (k+ k-2k)r ] |
^
av
of N arises because
atom
atoms
of the
in
the
coherence
I(k) w i l l be a m a x i m u m w h e n k = 2k
second
laser
beam
In order for
- k.
is at an angle @ from
t h e n the e c h o w i l l be at a n g l e 6
interference
effects
from
in
the
the
second
to be negligible,
the condi ti on
L
[ I / (1-e2 )
where
this
so
L
is the
experiment,
the
-
1]
length
are
crossing
angle,
1°
® = --1.5
the
the
is
the angle
n rad
x ----- = 0.012
180°
effects
is being
lost.
<< I.
E q .29
crystal,
must
be met.
and 1/X = (1.5)°(20560
angle
negligible
so. interference
intensity
of
L = 0.2 cm
con di tio n on
effects
/ X laser
such
that
& < < 0.013
in the
crystal
the
For
c m - 1 ),
interference
rad.
With
a I0
is
rad
are not negligible,
and some echo
24
Laser
Intensities
From
Ab e11 a, e t a I ^^ , the p o w e r
a n/2 p u l s e
P/A
where
square
Requir ed For n/2 And n, Pulses
=
density
required for
is
n^ech/ S X 3T 2 W
t is
of
pr ob ab il i ty
the
pulse
th e
index
of
Eg.30
duration
of
tr ansition
(S6 IO
*
refraction.
from
the
sec),
and
excited
s
W
to the
i s the
is
the
ground
state.
The
oscillator
strength for
the parity - forbidden ?Fg
r
-
transition
empirical
in
data
for
terbium
the
is
calculated
absorption
from
the
coefficient
and
inh om og en e ou s I i new i dth using the f o r m u l a ^ ^
I S m c s rt
f = :-----2
—
Tte2 I
where
AE
concentration.
constants
at
a AE
----:
(n2 + 2 )2
n is the
coefficient,
n
-----
N0
i n d e x of r e f r a c t i o n ,
is the
The
linewidth,
a is the a b s o r p t i o n
and N q
is the T b ^ + ion
ratio
of
value
of
the
fundamental
the be ginning
of
the equation is 8.21 * I O ^
cm~^/eV m m - * . Thus
( I . 5 ) ( 3 5 .8mm- 1 )(5* 10- 4 eV)
f = (S1I 6I O 1 ^cm 1 Z e V m m - 1 )
..... ' ■■
— (1 .52 + 2 )2 (1 .3 6 ° 1 0 2 2 cm- 1 )
f = 8.9 6 IO - *
25
Fr o m
this value
pr o ba bi lit y
of oscillator
is easily
strength,
ca lculated using
the
transition
the formu la
S jt2O 2 e2 f
W = --------me
where
o is the
energy
difference between
states
(20560
cm 2 ) and e2 /mc = c * Bohr radius = r 0 c. So
W = 8n 2o 2 r0 cf
W = 2.5/sec
From
e q u a t i o n 30,
pulse
the p o w e r
required
P/A
=
s e c t ! onaI
area
of n(50p)2 , the re quired laser p o we r
(4»
2.5
KW.
2.5KW)
The
10
Watts/jim2 . For
p ov er
KW.
required
for
a
beam
for
is
be
0.32
density
the
n
a n/2
cross
-
would
pulse
is
26
CBfAFTSS 4
TBBBI¥n STBWCTBFBl AHB TBAMSITSOHS
The
Tb ^ +
ion
has
62
electrons
in
the
l s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 10 4 s 2 4 p 6 4 d 1 0 5 s 2 5 p 6 4 f 8 .
lines
originate
configuration.
4f
from
Because
configuration
en v i r o n m e n t
lines
are
In
by
nearly as
accordance
shielded
and
within
from
5p closed
sharp as those
with
spectral
Hand's
the
the
isolated
r u l e ^ 2 ^,
the
in the
crystalline
shells,
of
4f
the
the e i g h t o u t e r e l e c t r o n s
are
the 5s
transitions
Th e
shells
spectral
atoms.
lowest
energy
s t a t e has S = 3, L = 3, a n d J = 6 . T h u s , the g r o u n d s t a t e
of
T b 8+
is
^F ^ . The
experiment
hc°(20,560
Because
are
these
electric
with
dipole
for
ah
of odd
extent)
of
(Al
transitions
+ 1 ),
#
t r a n s i t i o n s ^ 8 ^.
transitions
Ca l c u l a t i o n
^D^
forbidden
dipole
perturbation
small
is
used
this
echo
energy
of
c m - 1 ).
parity
electric
transition
with
is
the
transition
of c o n f i g u r a t i o n mixing,
the y
The
believed
parity,
two
mixing
strengths
must
be
physical
to
be
the 4f
adjacent
which
a m o n g 4 f states
an
forced
origin
external
orbitals
1- v a l u e s
requires
is difficult
of
(to
a
2 and 4.
a kno w l e d g e
to determine.
27
However,
s e paration
c a n be done
entirely
the L i Y F ^ :Tb ^
S4
of
symmetry
respect
rules
transition
b e t w e e n T 1 states
transition
used
for
forbidden
transitions
theoretical
grounds.
In
the Tb^"*" ion is at a s ite h a v i n g
selection
t r a n s i t i o n 7F g f 2 ~
and
on g r o u p -
crystal,
(with
allowed
to
the
symmetry
crystal)*-1 4 ^.
give
an
allowed
a n d T 2 states.
5 l)4 r i is n ~ p o l a r i z e d .
in this experiment.
The
Thus,
This
n
the
is the
28
CMAPTSS S
EEPBBIEISOTAL SETUP
A diag r a m
of
the
setup for
the
photon
echo
e xperiment
is s h o w n in f i g u r e 5. T w o p u l s e d n i t r o g e n — l a s e r - p u m p e d
dye l a s e r s are u s e d to s t i m u l a t e o p t i c a l
the crystal
L i T F ^ : T b ^ + . The
transitions
ti m i n g b e t w e e n
the
two
in
lasers
is contr o l l e d el e c t r o n i c a l l y and scanned a u t o m a t i c a l l y by
a
computer.
The
crystal
temperatures
(1 . 4 0 K
supe r c o n d u c t i n g
ma g n e t
to
6 0KG. T h e echo,
stimulating
scattered
detector.
This
and
which
is m u c h
pulses
light
to
techniques:
spatial filtering,
cell.
4°K)
must
produce
less
be
liquid-helium
surrounded
by
a
fields
up
intense
isolated
t h a n the
from
saturation
by u s i n g
three
the
of
the
kinds
of
p o l a r i z a t i o n discrimination,
and t e m p o r a l
E a c h of t h e s e
to
can
prevent
is a c c o m p l i s h e d
discrimination
Pockels
cooled
coil
which
laser
laser
-
is
d i s c r i m i n a t i o n using
techniques
is d i s c u s s e d
a
in
d e t a i l later. Th e e c h o is d e t e c t e d by a p h o t o m u l t i p l i e r
tube
and
the
electrical
boxcar averager whose
signal
from
the P M T
gate is adjusted by
is sent
the
to a
computer
to
r e m a i n on top of the e c h o d u r i n g the s c a n of the lasers'
timing. The
averaged
signal
from
the boxcar
is
sent
to an
29
a n a l o g — to — d i g i t a l
acquisition
laser
converter which
computer.
pulse
is p a r t
T h u s , a plot
separation
is
of
echo
obtained
of
the
data —
intensity
vs.
somewhat
automatically.
of
The
following
the
lasers,
electronic
chapter
the
contains
discrimination
and c o m p u t e r
control
detailed
descriptions
techniques,
and
of the experiment.
the
30
lens I :
f=1.3 3 m
lens 2 :
f = l .33m
dy e
laser I
X /2 plate
p o larizer I
lens
N2
laser
I
---- ik
3:
33cm
N2
laser
2
L i Y F 4 :Tb
crystal
echo
lens 4:
del ay e d
trigger
generator
25cm
aperture
Babine t - S o l ieI
compensator
p o l arizer 2
lens
d/ a
c onverter
5: 30mm
25|i
pinhole
PDP-11
c omput er
lens
spatial
filter
6 : 36mm
a/ d
converter
X / 2 plate
I
polarizer
boxcar
averager
Pockels
phot om n l tipi ier
tube
3
cell
p o larizer 4
lens
7: 20 cm
spectrometer
Figure
5.
Setup for photon echo experiment
31
CffiAFTSS S
C O H P O M E Q T S OF S Z P S S Z H B M T A L SBTffiP
Lasers
The
lasers used
tunable
dye
pulsed
lasers
lasers
repetition
nitrogen
pumped
with
rates
lasers
stability
in this e x p e r i m e n t
by n i t r o g e n
pulse
widths
6 Hz.
of
typically
relative
to
two
laser
on the o r d e r of ~5 O n s e c / h o u r .
shot-to-shot.
impediment
to the
jitter was
to the
eliminated
at Bell
of
may
t hese
accuracy
by
synchronizing
line
a feedback
Laboratories
[I?],
5 nanoseconds
and
be
the
the
pu l s e .
First,
as
large
e cho
circuit
is a
as
20 n s e c
a
serious
decay
data. The
l e v e l (5 n s e c s h o t ­
of
long term
designed by
& block
there
were
triggering
circuit
of the t r i g g e r p u l s e f r o m
timing
Second,
problems
of
with
t i m e of f i r i n g of the
f r e quen c y E1 6 ] ^ The
is s h o w n in f i g u r e 6 . The
Part
in the
r e d u c e d to an a c c e p t a b l e
to- s h o t ) by
lasers
Both
which
are
trigger
laser
timing
They
problems
is a l o n g - t e r m
in the
drift
lasers.
The. t h y r a t r o n - t r i g g e r e d
have
the
of
there
jitter
are H a n s c h - type I*5]
diagram
of
functions
the
tw o
drift was
Chip
the
Carter
circuit
as f o l l o w s ^
the t r i g g e r g e n e r a t o r
I'
delayed and c o m p a r e d w i t h the actual light pulse
from
is
the
32
laser.
edge
If
photodiode
of the d e l a y e d
fired
and
the
too
early
an u p / d o w n
added
to the
after
the
delay.
In
keeps
the
generator's
pulse
trigger,
relative
counter
to
this
laser
in
the
before
laser
the
the
the
trailing
is k n o w n
trigger
to hav e
generator
pulse,
the p r o g r a m m a b l e
If the photodiode
edge,
manner,
the
increments
trigger pulse.
trailing
comes
counter
drift
pulse
comes
decrements
compensator
synchronization
delay
with
the
the
circuit
trigger
pulse.
progr ammable
delay
pulse from
trigger
trigger
to laser
de I ay
up/down
counter
photodiode
pulse
comparator
Figure
circuit
6.
Block
diagram
of
laser
d rift
compensator
33
The
laser
layout
of
consists
of
1) a dye cell
optically
light
a dye
laser
containing
pumped
by
the
Is s h o w n
7 D4 T M C o r g a n i c
nitrogen
and w h i c h fluoresces
in f i g u r e
7. E a c h
dye w h i c h is
laser's
ultraviolet
in the w a v e l e n g t h range 460 nm
- 517 nm,
2 ) a d i f f r a c t i o n g r a t i n g w h i c h acts as one m i r r o r of
the c a v i t y
wavelength
and w h i c h
range
can be
from
rotated
the
dye's
to
select
active
a narrow
range
(n X
=
2 d sinO) ,
3)
a partially
cavity's
other
coated
exit
mirror
acting
as
the
mirror,
4 ) a t e l e s c o p e b e a m e x p a n d e r w h i c h e n l a r g e s the b e a m
to cover
a larger portion
of the
5 ) an a m p l i f i e r dye cell
dye
cell
and w h i c h
grating
acts
telescope
beam
expander
grating,
identical
to a m p l i f y
osc.
dye
cell
and
to the o s c i l l a t o r
the light.
exit
mirror
amp.
dye
cell
dye
laser
light
laser
beam
Figure
7.
N i t r o g e n - l a s e r - p u m p e d dye
laser.
34
Lenses and Waists
The
were
focal
chosen
crystal
and
lengths
to
of
provide
allow
for
lenses
I » 2,
and
a beam
waist
of
a
crossing
f o r m u l a relating b e a m
w a i s t at
to
collimated
the
waist
of
the
the
~ 25
angle
focal
beam
3 in f i g u r e
of
point
before
p
5
in
the
~ 1 0.
The
of
a lens
the
lens
is(I*]
fX
(0of
Ba.31
Ttm0
where
f
is
the
focal
w a v e l e n g t h of the light,
lens.
This
formula
in the c r y s t a l ,
by
measuring
laser b e a m
length
of
the
lens,
X
is
and W q is the w a i s t b e f o r e
is reversible.
To
calculate
the
the
the
waist
the b e a m w a i s t in the dye l a s e r w a s f o u n d
the
far-field
angle
of
divergence
of
the
and using the formula^®-*
X
“ ° = ----- T
Jtnt an®
where
n is the
index of r efraction of air.
At X = 478 nm,
B w a s m e a s u r e d to be 0.06*, so W q in the l a s e r is 140 jim.
For
this
Mq , the confocal p a r a m e t e r
ntoO 2
.z = '■■'■■■— = 0.13
X
Bi
rig]
is1
35
so
the
be
>
focal
length
0 0.13
5
m
of
=
lenses
0.65
m
2 was
I and
for
the
beam
c o l l i m a t e d a f t e r the lenses. A v a l u e
was
chosen
for
optimum
b eam
the
of
after
I and 2 are a>£ = 1.5 m m
f o c a l l e n g t h of lens 3 w a s
the c r y s t a l
size for
the
to
be
to
well
of f^ = f% = 1.33 m
after
e q u a t i o n 31,
lenses
waists
size
required
the leases.
collimated
laser
Using
beams
for X = 486 n m . The
c h o s e n to p r o v i d e
a waist
in
of a p p r o x i m a t e l y 25 p, w h i c h is the o p t i m u m
the echo experiment.
The
lens
chosen had. a focal
l e n g t h of 33 cm, w h i c h g i v e s a w a i s t in the c r y s t a l of 35
p. F o r a I 0 c r o s s i n g angle,
the d i s t a n c e b e t w e e n the t w o
beams
tan 0 = 6
at
lens
consistent
overlap
The
3 is
with
so that
focal
x =
the
they
length
fg
requirement
can be
of
ram.
t hat
spatially
lens 4 w a s
This
the
value
beams
an
aperture
could
select
the
not
separated.
chosen
such
that
d i s t a n c e b e t w e e n the b e a m s a n d the e c h o is l a r g e
so
is
echo,
and
yet
the
enough
small
e n o u g h so the e c h o an d s e c o n d l a s e r p u l s e b o t h are able
to
pass
through
compensator,
of
the
th e
and Pockels
collimated
distance b e t w e e n
openings
in
cell. W i t h
f 4 = 25
beams
the b e a m s
after
len s 4
is 0.4
cm.
the
polarizers,
cm,
the waist
is 0.1 cm
and the
36
The
len s
spatial
have
filter
lens
approximately
(lens 5) and its r e c o l l i m a t i n g
the
c o m b i n a t i o n does not change
7 has
the
a focal
slit
of
the
grating of the
x =
Thus,
distance
same
spectrometer.
cm.
size
of
At
its
The b eam
so
focal
size
at
the
Lens
plane
the
is
first
is
slit to grating distance
— --' .
lens 7 to slit distance
the
length,
the b e a m waists by much.
of 20
spectrometer
focal
the
( 1 0 0 c m / 2 0 c m ) @ l . l m m = 5.5 mm.
o (size before
spot
on
the
lens
7)
grating
is
37
P o l a r i z a t i o n d i s c r i m i n a t ion technique
The
polarization
discrimination
8 , Th e
technique
linearly polarized
direction
5D ^ P i
is a b s o r b e d
transition
transition,
crystal's
of
E
due
to
the
the
polarization
after
The
of
amount
crystal
the
along
forms
crystal,
the
the
X /2
c-axis,
c-axis
from
a
that
discrimination
blocking
the
crystal.
If,
absorbing
one
com p o n e n t
of
a slight phase
the
of both
and
a
w i l l be
c o mponent
The
echo's
only
the
the
can
the
echo's
two
be
laser
used
to
the laser beams.
is l i m i t e d by
of
also causes
the
since
of
n
then after
Because
polarizer
the echo while
along
plates)
of
a
a X / 2 plate
crystal.
echo.
is
of the t w o b e a m s
the
the
the
of
absorption
of
in one
polarization
is rotated by
of 0 f r o m
c-axis
birefringence
selectively
If
is
Th e
polarized
a discussion
different
of
in
light
crystal
be
polarized
L i Y F 4 : T b 3+
is
selectively pass
light
terbium
selective
will
light
polarization
beams
for
the
transition
an echo.
the p o l a r i z a t i o n
along
absorbed
A
the
the
and p r o d u c e s
only
to an a n g l e
crystal,
rotated
is,
before
appendix
polarizer
the
of
when
only
c - a x i s is absorbed.
laser b e a m s
(see
that
works
so that
was
^ 1 9 J. It is i l l u s t r a t e d in
b r a i n c h i l d of Dr. R u f u s C one
figure
technique
in
the
the amount
addition
E— field,
shift b e t w e e n
the
to
the
two.
38
then
the
beams
polarized,
The
bnt
analyzing
after
will
regardless
Soliel
compensator
of
the
of
will
its
must
birefringence.
compensator.
Once
polarizer
completely
there
is
not
be
used
the
linearly
also
be
able
to
Therefore,
to
linearly
light
is
polarized,
block
the
f luctuating
block
the
a Babinet-
compensate
A p p endix A contains
to
however,
will
not
angle.
be
elliptically
can
crystal
i n s t e a d be e l l i p t i c a l l y p o l a r i z e d .
polarizer
beams,
crystal's
the
for
the
a d i scussion
changed
an
laser
from
analyzing
beams.
birefringence
(If,
caused
by r a n d o m s t r a i n s in the c r y s t a l , the l i g h t w i l l s till be
slightly
completely
500
elliptically
blocked.)
improvement
polarized
Using
thi s
in echo/laser
and
will
technique,
can be
not
a factor
obtained.
be
of
< I /T h e laser pulses are blocked
E L N k w ith a 50% loss of echo power.
The analyzing polarizer
is perpendicular to
.
The echo has same polarization as absorbed light.
6 / / -
The crystal selectively absorbs light polarized along the
c-axis
e ffe ctive ly rotating the polarization
of the laser pulses to 45' from vertical.(tan 6 = =
VLVLT )
w
'O
L
B abinet-S oleil compensator.
0
c - axis o f crystal is vertical.
c a
Input light from lasers is linearly
polarized at angle 6 from vertical.
y
Figure 8 . Polarization discrimination technique
40
Spatial
By
filtering
combining
spatial
a
filtering,
t o-laser
ratio
crossed-beam
a significant
at
the
geometry
with
improvement
detector
is
tight
in the echo—
obtained.
The
laser
b e a m s are c r o s s e d at an a n g l e of ~ 1 ° w h i c h is the o p t i m u m
angle
for
echo
and
being
yet
because
of
itself
is
a ble
not
to
use
have
the
interference
very
an
aperture
echo
effects.
effective
in
the
lasers.
placed
at
The
spatial
the f o c a l
filter
point
of
select
intensity
The
the
a convex
filter
echo-to-laser
scattered
consists
the
destroyed
spatial
imp r o v i n g
r a t i o at d e t e c t i o n by e l i m i n a t i n g
to
light from
of
a pinhole
lens.
Only
light
p a r a l l e l to the o p t i c axis of the lens is a l l o w e d t h r o u g h
the
pinhole.
The
most
of
the
echo
is
given
length
of
1.1
mm,
the
lens
the
and
pinhole
still
is
the
waist
filter. For
and
X =486
p. A s e c o n d
lens
which
passes
D = 2 X f /M q , where
and M q
mm,
of
light
spatial
30
is 25
blocks
99 % of the
f is
the
f ocal
of the c o l l i m a t e d
this experiment,
nm,
so
is u s e d
the
Wq =
pinhole
to r e c o l l i m a t e
beam.
Removing
in
by^®-^
the
f =
diameter
the
scattered
the
b e a m before
diameter
the
stray
the p i n h o l e
laser
effectiveness
light
of
the
c a u s e s a f a c t o r of 5 0 0
the
detector.
spatial
This
filter.
increase
dem o n s t r a t e s
41
Pockels
The
Cell
Pockels
cell
between
crossed polarizers
acts
as a
v e r y f a s t g a t e w h i c h o p e n s b e f o r e the a r r i v a l of the e cho
bu t a f t e r
fast,
the a r r i v a l
hig h voltage
a K D 0P
pulse
crystal^^,
induced
which
a l l o w ing
it
is applied
to
a birefringence
rotates
through
is discussed
of the s e c o n d l a s e r pulse.
the
the
second
the
7.1
the
circuit
built
of less
pulses. The
avalanche
is
9.
at ~
O
Initially,
and
the
across
of
light
rotated,
in
at X = 1.064 pm
is
is
linear
so
The
below
the
Pockels
passing
the
w ith
time b e t w e e n
a one-sided v e r s i o n of an
circuit
used
collector-emitter
capacitor
the
is 3.2 KV. A
a 3.2 KV pulse
the m i n i m u m
in
is
in a
shown
in
p o i n t A an d p o i n t B in the c i r c u i t
and the
is just
voltage
the
cell
c i r c u i t u s e d to s e l e c t p u l s e s
laser^^.
3.2 KV
V
supply
than 5 nsec,
transistor
transistor
at
could
echo,
used
required
design f o l l o w e d was
mode-locked
figure
which
the
is
(Birefringence
the h a l f - w a v e v o l t a g e at 48 6 n m
was
a risetime
voltage
cell,
crystal
of
Pockels
the h a l f - w a v e voltage
wavelength,
in the
polarizer.
in appendix A.) For
Since
Pockels
polarization
this experiment,
K V.
the
When a
laser
C
breakdown
is
cell
through
beams
voltage
limit.
charged
is O V,
the
are
to
the
Pockels
blocked
by
of
each
Point B
3.2
KV.
is
The
polarization
cell
the
is
not
second
42
polarizer.
When
a trigger
is applied
down,
causing
to one
all
of
transistors,
it
transistors
s i m u l t a n e o u s l y b r e a k down. The result
to
breaks
pulse
the
v e r y fast g r o u n d i n g of p o i n t A in the c i r c u i t .
voltage across C cannot change
drops
the
to -3.2 KV r e l a t i v e
Pockels
cell
is
K V,
which
The p o l a r i z a t i o n of the e c h o
and
echo
passes
through
the
is
its
across
half-wave
is r o t a t e d by 9 O 6
second
polarizer
unattenuated.
12 MO
A
470
IOOOpF
B
68 KO
1 6MQ
16M0
1 6MQ
Trigger©— #--U-
Figure
1 6M0
9. Pockels
cell
is a
S i n c e the
The v o l t a g e
voltage.
the
other
instantaneously, point B
to ground.
3.2
the
trigger
circuit
43
Electronic
and C o m p u t e r Control
A PDP-11
computer
program
of the E x p e r i m e n t
controls
the
timing b e t w e e n
the t w o l a s e r s an d the t i m i n g of the P o c k e l s cel l t r i g g e r
circuit
by
A s sociates
as part
controlling
programmable
of the delayed
proportional
1000
the
nsec
to
the
for
a
connections
delay
trigger
input
10
within
input
boards ^23 ] ^
generator,
voltage.
Volt
voltage
The
input.
the d e l a y m o d u l e
on
Evans
These
boards,
provide
a delay
range
A
two
of
delay
diagram
is s h o w n
of
is
the
in f i g u r e
10 .
Trigger
Trig
in
Trig
out
+
To
La se r#2
Programmable
Del ay
Board
From computer
d/a converter
I
Manual laser
de I ay
potentiometer
To
Laser#l
V 1n
i
I
Trig
out
Trig
in
Progr ammable
Delay
Board
Manual Poc ke I sj
4— (,— w —
cell delay
potentiometer
P
Figure
10.
V in
Delayed
trigger
generator.
k To
r Pockels
cell
44
As
the
delay b e t w e e n
computer,
Pockels
to
the
signal
gate
scanned by
is
boxcar
is s h o w n
B
the
same
scanned by
the
laser
the
and
In
this
triggers at the same
time
the
portion
amount.
timing
of
is m e a s u r e d
contains
also
scans
time
by
of the b o x c a r
during
the boxcar.
which
For
to occur
the
a ful l
and
The
language.
in figure
of
simultaneous
program
the
is
The basic flow
11 .
this
to the echo
lasers.
listing
plots
a
and is
of
the
computer
the laser timing and the boxcar
reads
averager.
programming
always
scan range
which
which
second
the
controls
the c o m p u t e r
the entire
program
the
is
the gate is a p p r o x i m a t e l y 20 nsec w i d e
Appendix
an d
a lso
to the boxcar
experiment,
over
cell
by
lasers
echo.
computer
The
two
between
scanned
the Pockels
relative
gate.
delay
cell, is
manner,
The
the
the
voltage
written
from
in
gate
the
the
C
chart of the program
45
CALIBRATION^
Find
voltage
pulses_____ are
Find
boxcar
at which laser
simultaneous
voltage
gate
i
which
lasers
Apply voltage for full-scale
delay
betw e e n
the
lasers
Find voltage
gate
at which boxcar
echo
SCANNING
Set
Wait
reach
Read
and
initial
delay
and gate
for
boxcar voltage to
eqnil ibrinm_____value
boxcar
voltage
store
in
data
delay
Increment
array
and__ ga te|
can finished
nough ScansJ.
Store
Figure
11.
data
Flowchart
program
array
disk
of d a t a - a c q u i s i t i o n and c o n t r o l
for
photon
echo
experiment.
46
CHAPTER 7
A
Each
run,
adjustment
figure
or
of
12).
M T
experimentation
the
If
thi s run,
wavelengths
the
in the
they
dye
spatial
power,
an d
are
amplifier
into
the
cell,
laser
the
the vertical
connected
linewidth
of
dye
the
lasers
an
(AX) by
Perot
e talon.
laser
is adjusted,
(see
have
not
the c o r r e c t v a l u e
the m i c r o m e t e r -
factor
laser
the
of
the
of the
of the
laser
beam,
the
in
laser.
o rientation
grating,
in
laser
With
the
beam
power
oscilloscope,
are
all
spatial
with
and
of
the
dye
and the vertical
exit mi r r o r
monitoring
relative
to
of
the
lasers
the h e i g h t of N 2 l a s e r b e a m
cell,
angle
angles
visually
monitoring
dye
the
is an i m p o r t a n t
linewidth
o s cillator
with
pumped
changing
cavity
dye c e l l r e m o v e d ,
and horizontal
while
set by
starts
a n g l e s of the g r a t i n g s . Th e o p t i c a l
quality
in
day,
set to a p p r o x i m a t e l y
adjusted horizontal
path
N 2— l a s e r
two
the
previously been
for
C © R T W © )' IBS TH E LA®
adjusted
quality,
a photodiode
monitoring
l aser
looking at the ring patt e r n of a Fabry-
After
the
oscillator
the a m p l i f i e r
portion
dye cell
its o r i e n t a t i o n is adjusted for m a x i m u m
of
the
dye
is r e p laced and
power.
47
photo­
diode
oscillo­
scope
exit
mirror
cell
osc
dye
cell
telescope
beam
expander
diffraction
g r a ting
laser
partial
mirror
exit
mirror
cell
osc
dye
cell
telescope
beam
e xp ander
diffraction
grating
laser
partial
mirror
photo­
diode
oscillo­
scope
lens I
f= 1 .3 3m
lens 2
FabryPerot
etalon
e talon
pattern
on wall
F i g u r e 12.
The t w o N 2 - I a s e r p u m p e d dye l a s e r s
m o n i t o r i n g p hotodiodes and F a b r y — Perot etalon.
with
48
After
optics
both
lasers
involved
crossing
the
are
adjusted
in sending
beams
in
the
In this manner,
the b e a m s
crystal
to the
are
the
de w a r
adjusted
and
(see
figure 13).
lens 8
lens 9
prism I
prism 2
I cm
polarizer
lens 10
mirror 2
razor
blade
crystal
Figure 13. The optics between the lasers and
the
crystal
Lenses
8 and 9» the
laser b e a m
collimating
lenses,
are
centered on the beams. Pr i s m s
I and 2 are a d j usted so th$
beams
respectively.
hi t
adjusted
the
prisms
3
along w i t h
dewar
(not
and
4,
the m i r r o r w h i c h
shown
in
figure
sends
13)
Prism
the
so t h a t
4
bea m
the
is
into
second
l a s e r b e a m g o e s s t r a i g h t up in t o the d e w a r , g e t s b e n t 90°
by
one
the
prism,
other
d own.
goes
prism
Once
through
an d
comes
will
go t h r o u g h
beam
(checked
razor
blade
plane
of
back
it and its face
with
back
arrangement
lens
10
is
portion,
shown
adjusted
of
the
bl a d e .
bI a de .
Figure
2 to the
the bea m
14
Using
in f i g u r e
the
13,
focused
beam
is
from m i r r o r 2 to the
along
is p e r p e n d i c u l a r to the
the
to
the
illustrates
dewar:
razor
crystal.
until
straight
s u c h that b o t h l a s e r s
the
the
blade
is
within
or
the
conjugate
is,
the
same
the
as
is m o v e d back
is at
function
f ocal
waist,
that
Lens 10
waist
external
the
is p o s i t i o n e d at the
external
90° by
is a l i g n e d as above,
that
distance
and forth
dewar
such
location
mirror
gets bent
the
reflections).
crystal
from
out
in the b e a m
c r y s t a l . Th e r a z o r b l a d e
that
crystal,
the s e c o n d la se r ' s b e a m
p o l a r i z e r I is p l a c e d
narrowest
the
of
the
razor
the r a z o r
50
a
b
e
spot
on
wall
Figure
14.
The
slicing
of the beam by the razor blade.
If the r a z o r b l a d e is at p o i n t a a n d s l i c e s the b e a m f r o m
top-to-bottom,
the
bottom-to-top.
slices
the beam
to-bottom.
and
If
razor
the
beam,
p r i s m 3 are
The
the
crystal.
This
a horizontal
is
is
e x tinguish
at
the spot
point
from
c
extinguishes
and
top-
is at p o i n t b , the w a i s t ,
entire
vertical
then adjusted
will
blade
top-to-bottom,
simultaneously.
using
the
If the r a z o r b l a d e
slices
the
spot on the wall
and
such
checked
and vertical
spot
extinguishes
horizontal
that
angles
of
cross
in
the l a s e r s
external
razor
to
blade
the
as
crystal
described
above.
After
waists
the
of
lasers
the
p r e p a r a t i o n for
laser
adjusted
beams
pumped
day,
to v a c u u m
and
the
crossing
and
in
the
crystal,
the
is complete
and the actual
l i q u i d h e l i u m m u s t be
the 100 l i t e r s t o r a g e
On the previous
occur
the expe r i m e n t
run can start. F i r s t ,
from
are
dewar
the ma g n e t
and c o o l e d
transferred
into the m a g n e t dewar.
d ewar
(see figure
15) was
to l i q u i d N 2 t e m p e r a t u r e s .
51
This was
done by f i r s t p u m p i n g
cold-trapped
space
and
diffusion
the
sample
helium
gas
liquid
N 2 in
amount
(~1 g a l l o n )
space
to
to
pump,
space
atmospheric
the
start
the v a c u u m
liquid
then
and
pumping
space.
and
That
the day of the run,
the
massive
them
then
hel i u m
both
gas.
sample
atmospheric
all
magnet
magnet
pressure
to
cause
transfer
magnet
pressure
helium
tube,
space.
removed.
sample
the
with helium
gas,
A needle
space
coils.
The
completed,
space
the
the
on
to
valve
needle
in the
start
coming
space
between
the ma g n e t
space
the flow
space.
valve
is p u m p e d to ~.3 m m
Hg.
is
out
sample
this
closed
the tubq
is
and the
space with
during
the magnet
transfer
and
the
into the
space
of h e l i u m from
is
of
lightly p u m p e d
After
dewar
is sufficient
is full,
the
and.
above
storage
is c o nnected to an inlet
sample
sample
space wit h
the t r a n s f e r can
dewar
the ma g n e t
this transfer to cause
to
Now,
.slightly
storage
is then opened to fill
liquid helium.
space
is p l a c e d
in the
tube
After
a small
has b een purged,
are
liquid
the
liquid Ng
spaces
begin. The t r a n s f e r t u b e
and w h e n
putting
the f i r s t s t e p in t r a n s f e r r i n g is to
Once
and
with
int o the m a g n e t
b l o w out any r e m a i n i n g liquid Ng in the m a g n e t
warm
a
magnet
ni g h t ,
of l i q u i d Ng w a s put
cooling
using
the
backfilling
pressure,
Ng
space
the
is
sample
52
vacuum
space
liquid Ng
space
r vacuum
space
magne t space
vacuum
space
s a m p Ie space
Figure
15. A d i a g r a m m a t i c
c r o s s-section of
the
dewar.
53
When
the
magnet
coil
and the
crystal
are
at l i q u i d -
:
helium
temperature,
magnetic
current
the
field
of
through
for
the
to
ratio
a c c uracy
the
the
field.
is
Then
the
monitored
con n e c t e d to a chart
is desired,
changing
First,
a boxcar
light
the
at
the
coil
to avoid large voltages)
crystal
transmitted
systematically
spectrum
taken.
the
s a m p l e - a n d - h o l d photodiod e
If greater
is
(slowly,
required
transmitted
absorption
interest
is r a m p e d up
current
an
to
laser
laser
wavelength,
l ight
with
a
recorder.
averager
the
to
is used
light.
By
a
of
plot
t r a n s m i t t e d l i g h t vs. w a v e l e n g t h is o b t a i n e d . F o r large
wavelength
angle
ranges,
a stepper
of the grating. Using
can be
obtained.
wavelength
is
However,
known
dye l a s e r is used.
motor
this method,
since
within
for
2 c m - *,
to
+
15
chamber
is
chamber
and
psi.
(nX = 2 d s i n ® )
in
the
the
a 300 c m - * range
this
experiment,
the
the
pressure-scanned
proportional
wavelength
chamber
20 psi or f i l l e d w i t h Nj
velocity
is
an d is c o n s t a n t
f r equency in the
pressure
Since
inversely
the
the horizontal
Its g r a t i n g is e n c l o s e d in a p r e s s u r e
c h a m b e r w h i c h c a n be p u m p e d to gas
drives
of
light
to the p r e s s u r e
determined
by
the
for c o n s t a n t 0
Th e
frequency
the
in the
grating
and d,
is inversely proportional
chamber.
in
the
to the
outside
the
c h a m b e r is the s a m e as that i n s i d e so, s ince the v e l o c i t y
54 .
of light
outside
outside
the
inside
the
the
chamber
chamber
is
chamber.
For
a b o v e , the w a v e n u m b e r
is
constant,
proportional
to
the
range
pressure
(I / X ) r a n g e
and
recorder,
monitoring
the
wavelength
cavity
measure
the
tube
to
that
connected
wavelength
angle is adjusted until
are
is
and
the
pressure
mentioned
set
light
so
peak.
on
that
A
the
the
second
the w a v e l e n g t h s
chart
laser's
spectrometer
to a p i c o a m m e t e r
the
the
the p r e s s u r e - s c a n n e d
transmitted
pressure
corresponds
photomultiplier
to
the
wavelength
is ~ 1 0 c m - *. Once
absor p t i o n pea k is found by scanning
laser
the
are
laser's
of the
and
used
grating
two lasers
identical.
W i t h b o t h l a s e r s set to the p e a k of the a b s o r p t i o n , we
return
check
using
to
to the
optics
is m a d e
the
of
external
liquid h e l i u m
of
the
the
crossing
(SPEX)
so
slit
that
and
razor blade,
temperatures
l o c a t i o n of the c r y s t a l .
adjusted
experiment.
the
The
echo's
is straight.
The
First,
waists
because
often
of
the b e a m s
cooling
the
dewar
the
physical
collection optics
are t h e n
path
changes
a quick
to
the
spectrometer
c ollection m i r r o r
(not
shown
in f i g u r e
5)
is t i l t e d to s e n d the
second laser beam
to
the
t h e n the c o l l e c t i o n lens
(lens 4 in f i g u r e
5)
its face p e r p e n d i c u l a r
to
slit,
is c e n t e r e d
the
beam.
on the b e a m
Th e
SPEX
lens
with
(lens
7
in
figure
5)
is
also
55
centered w i t h
is n o w
made
through
its face p e r p e n d i c u l a r
that
the
spectrometer
the
second
center
of
grating
close
Ba b i n e t - S o l i e l
compensator
the
location
expected
perpendicular
close
A
enough
check
the
to the
is m ade
- crystal
the
to
- BS
second
the
to
the
the
polarizer
echo
The
second
extin c t i o n
compensator
vertical
and
the
be
the two.
polarizer
arrangement.
vertical
its
on
faces
should
it through
from
(0°)
their
of
the
centered
laser
ratio
the
Nezt,
are
with
- polarizer
45°.
hitting
center.
the
beam.
is i n d e e d g o i n g
and
echo to just make
of
axis
beam
slit
and
of
polarizer
compensator's
laser
to the beam. A check
With
and
thickness
th e
set to
O , the f i r s t p o l a r i z e r is r o t a t e d to get an a p p r o x i m a t e
null
The
in the
light
compensator's
null,
This
then
is
the
without
the
now
beam
changed
to
and
lens
in figure
face
is
the
to
is
let
finally
The
jus t
Pockels
is
of
enough
cell.
500
of
The
for
better
a better
less
of
the
it w a s
second
of
the
c ompensator
the
laser
spatial
filter
and p o sitioned
and
null.
of
than
the
spatial
to the b e a m
a
intensity
adjustment
centered
is p e r p e n d i c u l a r
the
thickness
facilitate
5,
adjusted
is r o t a t e d for
a factor
polarizers.
filter
its
until
light
through
5
thickness
polarizer
iterated
transmitted
is
t r a n s m i t t e d t h r o u g h the a r r a n g e m e n t .
such
lens,
that
slid b ack away
56
from
the
pinhole
the p i n h o l e .
vertically
through
position
The p i n h o l e
and
the
pinhole.
point
lack
centered
of the l ens
of
an A i r y
through
the
adjusted
in
around
the
The
straight
lens
4
then
the pinhole
t his
manner
pinhole
until
avoid
the
(lens
the
beam
steadily
so
is
the
checked
by m o v i n g
on
its
centered
the
first
on
collimating
the
doe s
go
travelling
laser
beams,
t h r o u g h the p i n h o l e .
the
the
If not,
through
moved
the f ocal
beam
position
burning
must
once
the
and
made
again
goes
the focus of
spatial
until
If lens 4
laser
be
holes
Nezt,
centered
stage
going
other
beam.
beam
first
goes
is adjusted
on the
entire
beam.
on
e v i d e n c e d by the
focussed
6)
bea m
finally
as
pinhole
focus
and m o v e d
the
posit i o n
pattern
to
with
lens
to
The
of
i n t o the S P E X slit. At t his point,
is
beam
is
not
in p l a c e
some
is at the p i n h o l e
vertically
the
lens
on the b e a m
pinhole.
perpendicular
until
diffraction
recoil!mating
it w i l l
is t h e n put
horizontally
t o w a r d the pinhole while
to r e m a i n
so tha t
filter
lens
is
5
is
correctly
beam
will
go
l e n s 4 is r e f o c u s s e d u n t i l
the
pinhole.
Now
the
spatial
f i l t e r is r e c e n t e r e d on the s e c o n d l a s e r b e a m a n d the X/2
plate,
faces
Pockels
cell
perpendicular
positions
such
that
and p olarizers
to the
are put
in w i t h
second,laser beam
they will
pass
both
the
their
and t h e i r
second
laser
57
beam
3
and
set
the echo
on
maximum
horizontal,
light
4 is set
Pockels
null
through
passes
is
to let a small
not,
tha t
lens
are
now
for
voltage
and
potentiometer
(see
two
voltage
is
both
the
the
compensator
This
the
until
polarizer
face
until
and
cell
of
the
a quick
and
and
spatial
expected
echo
the
best
check
slit.
If
polarizers
filter
is
location,
and
its p o l a r i z e r s
are
com p l e t e s
the
adjustment
of
the p h o t o n e c h o
is
to the detector.
being
complete,
on
the
trigger
figure
turned
the
is
and
triggers
after
on
adjusted
the
cell
seen
th e
PMT
on
laser
for
tube
the
trigger
a delay
second
cell
generator's
is a d j u s t e d u n t i l
the
an
delay
nsec. Then the Pockels
Pockels
just
of
generator's
10)
on
photomultiplier
output
lasers of ~50
be
Then
axes
Pockels
delay p o t e n t i o m e t e r
should
3.
c e n t e r e d on the
cel l
echo
rotated
of
through
Pockels
The
is
p o l a r i z e r 4 is r o t a t e d
null,
observing
The
b e t w e e n the
The
turning
oscilloscope.
high
to
null.
preparations
by
plane
light
best
f r'dm crystal
obtained
the
is s t i l l
t r anslated
optics
All
of
the b e a m
for best
plate
At t his p o i n t ,
the B a b i n e t - S o l iel
adjusted
X/ 2
along
7 is adjusted.
vertically
the
and
amount
readjusted
apertures. W i t h polarizer
through p o l a r i z e r
tilted
is o b t a i n e d .
is m a d e
the
to v e r t i c a l
cell
their
laser
oscilloscope
the
pulse.
now.
The
58
crossing
angle
is
adjusted
discrimination
is
by p r i s m
adjusted
until
3,
and
stage
of
echo/laser
is
is the m e a s u r e m e n t
of
maximum
each
obtained.
The
the
finale
echo
as
of
this
experiment
a function
of
the
delay b e t w e e n
lasers.
is a c c o m p l i s h e d w i t h the h e l p of the c o m p u t e r .
d/a
output
generator,
gate
control,
run.
constant
limited.
the
computer
connected
to
W i t h one
the
trigger
the o t h e r c o n n e c t e d to the b o x c a r ' s e x t e r n a l
computer's
is
from
This
and
a/d
input,
Because
of
the
of
the
boxcar
output
the p r o g r a m
limitations
boxcar
The m a x i m u m
connected
described
on
the
averager,
the
to
the
in figure
averaging
scan
11
time
speed
is
scan speed is calculated b e l o w ^ ^ .
5 c ETC * Delay
range
MS T = M a x i m u m Scan Time ■=
A p e rture
ETC = Effective
Time
duration
Constant =
M164 Time
OTC = O b s e r v e d Time
Constant
Constant =
(Aper dur)*(Trig
= 8 sec
OTC =
20 ns
* 6 Ez
MFTC = M a i nframe
Time
Constant = I sec
rep
rate)
59
ETC = V (8 s e c )^ + (I s e c )^ = 8 sec
Therefore,
MST =
So
5 6 (8 s e c ) 0Del ay range
.........
= 2sec/nsec
20 nsec
the
fastest
nano second
While
the
computer
one
can
put
data being plotted,
luck holds,
in
the
lase r s .
helium
home .
rate
is
2
seconds
range
for
every
scanned.
the
data,
scan
0 delay
same
is
scanning
one's
fee t
the
up,
As m a n y
re l a x ,
and reading
watch
the
and w a i t for the end of the scan. If
data for several m a g n e t i c
night
timing
by
scans
in the m a g n e t
changing
are
dewar
the
fields m a y be taken
wavelengths
t a k e n as p o s s i b l e
runs
of
the
before
the
out a n d it's t i m e
to go
60
CSAPTEE 8
MTA
Echo
intensity
ABto C®mXSSI©KI§
vs. p u l s e
separation was
measured
the f i e l d s 45 K G , 25 K G , an d 15 KG. W h e n t h e s e
taken,
th e
experimental
polarization
of
echo
discrimination
intensity
photomultiplier
separation.
for
the
was
echo
close
100
averager
decayed
as
output
was
are
s hown
photomultiplier
to the m a x i m u m
on the
made
of
the
in figures
16
and
and
65
voltage
and
the
nsec,
was
1700
sensitivity
lack of
measurement
difficult.
a strong
A
boxcar
for good signal-to-
If the e c h o s i g n a l w a s I O Q m V at 50 n s e c and
ex p ( — 4t/T2),
with
Tg
=
100
nsec,
then
s i g n a l w o u l d be o n l y 15 m V at 1 0 0 nsec. T h e r e f o r e ,
desirable
the
the
tube
ne e d e d
at
of
The
was
ratio
traces
oscilloscope.
> 10 m V
the
at 50 n s e c p u l s e
nsec
allowed,
include
obtained
4:1
35
d ata w e r e
The m a x i m u m
oscilloscope
The
input
n o i s e r atio.
detector
of
not
intensity
separations
mV/div
signal
tube
technique.
laser
tube
did
pulse
respectively.
Volts,
to
Photographs
photomultiplier
17
setup
at
to scan from
50 nsec
to 100 nsec,
the
it is
then turn the
P M T v o l t a g e up to i n c r e a s e the s i g n a l to 100 mV, a n d s c a n
61
laser I
laser 2
—
Figure
16.
Pulse
echo
separation - 35 nsec
Figure 17. Pulse separation = 65 nsec
62
100 n s e c to 150 nsec,
expe r i m e n t
for
two
tube voltage was
at the
reasons.
was
enough
was
increased.
these
15 KG w e r e
the
separations.
the d e f e c t o r f r o m
to cause
Despite
First,
saturation
the a m o u n t of
the s e c o n d l a s e r p u l s e
scans
t a k e n w i t h the p u l s e
a l l o w e d voltage
Second,
in the
difficulties,
in this
photomultiplier-
already near its m a x i m u m
shorter pulse
light reaching
etc. T h i s w a s not p o s s i b l e
PMT
if its voltage
at 45 KG,
25 KG,
and
separation ranging from
50 n s e c
to 10 0 nsec. By s u b t r a c t i n g the z e r o (the b o x c a r
voltage
output
when
no
light
was
hitting
p l o t t i n g the d a t a on a s e m i - l o g a r i t h m i c
plot
was
obtained
calculated.
15
KG
from
Because
data,
no
which
a good
decay
the
zero
constant
was
not
PMT)
scale,
decay
was
the
time
a linear
could
obtained
calculated
an d
for
for
be
the
this
field. The s e m i - l o g a r i t h m i c p l o t s of e c h o i n t e n s i t y vs.
pulse
separations
shown
in
p l o t s are
the
data.
nsec wa s
figures
for
18
the
and
fields
19.
Th e
the c o m p u t e r - g e n e r a t e d
From
these
straight
25
and
straight
45
lines
KG
a decay
Thus,
time
are
in the
l e a s t - s q u a r e s f its
lines,
o b t ained for both fields.
KG
of
of 27.5
T 2 ~ I 10 nsec at
25 KG and at 45 KG.
™ i
63
120ns
120ns
Figure 18. S e m i - l o g a r i t h m i c
pulse separation at 25 KG.
plots
of echo
intensity
vs.
64
110ns
110ns
Figure 19. S e m i - l o g a r i t h m i c
pulse s e paration at 45 KG.
plots
of echo
intensity
vs.
65
An a t t e m p t
was made
to o b t a i n an e c h o at z e r o field.
No echo was
seen,
implying
time
field
is d e c r e a s e d b e l o w
as
the
the e x p o n e n t i a l
pulse
th e
sepa r a t i o n
dephasing
dependent. For
to 50 nsec,
would
old
decay,
is
decrease
the
not
time,
by
intensity,
echo
s i mply
but
example,
the echo
a decrease
in the
15 KG.
amplitude
inversely
instead
is
On
cell
more
was
used.
the
a factor
one
of e^,
would
have
or 7.4. To o b t a i n the
to
decrease
this
20
shows
the
PMT
trace
of
of
improved
with
the
pulse
the m i n i m u m
of the
at
photograph
output
25
when
16
KG
this
and
as
at
the
of
the
technique
15
KG
were
and 22. Both decays
indicated
the s e m i - l o g a r i t h m i c plots.
whether
ratio
and 17 d e m o n s t r a t e s
s h o w n in figures 21
behavior
discrimination
the e c h o : l a s e r
a
figures
Scans
no n - e x p o n e n t i a l
time
polarization
Figure
Comparison
nonlinearity
at
the
This
p e r f o r m e d and are
a
strongly
due to the r i s e - t i m e
used.
improvement.
show
to
intensity at a separation of 50 nsec
runs,
technique was
o s c i l loscope
at a c e r t a i n
trigger.
later
immensely.
of
if T 2 is d e c r e a s e d f r o m I O O n s e c
s e p a r a t i o n is 15 n s e c
Pockels
Because
proportional
s e p a r a t i o n to 25 nsec. Fo r this e x p e r i m e n t ,
pulse
dephasing
non-exponential
by
the
It is u n k n o w n
decay
is
an
66
I
Figure 20.
40ns
Photon echo signal wit h pol a r i z a t i o n
d i s c r i m i n a t i o n technique.
120ns
Figure 21. Echo intensity vs. pulse separation at 25 KG.
67
Figure 22. Echo intensity vs. pulse separation at 15 KG.
artifact
a real
of
the
experimental
effect.
component
Unfortunately,
in one
investigation
None
of
of
the
behavior.
decay
curves
splittings
are m u c h
to hyperfine
modulation
too
by
of
it
former
to
be
until
beyond
et
case
is
or
the
and
the
nsec
indicated
splitting
a It 2 6 ^ .
and
MHz
100
of
Using
pulse
by
the
a
found
A/h
=
splitting
of
the
s eparation
of 3.13
6.26
ground
of
f r e quency
difference
of
0.06
state
+ 0.03 GHz.
of the e x c i t e d state,
number
+
the
GHz,
i nto
Depending
ground
the o p t i c a l
compo n e n t s
splittings
state
consist
of f o u r
splitting
was
not
lines
split,
with
is m u c h greater
the
with
a
a
splitting
s p e c t r u m m a y h a v e any
arising
from
the
superimposed
u n p e r t u r b e d s p e c t r u m . The s i m p l e s t
excited
that
implies
levels
on
the
hyperfine
which
four
the
MHz
detected,
= 10
of
either
t h a n I / Sn s e c = 20 0
hyperfine
Laursen,
indication
H a m i l t o n i a n of H = A S z I z w i t h S = 1/2 and I = 3/2,
group
is
an electronic
Thus,
than 1/lOOnsec
occur
the
show, any
fast
less
not
The
measurements
state
is
doe s
separations.
of
splittings.
greater
are m u c h
modulation
a failure
this
due
splittings
or w h e t h e r
of the n i t r o g e n lasers p o s t p o n e d further
modulation
the.
technique,
then
sum
upon
and
the
case w o u l d be if the
the
a separation
spectrum
of
3 GHz.
would
This
than 20.0 M H z , so any m o d u l a t i o n
69
w o u l d not be r esolvable w i t h o u t
The
ultimate
goal
of
measurements
of e c h o
lower
and
this
of
fields
crystal
terbium.
and
to
on
From
the
this
experiment
obtained.
1% u s e d
due
a larger
crystals
field
ions
with
with
and
include
the
the
on
concentrations
dephasing
electrons
fluoride
become
interaction
dependence
at
give
insight
of
high
(like the
magnetic
different
These
would
c o n c entrations
concentrations.
dependence
int o
could
the t e r b i u m
increases,
important.
the
and
Thus,
of the d e p h a s i n g
interaction
of
the
of
the
ions w i t h their environment.
A significant
step t o w a r d
goal
apparatus
an d p r o c e d u r e by w h i c h v e r y w e a k p h o t o n
could
documented
apparatus
was
be
the
the a c c o m p l i s h m e n t
ultimate
signals
of
is p r o b a b l y
nuclear
at m e d i u m
the c o n c e n t r a t i o n
would
terbium
temperatures
temperature
the
T b ^ + spin d i ffusion
studying
times
of
neighboring
mechanisms
Tb ^ + :Tb ^ +
to
higher
^ As the c o n c e n t r a t i o n
dephasing
of
separation
dephasing m e c h a n i s m s
experiment),
interaction
moments
range
extend the
At l o w c o n c e n t r a t i o n s of t e r b i u m
in thi s
to the
is to
i n t e n s i t y vs. p u l s e
T 2 » details of the h o m o g e n e o u s
be
shorter laser p u l s e s .
measured.
in this
thesis
and pro c e d u r e
development
This
so
that
of
experimental
accomplishment
others
has
can use
to obtain the complete
e cho
been
similar
data.
REFERENCES
71
1.
N. A. K u r n i t, I. D. A b e l I a, a n d
S. R. H a r t m a n n ,
Phy s .
S. R. H a r t m a n n ,
Phy s .
Rev. L e t t e r s 13 , 567 (1964).
2.
I . D. Ab e 11 a , N. A. K n r n i t , a n d
Rev. 141,
391
3.
R. H.Dic k e ,
4.
L. ($.
6.
C.
K.
2022
N.
20,
1087
Jos
B . W.
Wiersma,
7.
P h y s. Rev.
Lambert,
Rev. A4,
5..
and
C o m p a an,
and
R.
E.
Morsink,
N.
Takeuchi,
New
York
10.
M . F . Joubert,
A b e l l a, Phy s .
Ph y s . Rev.
J . Aartsma,
and
Lett.
and
D o u w e A.
S.
R.
Hartmann,
P h y s.
E l ec t r o d v n a m ic s . John Wiley
(1975).
P h y s . T o d a y I Iu, 4(1953).
B.
Jacquier,
B o u l on, J . P h y s i q u e 4 3., 893
J . F.
I. D.
91 (1972).
E. L. Hahn,
L.
and
Che m. P h y s . Lett. 49,, 34 (1 977).
9.
11.
(1954).
Slasher,
Thijs
Jackson, C l a s s ical
Sons,
99
(1968).
S. C h a n d r a ,
J. D.
A.
93,
(1971).
Patel
L ett. 41A,
8.
(1966).
Broer,
C.
R.
Moncorge,
and
6.
(1982).
J . Gorter,
an d
J.
Hoogschagen,
P h y s i c a 11, 231 (1945).
12.
Renata
Reisfeld
and C h r i s t i a n
K. J o r g e n s e n ,
jind E x iCjLiIiB d SjLJiJLiLS. S Z Rji re. Earths,,
New
York
(1945).
L jiiSiBiTiS
Springer-Verlag,
72
13.
S.
H u f n e r,
Optical
Spectra
Compounds. Acedemic
14.
H. P. C h r i s t e n s e n ,
15.
T . W. H a n s c h ,
16. F . L.
of
Press,
Ne w York
Appl. Opt. 11 , 895
18.
A m n o n Yariv,
Sons,
19.
(1972).
and R. W. G row,
(1982).
discussion.
New
Private
Earth
(1978).
0. P. G a n d h i ,
Rev. Sci. I n s t r u m . 5.3., 7 0 8
Private
Rare
Phy s . Rev. B 7, 4 0 6 0 (197 8).
S c ho w , A. R i a z i ,
17.
T ransparent
Q u a n t u m E l e c t r o d y n a m ics, John
York
Wiley
and
(1975).
discussion.
2 0 . 0 j)_t_i_c_s. Guj. d
2
(M e I I e s Gr lot,
Irvine,
California,
1975).
21.
L.a.s.e r FjoiCUji Bujjjrj G u j d j ( A d v a n c e d T e c h n o l o g y G r o u p
Publications,
22.
Isao
Littliton,
Matsushima,
M a s sachusetts,
Takeshi
Kasai,
Rev. Sci. I n s t r u m . 5.2., 1 8 6 0
23.
Model
4141
Associates,
24.
Model
162
R.
346
26.
Berkeley,
Signal
Corporation,
25.
Programmable
USA,
M . Macfarlane
Masaaki
Yano,
(19 81).
Time
Pelay
California,
Averager
and
1983).
Module
(Evans
1982).
(Princeton A p p l i e d Research
1975).
and R. M . S h e l b y ,
Opt.
C o m mun.
4.2,
(1982).
I. Laursen
and
L.
M . Holmes,
P h y s . 7 , 3 7 6 5 (1974).
Phy s. C : Solid
S tate
73
APPENDICES
74
APPENDIX
A
CRYS T A L A N I S O T R O P Y
7-5
If
a
crystal
parallel
is
a n i s ;;o t r o p i c ^2 7 J ,
then
D
is
not
to E and e is a tensor
D i = 5 8 ij E ij
A choice
of axes,
can be made
D
Z
== 8 _
X
such
c a l l e d the p r i n c i p a l
that
e is diagonal
d i e l e c t r i c axes,
so that
X
D vr = 6 _ E„
y
d z
-
y
y
8Z
EZ
If l i n e a r l y p o l a r i z e d
axis,
D
x
the
components
going
suffer
the
is p r o p a g a t i n g
of the D vector
along
the z
are
= a cos tot
Dy = b cos
When
light
two
Mt
through
different
rays
the
phase
crystal,
changes
are d i f f e r e n t
L is the length of the
© = 2 n L ('/Ts -
is introduced by
the
two
because
components
the
will
v e locities
of
(v x = c /'/Tx , v y = c /V T y ) . If
crystal,
) / X
the c r y s t a l .
a difference
in phase
of
Eq . Al
76
A half
-
introduces
If the
45°
light
crystal
of
90°
is an a n i s o t r o p i c
shift
of n b e t w e e n
the
crystal
two
which
components.
is l i n e a r l y p o l a r i z e d at an a n g l e
a principal
the
rotation
plate
a phase
i nput
from
then
wave
axis
will
(Dx =
cause
an
(Dx = acoswt,
acoswt,
Dy =
effective
of
acoswt),
polarization
Dy = - acoswt).
y
Figure
plate.
By
23.
The
varying
the
polarization
angle
of
the i nput p o l a r i z e r ,
be
the
rotation
principal
by
a
axes
half-wave
relative
to
any l i n e a r o u t p u t p o l a r i z a t i o n can
obtained.
A
Pockels
cell
which a voltage
shift
the
difference
components
be come s
a crystal
applied
proportional
between
is
to
along
that
the z axis
voltage
x and y c o m p o n e n t s
in
the
index
is proportional
of
(K D P , A D P ,
of
the
to
causes
be
input
refractions
to the voltage,
or
KD*P)
in
a p hase
introduced
light.
of
the
The
two
so equation Al
77
e ~ v / x
T hus,
the
voltage
re t a r d a t i o n
echo
required
is proportional
experiment,
produce
a half -
light,
of
has
the
its
pulse.
at
the p h a s e
-
certain
For
pulse
Soliel
difference
an
angle
It
then
blocked
compensator
the photon
The
of 4 5 °
can
wave
sufficient
polarization
pol a r i z e r w h i c h had p reviously
A Babinet
a
to wavelength.
voltage
polarized
axis,
application
fast
produce
w a v e r e t a r d a t i o n is a p p l i e d .
linearly
principal
a
to
input
from
rotated
pass
to
the
upon
through
a
it.
is a d e v i c e
in w h i c h
can be set to any d e s i r e d value.
It
c o n s i s t s of t w o w e d g e s of an a n i s o t r o p i c c r y s t a l w i t h a
b lock of the same
perpendicular
Figure
Sliding
th e
crystal m o u n t e d w i t h
to that
24.
compensator.
of
At
compensator.
a l o n g t h e i r p l a n e of c o n t a c t
th e
full
axis
the wedges.
The Babinet - S o l i e l
the w e d g e s
thickness
of
its principal
wedge
—
wedge
thickness,
portion
the
phase
changes
of
the
shift
78
introduced
by
the
by
the wed g e s
block,
so
the
thicknes s decreases,
wedges
and
device,
from
the
linear
the
is
t otal
so
t otal
phase
does
phase
polarization
to e l l i p t i c a l
cancelled
of
the
e l l i p t i c a l l y pol a r i z e d
linearly
light.
polarized
so
shift
phase
that
shift
increases.
input
light
that
a
As
the
through
the
With
c a n be
or f r o m e l l i p t i c a l
laser
introduced
is zero.
shift
In the p h o t o n e c h o e x p e r i m e n t ,
the
by
changed
to linear.
the c o m p e n s a t o r
from
this
changes
the crystal back to
polarizer
can
block
the
79
APPENDIX B
D A T A - A C Q U I S ITION AND CONTROL PROGRAM
FOR PHOTON E CHO EXPE R I M E N T
80
/o COMPUTER PROGRAM FOR SCANNING TIMING BETWEEN TWO LASERS,
SCANNING BOXCAR GATE, AND READING, PLOTTING, AND STORING
SIGNAL FRCM BOXCAR AVERAGER
*/
^include
//include
//include
wda.c"
wItc. cw
wad.cw
/* OTHER FILES REQUIRED BY PROGRAM */
main()
{
long
int
int .
int
int
int
int
int
int
int
char
float
long
int
char
long
int
char
FIO
int
INITTC),DRWABS(),MOVABS();
x,y;
xO = 30;
yO = 200;
yl = 190;
xmax = 660;
ymax = 730;
nsec,ns,endnsec;
zero,zer,cal,calib;
range,delta,scans;
c[2];
boxcal,boxcar,endboxcar,box;
delay,old;
i,j;
bell[2];
data[512];
bin;
filnam[20];
fio;
baud = 2400;
bellC0] = 07;
bell[I] = 0 ;
Z0 ASCII CODE FOR BELL »/
putinitC);
getinitC);
Z0 INITIALIZATION »Z
for(;;)
{
Z0 MAIN LOOP OF PROGRAM o/
for(i=0;i<512;i++)
Z0 INITIALIZE DATA ARRAY <7
data[i] = (long)O;
}
81
putfmt(wWhat is the name of the data file?\nw);
ge tfmt (”%p \nw,f11 nam);
if(!fcreate(&fio,filnam»I))
{
putfmt(wError; can't open %p\nw,filnam);
return;
}
/o INITIALIZE PLOT */
fcall(INITT,I,&baud);
Z0 BOXCAR GATE AT INITIAL VALUE 0/
putvoltd ,204?);
Z0 FIND VOLTAGE FOR SIMULTANEOUS PULSES <V
putfmt(wType delay for simultaneous pulses\nw);
getfmt(w%i\nw,&zero);
whil'e(zero>=0)
{
zer = zero;
putvolt(0,(int)(2047.0 - 204.8»(zerZ100.0)));
putfmtC Type delay for simult. pulses\nw);
getfmt(w%i\nw,&zero);
}
Z0 CALIBRATE BOXCAR GATE <7
putfmt("Boxcar gate on lasers via %% initial\nw);
putfmt(w(Check with scope)Xnw);
putfmtC Type calibration delay in ns\nw);
getfmtC%i\nw,&calib);
if(calib>=0)
cal = calib;
putvolt(0,(int)(2047.0 - 204.8»((cal+zer)Z100.0)));
putfmt("Type %% to put boxcar gate on echo\nw);
putfmtC (Use decimal point)Xnw);
getfmtC%f\n", Aboxcal);
while(boxcal>=0)
{
box = boxcal;
putvolt(I,(int)(2047.0 - 204.S1KboxZI0.0)));
putfmt("Type %% to put gate on echoXn");
getfmt(w^fXnw,Aboxcal);
}
82
putfmt(wSCAN OF LASERS AND BOXCARXn");
putfmt("Type initial delay in ns\n");
getfmt("?i\n",&nsec);
putfmt("Type scan range in ns\n”);
getfmt("%i\n".,&range);
putfmt("Type delay increment in ns\n");
getfmt(”%i\n”,&delta);
putfmt("How many ticks between delay increments?\n");
getfmt("%l\n"»Adelay);
putfmt("How many times do you want to scan?\n");
getfmt("%i\n",Ascans);
for(j=1;j<=scans;j++)
t
/o ENOUGH SCANS? «/
c[0] = '0';
putfmt("CR -> scan, Q CR -> quitXn"j;
getfmt("%p\n”,c);
if (c[6] == 1Q1 11 c[0] == eq*)
break;
/o INITIALIZE DELAY BETWEEN LASERS tV
putvolt(O ,(int)(2047.O - 204.8°((ns+zer)/100.0)));
/0 INITIALIZE BOXCAR GATE <V
boxcar = nsBbox/(cal);
putvolt(I,(int)(2047.0 - 204.8°boxcar/10.0));
/o ERASE SCREEN °/
fcall(INITT,1,Abaud);
old =. ticks((long)0);
/0 WAIT 0/
while(ticks(old)<(5°delay));
/o DRAW AXES *>/
fcalI(MOVABS,2,AxO,Aymax);
fcall(DRWABS,2,AxO,AyO);
fcall(PRWABS,2,AXmax,AyO);
fcall(MOVABS,2,AxO,AyO);
83
/*> THE SCAN o/
for(i=ns;i<=ns+range;i+=delta)
{
/° SET DELAY BETWEM LASERS *7
putYOlt(0,(int)(20Vf.0-20M.8»(1+zer)/100.0));
/a SET BOXCAR GATE tV
boxcar = I^boxZ(Cal);
putvoltd ,(int)(2047<>0 - 204.8°boxcar/10.0));
/a WAIT o/
old = tloks((long)0);
while(ticks(old)<delay);
Z0 READ DATA <7
bin = (i-ns)Zdeita;
data[bin] -= (long)getvolt(O);
Z0 PLOT DATA <7
x = (int)( (float)630l>(i-ns)Zrange + 30);
y = (int)((float)53P°data[bin]Z(j°2047)+200);
fcall(DRWABS,2,&x„&y);
}
endnsec = i-delta;
endboxcar = endnsec0boxZcal;
Z0 GET ZERO READING 6Z
putfmt(mSpXnXn",bell);
putfmt(”Put card in for zero. CR to cdntinueXh");
^etfmt(wXnw);
for(i=1;i<=5;i++)
{
Z0 WAIT *Z
old = ticks((long)O);
while(ticks(old)<delay);
Z0 READ ZERO o/
data[rangeZdelta + i] -= (long)getvolt(O);
}
}
84
/0 WRITE DATA TO FILE 0/
putf( AficVAn");
for(i= O;i<(range/delta +12)/8;i++)
{
pufcf(6fio,"A\06041 $+\06061 %+\0606l %+\0606l
%+\0606l %+\0606l %+\0606l %+\0606l %+\0606l\n",
(long)(8ei),data[8ai+0],data[8si+1],data[8ai+2],
data[8si+3],data[8°i+4],data[8°i+5 3,data[8°i+6],
data[8Qi+7]);
}
/0 CLOSE FILE 0/
fclose(Afio);
putfmt("%p\n",bell);
putfmt("Scan finished. Delay = %i nseo.\n",endnsee);
putfat("boxcar gate = %% initial + %4.2fg$\n",endboxcar);
}
MONTANA STATE UNIVERSITY LIBRARIES
stks N378.D246@ Theses
Photon echoes o f terbium in lith iu m y t t r
RL
3 1762 00178771 O
*
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