Spectroscopic evaluation and fluorescence dynamics of several erbium doped laser materials
by Bruce Farris
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of
Philosophy in Chemistry
Montana State University
© Copyright by Bruce Farris (2001)
Abstract:
Spectroscopic analysis and kinetic modeling of relaxation processes have been completed for 2.1% and
4% Er:YAG, 2% Er:YSO, and 0.5% Er:LuAG at room temperature. Simultaneously time and
frequency resolved emission spectra were obtained using step scan Fourier transform spectroscopy.
Experimental emission lifetimes of the 4I13/2, 4I11/2, and 4S3/2 levels were obtained directly from
these spectra. The 4I13/2 lifetimes were found to be 6.6, 6.5, 8.6, and 5.7 msec for 2.1% and 4%
Er:YAG, 2% Er:YSO, and 0.5% Er:LuAG, respectively. Lifetimes of 67.3, 10.5, and 67.7 μsec were
observed for 4% Er:YAG, 2% Er:YSO, and 0.5% Er:LuAG, respectively for the 4I11/2 state.
Experimental lifetimes of 10.7, 3.5, and 12.3 μsec were observed for the 4S3/2 state in 4% Er:YAG,
2% Er:YSO, and 0.5% ErrLuAG.
Quantum-mechanical calculations using Morrison’s code were performed and theoretical radiative rates
were generated. Comparison of these values with experimental emission lifetimes allowed
determination of non-radiative relaxation rates and branching ratios.
Additionally, an experimentally based new method of determining non-radiative rates was developed
that utilizes the intensity "volume" under the three-dimensional emission bands. The two methods
provided good agreement.
Finally, the rate constants obtained by these methods were plugged into a set of coupled differential
equations and. numerically solved using the fifth order Runge-Kutta method in Mathematica. Good
agreement with the experimental temporal behavior was obtained (including emission rises as well as
decays) indicating the validity of the results. Non-radiative vs. radiative branching ratios were
approximately equal for the first excited state 4I13/2, but higher electronic states typically showed >
99% of the population relaxing via the non-radiative channel. SPECTROSCOPIC EVALUATION AND FLUORESCENCE DYNAMICS
OF SEVERAL ERBIUM DOPED LASER MATERIALS
by
Bruce F arris
A dissertation subm itted in p a rtial fulfillm ent
of the requirem ents for the degree
of
Doctor of Philosophy
in
C hem istry
MONTANA STATE UNIVERSITY
Bozeman, M ontana
Ju ly 2001
11
3 )3 ^
f 3L4ct\
APPROVAL
of a dissertation subm itted by
Bruce F arris
This d issertation has been read by each m em ber of the dissertation
com m ittee and has been found to be satisfactory regarding content, English
usage, form at, citations, bibliographic style, and consistency, and is ready for
subm ission to the College of G raduate Studies.
Lee Spangler
(Signature)
7^/5'-O/
Paul Grieco
D ate
Approved for the College of G raduate Studies
Bruce McLeod
(Signatdre)
Z
Date
STATEMENT OF PERM ISSIO N TO USE
. In
presenting
th is
dissertation
in
p a rtia l
fulfillm ent
of the
requirem ents for a doctoral degree at M ontana S tate U niversity, I agree th a t
the L ibrary shall m ake it available to borrowers under rules of the Library. I
fu rth er agree th a t copying of th is dissertation is allowable.only for scholarly
purposes, consistent w ith “fair use” as prescribed in the U.S. Copyright Law.
R equests for extensive copying or reproduction of th is dissertation should be
referred to Bell & Howell Inform ation and Learning, 300 N orth Zeeb Road,
A nn'
Arbor,
M ichigan
48106,
to
whom
. I
have
granted
“the exclusive rig h t to reproduce and distribute my abstract in any form at in
whole or in p a rt.”
S ignature
ACKNOWLEDGEMENTS
I would first like to th an k my advising professor, Dr. Lee Spangler, for
his support of my graduate Study.
Not only did he serve as an excellent
teacher and research m entor, bu t he dem onstrated extraordinary flexibility
and support as I completed th is w ork.after accepting an opportunity to teach
a t L ansing Com munity College in Michigan. I would also like to th an k the
other m em bers of my committee; Professors Edwin Abbott, P a trik Calks,
Rufus Cone, Jack Riesselm an, and J a n S u n n er. This work would not have
been possible w ithout the other m em bers ofThe Spangler research group;
particu larly Rich Martoglio, Tony Sm ith, and W endi Sonnenberg for th eir
involvem ent in other phases of the project.
Additionally, I would like to .
express my gratitude to .Dr. Reed Howald for his help and guidance on the
kinetics portion of th is work.
T hank you to Dr. Michele McGuirl for the
friendship and opportunity to discuss the project w ith a scientist th a t w as not
fam iliar w ith my work, providing a valuable perspective.
Also, I would like to th an k my paren ts, Joe and W iletta Farris, and
my parents-in-law , Merle and Bonnie Johnson, for th eir support during m y
tim e in M ontana. T hanks also to Benny F arris for m otivating me to get away
occasionally for m uch needed “sanity breaks”! ' Most of all, I’d like to th a n k
my wife, Leslie, for h er help, support, and extraordinary patience while I was
completing th is work.
- ,
TABLE OF CONTENTS
1. INTRODUCTION ................. :.................................. .................................;............1
Energy Levels in E rbium L aser M a te ria ls........................................................ I
C haracterization of L aser M a te ria ls .................................................................. 4
Q uantum -M echanical M odel......................................
7
2. EXPERIMENTAL PR O C E D U R E ............................................:................. .......n
' Overall M ethod of A nalysis........... ....................................................................H '
S am p les..................................................................................;............................. ig
Equipm ent and Techniques ..........
14
A bsorption......... ......................................... ............;........................................15
E m ission...............................................................'.............................................17
Tem porally Resolved E m issio n .................:.................................................. 18
Tem porally Resolved FT Spectroscopy............................ !..............................20
3. EXPERIMENTAL RESU LTS................................
27
A bsorption.................................................. ........................................................... 27
Continuous E m ission.................... ............................................ ....... ".................gg
IR E m ission........................................................................... ;.......................... gg
Visible E m issio n ...........................;........................................... ......................41
Tem porally Resolved E m ission ...............................................................■
..... 41
E rrY SO ..............................
;..49
ErrYAG..................................................................................
56
ErrLuA G .......................................................... ...............................•................. 60
4. DISCUSSION OF R E SU L T S..........................................................
....72
G eneration of E xperim ental Decay C u rv es....... ...........................................72
Discussion of Long Term C urvature in Log P lo ts........................................ 73
Kinetic M odeling of E xperim ental D ata....;........................................ ...........77
4111/2 E xcitation.........................................................;....... .•..............................78
Sum m ary of Total Volume Modeling M ethod.................................
85
4S 3/2 E x c ita tio n ............ ....................................................
91
5.
SUMMARY...,................... ;...........................................'......... ;......................... 104 ‘
. REFERENCES CITED ....'......................... ........................ ............................ 109
APPENDIX A: PROCEDURES USED IN QUANTUM-MECHANICAL MODELING
OF LASER MATERIALS......... .......................................................................... 115
vii
LIST OF TABLES
Tabl®
Page
1. Excitation Frequencies for 2% E r:Y SO ........................................................49
2. Excitation Frequencies for 4% Er:YAG.......................
56
3. Absorption T ransition Frequencies in E r3+
Doped M aterials (cm-1) ..................................................................................68
4. E xperim ental Lifetim es of T ransitions in
E rbium Doped M a te ria ls.................... ...................................................;..71
5. R adiative B ranching Ratios Calculated from
E xperim ental D a t a ....................................................................................... 71
6. B ranching Ratios for Decay from 4L 1/2
Manifold in 4% E rY A G ............................................................................... 83
7. B ranching Ratios for 4L m M anifold in 2%
E rY SO or 0.5% E rrL uA G .................. '....................................... ................ 83
8 . Rate C onstants for Relaxation following 4111/2
Pum ping (psec-1) ............................................................................................84
9. B ranching Ratios for Decay from the 4S3/2
and 4111/2 Manifolds in 4% E rY A G
following 4S3/2 E x cita tio n .......................................................... :................ 94
10. B ranching Ratios for Decay from the 4S3/2
and 4111/2 Manifolds in 2% E rY S O following
4S3/2 E x c ita tio n ..................... ;....................................... ...............................95
11. B ranching Ratios for Decay from the 4S3/2
and 4111/2 Manifolds in 0.5% ErrLuAG
• following 4S3/2 E x citatio n ............................. ...............................................96
12. R ate C onstants for Relaxation following 4S3/2
Pum ping (psec1) ................................................ i.......................................... 97
Vlll
LIST OF FIGURES
Figure
Rage
1. Free Ion Energy Levels for the E r3+ Io n ................................................ ;...... 3
2. E xperim ental Set-up for A bsorption and
E m ission ....................................................
16
3. Response Curves for D etector/B eam splitter
C om binations.........................................................
.17
4. Form ation of the Interferogram in Step-Scan
M o d e.................................
22
5. Time Resolved Spectrum Exhibiting
Broadband System atic Noise.......... ............................................................24
6. Illu stratio n of System atic Noise Introduced
by Source Pulse In sta b ility ......................................................................... 25
7. IR (top) and Visible Absorption Spectra of
ErrY AG......................................................................................
29
8. IR (top) and Visible Absorption Spectra of
E rrLuA G .......................................................................................
9. U noriented IR (top) and Visible Absorption
Spectra of ErrY SO ............................
30
31
10. IR Absorption Spectra of ErrYSO for Light
Polarized P arallel to the <010>, Bi, and D2
D irectio n s.................................................................
11. Visible Absorption Spectra of ErrYSO for
Light Polarized P arallel to the <010>, Di,
and D2 D irections...................................................................................
12. Absorption T ransitions Observed in Figures
7-11...........................................................
13. R adiative Em ission T ransitions Observed
for 4Fg/2 and 4F t/2 E x citatio n ....................................................
33
.34 i
38
14. IR Em ission Spectrum of Er:YAG Pum ped to
the 4Fg/2 M anifold...........................................................................................39
15. IR Em ission Spectrum of Er:YSO Pum ped to
the 4Fg/2 M anifold...................................... ..............;................................. .'49
16. IR Em ission Spectrum of Er:YAG Pum ped to
the 4F 7/2 M anifold................................................. ........................................42
17. IR Em ission Spectrum of ErfYSO Pum ped to
'th e 4F 7/2 M anifold................... ;..................................................................... 43
18. Visible Em ission Spectra of ErfYAG (top)
and ErfYSO Pum ped to the 4F 7/2 M anifold..............................................44
19. Tem porally Resolved Em ission of Er:YSO
Following Excitation to the 4Ii3/2 M an ifo ld ............................................46
20. Tem porally Resolved Em ission of Er:YSO
Following Excitation to the 4111/2 M an ifo ld .............................................47
21. Tem porally Resolved Em ission of Er:YSO
Following Excitation to the 4F 7/2 M anifold...............................................48
22. Decay (top) and Log of Intensity for th e 4Ii3/2
-> 4115/2 T ransition in 2% E r:Y SO ,.............................................................51
23. Decay of Em ission T ransitions O riginating
from the 4111/2 M anifold in 2% ErfY SO ......................................................53
24. Log of Intensity of Em ission T ransitions
O riginating from the 4111/2 M anifold in 2%
ErrYSO..............................................................................................................54
25. Decay.(top) and Log of Intensity of Em ission
T ransitions O riginating from the 4S3/2
Manifold in 2% ErrYSO............................................................................... 57
26. Decay of the 4113/2 -> 4Ii5/2 T ransition for 2.1%
and 4% ErrYAG.............................................................................................. 58
27. Log of In ten sity of the 4113/2 -> 4115/2 Em ission
T ransition in 2.1% and 4% E rrY A G .......................................................... 58
28. Decay (top) and Log of In ten sity of Em ission
T ransitions O riginating from the 4I n /2
Manifold in 4% ErrYAG.......................................................................... '...61
29. Decay (top) and Log of Intensity of Emission
T ransitions O riginating from the 4Ss/2
M anifold in 4% ErrYAG.............................................................................. 02
30. Decay of the 4Ii3/2 4115/2 T ransition for '
0.5% E rrL uA G .......................................................... ;.............. ’....................33
31. Decay (top) and Log of Intensity of the 4L m
4115/2 T ransition in 0.5% ErrLuA G ........................................................ 05
32. Decay .(top) and Log of Intensity of the 4Sgyg
4113/2 T ransition in 0.5% ErrLuA G ........................................................ 00
33. Rise of the 4Iigyg -> 4Lsyg T ransition for E r3+
in Three C rystal H osts................................................................................. 07
34. Illu stratio n of Integration M ethod Used to
G enerate Decay T ra c e s.................................................................................74
35. C urvature Observed in Logarithm ic Plots of
Em ission D ecay s........................................................... ;..............................75
36. D ata from Figure 35 After Addition of a
C onstant Value to the Decay Trace D ata for
B aseline C orrection....................................................................................... 77
37. Decay Processes Following 4L m E x citatio n ............................................... 78
38. L inear (top) and Log Plots of Decay from the
4113/2 Manifold (Level I) of 4% ErrYAG.................................................... 87
39. L inear (top) and Log Plots of Decay from the
4111/2 M anifold (Level 2) of 4% ErrYAG......................................................88
40. Logarithm ic Plots of Decay from the 4111/2
M anifold (top) and 4Ii3/2 Manifold of 0.5%
E rrL uA G ......................................................................
89
41. Logarithm ic Plots of Decay from the 4I im
M anifold (top) and 4Ii3/2 M anifold of 2%
ErrYSO.............................................................................................. ...............Q0
42. Decay Processes' Following 4Ssti E x cita tio n ..............:.............................. 93
43. L inear (top) and Log Plots of Decay from the
4113/2 M anifold (Level I) of 2% ErrY SO .....................................................99
44. L inear (top) and Log Plots of Decay from the
4111/2 Manifold (Level 2) of 2% ErrY SO ......... '................................... ..... 100
45. L inear (top) and Log Plots of Decay from the
4S3/2 M anifold (Level 5) of 2% ErrYSO.....................................................101
46. Log Plots of Decay from the 4I is/2 (top), 4L 1/2 ■
(center), and 4S3/2 Manifolds of 4% ErrYAG................................. ;.......102
47. Log Plots of Decay from the 4113/2 (top), 4111/2
(center), and 4Ss/2 Manifold of 0.5%
E rrL uA G ...........................................................
..103
A l. Flow sheet for Calculations W hen
E xperim ental Energy Level D ata a r e '
A vailable...............................
A2. Flow sheet for Calculations W hen
E xperim ental Energy Level D ata for the
Rare E a rth Ion in a Sim ilar Host are
A vailable........................................
118
"
119
Xll
ABSTRACT
Spectroscopic analysis and kinetic modeling' of relaxation processes
have been completed for 2.1% and 4% Er:YAG, 2% Er:YSO, and 0.5%
ErrLuAG a t room tem perature. Sim ultaneously tim e and frequency resolved
em ission spectra were obtained using step scan Fourier transform
. spectroscopy. E xperim ental emission lifetim es of the 4L 3Z2, 4I uz2, and 4S3z2
levels were obtained directly from these spectra. The 4L 3z2 lifetim es were
found to be 6.6, 6.5, 8.6, and 5.7 msec for 2 .1% and 4% ErrYAG, 2% ErrYSO,
and 0.5% ErrLuAG, respectively. Lifetim es of 67.3, 10.5, and 67.7 psec were
observed for 4% ErrYAG, 2% ErrYSO, and 0.5% ErrLuAG, respectively for the
4111/2 state. E xperim ental lifetim es of 10.7, 3.5, and 12.3 psec were observed
for the 4S3z2 state in 4% ErrYAG, 2% ErrYSO, and 0.5% ErrLuAG.
Q uantum -m echanical calculations using M orrison's code were
perform ed and theoretical radiative rate s were generated. Comparison of
these values w ith experim ental emission lifetim es allowed determ ination of
non-radiative relaxation rates and branching ratios.
Additionally, an experim entally based new m ethod of determ ining nonradiative rates was developed th a t utilizes the intensity "volume" under the
three-dim ensional em ission bands.
The two m ethods provided good
agreem ent.
Finally, the rate constants obtained by these m ethods were plugged
into a set of coupled differential equations and. num erically solved using the
fifth order R unge-K utta m ethod in M athem atica. Good agreem ent w ith the
experim ental tem poral behavior was obtained (including emission rises as
well as decays) indicating the validity of the results. Non-radiative vs.
radiative branching ratios were approxim ately equal for the first excited
state 4Ii3/2, bu t higher electronic states typically showed > 99% of the
population relaxing via the non-radiative channel.
I
CHAPTER >1
INTRODUCTION
E rbium has been found to be a useful and versatile active atom for a
variety of laser applications. It is capable of lasing n ear 3 pm, a desirable
w avelength for laser surgery, arid around 1.7 pm in the eyesafe region (1,2).
L asing has also been achieved a t several IR and visible w avelengths via
upconversion processes (2-8).
E rbium is often doped into solid-state
crystalline hosts, which offer the benefits of durability, reliability, and
convenience. For these reasons, there has been considerable in terest in solidsta te lasers using erbium as the active ion. Since the optical properties such
as absorption and em ission influence laser perform ance, spectroscopic
investigation has been carried out on erbium in a wide variety of crystalline
hosts (9-56). Additionally, lasing has been achieved in m any solid state laser .
m aterials using erbium as the active lasing species. (2-11, 36, 45, 54-76).
Energy Levels in E rbium L aser M aterials
• Erbium , like all rare earth elem ents typically used as active lasing
species, is m ost commonly incorporated into crystalline hosts in the triv alen t
2
state. The resu ltin g ground electron configuration of the E r3+ ion is a xenon­
like electron stru ctu re w ith 11 additional electrons in the partially filled 4f
orbitals ([Xe] df11). The energy level stru ctu re of the erbium ion in crystalline
hosts resu lts from the Coulombic interaction of the 4f electrons w ith the
nucleus and w ith each other, spin orbit coupling, and the host crystal field
(77).
The energy levels of the free E r3+ ion resu lt from the electrostatic
interaction and spin orbit coupling. The Coulombic interaction yields 2s+1L
term s, which are fu rth er split into J sta te s by the spin-orbit interaction.
The
resu ltin g free ion energy level diagram w ith the 2^ 1L j states is shown in
Figure I, which includes the ground 4115/2 state through the seventh excited
state (4F 7/2). In actuality, the levels shown in Figure I are not pure 2s+1L j
states, due to m ixing of the levels. Each of the levels referred to by the 2s+1L j
term symbol has other sta te s mixed in. As can be seen by exam ination of the
diagram , the free ion level separation is typically thousands of w avenum bers
and tran sitio n s betw een the free ion levels are in the visible and infrared
regions of the spectrum . .
The host crystal field fu rth er splits each 2s+:LLj term into a manifold of
S tark levels. The degeneracy of the 2s+1L j sta te s is 2J+1, and th is degeneracy
is reduced by the crystal field splitting (77). The splitting is relatively sm all,
w ith an overall m anifold w idth of typically several hundred cm-1, and a
splitting betw een adjacent S tark levels of norm ally 50 - 100 cm-1, due to the
shielding of the 4f electrons from the host crystal field by the outer 5s and 5p
3
E n e r g y ( c m 1)
4F 5/2
------------------------------------------------------- 22100
4F 7/2 ■ -----------------------------------------------— - 20500
2Hnz2------- ------------------------------------------------- 19200
4Ssz2 —;
:--------------------------------------------- -— 18300
4Faz2
15300
4l9Z2 ------------------------------------------------ :---------------------------------
12500
4I iiZ g -------------------------------- :-------------------------------------------------- 1 0 2 0 0
4 IlSZ 2
6700
4 IlSZ 2
0
Figure I. Free Ion Energy Levels for the E r3+ Ion.
4
electrons. The S tark level splitting can be exploited w hen a laser operating
a t a specific w avelength is desired.
Coarse w avelength selection can be
accomplished by choice of a specific rare e a rth ion, and fine “tuning” can be
accomplished by selection of a host crystal th a t shifts the S tark levels such
th a t the m aterial is suited, for the desired application. For th a t reason, S tark
level assignm ent for different rare e a rth ions (including E r3+) in crystalline
hosts h as historically been of interest, and a significant am ount of work has
been perform ed in th is area for e rb iu m (l,2,4,6,13,14,16,19-21,23,26,33,36,3841,46,52,54,65).
C haracterization of L aser M aterials
The potential perform ance and operating characteristics of laser
m aterials are to a large extent due to th e ir spectroscopic properties.
absorption spectrum
will determ ine p o tential pum p
The
w avelengths and
influence pum p efficiency, since pum ping will be more efficient at strong
absorption frequencies. After excitation has ta k e n place, relaxation can occur
via spontaneous radiative emission, nonradiative decay processes such as
m ultiphonon relaxation, and/or ion-ion energy tran sfer processes. Radiative
em issions can potentially be used as lasing transitions, and nonradiative
5
processes including energy tran sfer and m ultiphonon decay can enhance or
d etract from' laser perform ance, depending upon the. desired operating
characteristics and pum ping m echanism ..
To ascertain the effect of the aforem entioned properties on a p articu lar
lasing medium, it is useful to answ er the following questions:
o W hat is the S tark level stru ctu re of the m aterial?
o A fter excitation to a given m anifold of S tark levels, w hat levels are
involved in the relaxation process?
Are these levels populated
radiatively or nonradiatively? W hat levels are bypassed?
o
Once populated, how is an excited state manifold depopulated? Is
the manifold depopulated by radiative processes, nonradiative
processes, or both? How quickly is it depopulated?
°
W hat is the relative distribution (branching ratio) of the term inal
levels of tran sitio n s originating from a p articu lar manifold? W hat
are the branching, ratios of nonradiative relaxation processes?
o Do changes in dopant ion concentration (Er3+ in th is study) or host
crystal affect these population and/or depopulation processes?
o W hat, if any effects, are introduced by external factors such as
tem perature or excitation beam power?
E xperim ental m ethods or strategies th a t can answ er some or all of
these questions are of interest.
C haracterization of a m aterial involves
6
absorption spectra, em ission spectra resu ltin g from continuous excitation to
various manifolds, and tem porally resolved emission spectra following
m odulated or pulsed excitation t o ' the sam e manifolds.
The absorption
spectra
structure
are
useful
for
determ ination
of S ta rk
level
and
identification of both excitation frequencies for emission experim ents and
pum p
frequencies for laser
operation.
The
emission
spectra provide
inform ation about the relaxation m echanism s from p articu lar manifolds and
identify prom ising laser transitions.
The tem porally resolved emission
spectra provide inform ation about the dynam ics of both the radiative and
nonradiative relaxation processes, which is of in terest since the relative rate s
of population and depopulation of energy levels can affect w hether and how
well a m aterial can function as a laser.
For these reasons, these three
experim ental m ethods are all used in th is study.
.
While spectroscopic study is useful in providing the previously
m entioned inform ation, combining it w ith com plem entary tools such as
quantum -m echanical.m odels and kinetic modeling using rate equations can
increase its effectiveness. Em ission spectra alone allow for direct observation
of radiative emission tran sitio n s and provide some indirect inform ation about
nonradiative relaxation transitions. However, nonradiative relaxation rate s
and branching ratios can be gained by com paring experim entally determ ined
lifetim es (which r e s u l t ' from both radiative and nonradiative m eans of
depopulation) w ith eith er low tem p eratu re or theoretically determ ined
7
radiative m anifold lifetim es (which assum e radiative processes are entirely
responsible for depopulation).
In this study, theoretical lifetim es are
obtained from the quantum -m echanical model discussed in the next section.
A lternatively, the rate and the relative contribution of the nonradiative
relaxation processes can be determ ined by a newly developed m ethod
(presented in C hapter 4) th a t utilizes the in tensity “volume” under the three
dim ensional em ission bands of the tem porally resolved spectra.
The kinetics of relaxation can be investigated by using the rate
constants for the various relaxation processes (obtained by the experim ental
and/or theoretical m ethods discussed in the previous paragraph) following
excitation to a p a rticu lar level by solving sets of sim ultaneous rate equations
for the involved energy levels.
The com bination of experim ental methods,
quantum -m echanical modeling, and kinetic modeling using rate equations
can provide inform ation th a t none of the three m ethods can provide alone,
including
inform ation
about
processes
th a t
cannot
be
observed
spectroscopically.
Q uantum -M echanical Model
A quantum -m echanical model developed by M orrison et al was used to
calculate theoretical radiative lifetimes.. The model is described in detail in
Refs. 78-80 and is described briefly in th is section.
8
Free ion wavefunctions are determ ined using a H am iltonian th a t
contains electrostatic, spin-orbit , and
L3
interactions (81). The host crystal
field splits the free ion energy levels into S ta rk levels.
Calculations of the
S tark effects use a crystal field H am iltonian of the form:
H-CF = ^BnmCnm ( 0
(I)
/
where the
Bnm
'
are crystal field p aram eters which describe the effect of the
host crystal field on the free ion energy levels, the Cnm are spherical tensor
operator components, nm covers the values of n and m allowed by the
sym m etry of the site of the rare e a rth in the host crystal, and th e i
sum m ation is oyer all of the 4f electrons of the rare e a rth ion. The crystal
field p aram eters are related to the crystal field components
A nm,
which are
spherical tensor components of the host crystal field at the ion site, calculated
by a point charge lattice sum m ation. The
Anm
and
Bnm
are related by the
rad ial factor p,h which are prim arily functions of the rare e a rth ion, though
they can be influenced slightly by the host crystal, and the relationships:
(2)
B u m —P n A n m
( 4 ki + 2 4 h + 2 4 ? 4 + " O iz2
S ta rk levels for a rare e a rth ion in a p articu lar host crystal can be
calculated entirely from the theoretical model. To do so, host x-ray d ata can
be used to determ ine
Anm
and E quation 2 can be used w ith the newly
9
determ ined
A nm
to calculate the crystal field p aram eters I W
The calculated
Bnm can th en be used w ith E quation I used to determ ine the crystal field
H am iltonian. A lternatively, if experim ental energy levels are known, a set of
Bnm
can be fit to the experim ental data. The
used along w ith the
A nm
B nm
from the fit can th en be
for the host and E quation '3 to calculate pn, which
are th en used w ith
Anm
for a sim ilar host and E quation 2 to calculate
the sim ilar host.
By the second method, experim entally determ ined S tark
Bnm
for
levels for a rare e a rth ion in one host can be fit and used to calculate S tark
levels for the sam e ion in the other hosts from the sam e "fam ily" (i.e. the
garnets) providing b e tter results th a n purely theoretical calculations (I). The
M orrison model is lim ited to low dopant concentrations, as it does not include
the effect of the rare e arth dopant ion on the host crystal (which would
change the point charge lattice sum m ation and the
Anm
values).
The model can also be used to calculate theoretical emission transition
probabilities, which m eans th a t theoretical radiative lifetim es of a particu lar
energy level can be obtained as well, using th e Ju d d - Ofelt trea tm e n t (82,83).
In th is treatm en t, the Judd-O felt p aram eters
Qk
(which contain the host
dependent p a rt of the electric dipole line strengths) can be calculated (39,79)
and used to calculate the theoretical electric dipole line stre n g th SzdiJ- -» J ) .
The m agnetic dipole line stren g th S m d (J ^ J ), which is entirely dependent on
the rare e a rth ion w ithin the theory, is calculated from the theoretical
m agnetic dipole oscillator strength. The line strengths for each tran sitio n of
10
in terest ca.n th e n be related, to the E instein A. coefficients for the respective
transitions.
11
CHAPTER 2
EXPERIMENTAL PROCEDURE
In th is chapter, the overall approach to analysis of the m aterials, including
types of spectra collected and justification for collecting them , will be
explained.'
Next, the discussion includes a description of the analyzed
sam ples and the equipm ent and techniques utilized in obtaining the various
spectra.
The chapter concludes w ith a brief discussion of the tem porally
resolved Fourier Transform spectroscopic technique used to gather m ost of
the data.
Overall M ethod of Analysis
We are interested in the various population and depopulation
processes (both radiative and nonradiative) of the energy levels in' erbium doped laser m aterials.
The project goal w as to use several spectroscopic
techniques to gain relevant inform ation about these processes.
absorption spectra were .collected for each sample.
First,
A lthough absorption
spectra of laser m aterials are of fundam ental in terest since they provide
inform ation on potential pum p w avelengths for laser use, the
more
12
im m ediate and practical use for the purposes of th is project was to provide
appropriate excitation frequencies of the sam ples for collection of emission
spectra.
While absorption spectra for Er:YAG (4,6,26,34,35,37,38,46),
Er:LuAG (9), and Er:YSO (34,36,39,40), are well documented in the
literature,- it was still beneficial to ru n absorption on the specific sam ples
analyzed to account for any im purities or anom alies.
Two different em ission techniques were em ployed in th is project:
sta n d ard em ission and tem porally resolved emission. The stan d ard emission
spectra were obtained using a continuous excitation source, bu t source
m odulation was necessary to obtain the tem porally resolved spectra.
The
sta n d ard
and
em ission
spectra
were
obtained ' m uch
more
rapidly
conveniently th a n the tem porally resolved spectra and provided sufficient
inform ation to m ake them worthwhile, even though the spectral inform ation
w as reproduced in the •tem porally resolved emission spectra.
First, the
continuously pum ped emission spectra were useful for in itial identification
and assignm ent of em ission bands and th e ir relative contribution tow ard
depopulating highjer energy levels after excitation. Secondly, they provided
inform ation useful in the setting up of the tem porally resolved spectra. The
longer-lived em ission bands typically appeared stronger in the continuous
spectra, so some very rough qualitative inform ation on emission band
lifetim e could be extracted.
However, th is inform ation is by no m eans
conclusive since actual emission tran sitio n intensity m ust also be considered.
13
Finally, identification of emission bands by continuous spectra facilitated the
settin g ' up of appropriate w avelength ranges for tem porally resolved
spectroscopy.
In
addition to providing the
above-m entioned information,
the
tem porally resolved emission spectra provided emission decay rates and
allowed the deconvolution of m ultiple overlapping bands which were difficult
to distinguish in the spectral dimension. Second, since em ission bands w ith
the sam e decay profiles likely originate from the sam e manifold, it provided
more convincing evidence for the assignm ent of emission transitions. Third,
com paring
em ission
spectra
excited
at
different
frequencies
yielded
inform ation about the nonradiative processes involved in deexcitation.
Finally, the ability to pum p several excited states of the sam ple and observe
the “tu rn in g on and off” of emission bands helped identify the originating
m anifold of the em ission transitions. These capabilities will be dem onstrated
in C hapters 3 and 4.
Sam ples
Scientific M aterials Corporation provided the erbium -doped YSO,
YAG, and LuAG sam ples analyzed in th is project. The sam ples were discs
cut from crystals grown by Scientific M aterials and polished on the edges and
faces. They are approxim ately one-eighth inch thick and range from 0.5 to 1.5
14
inches in diam eter. E rbium ions occupy y ttriu m sites in both the YAG and
YSO crystal lattices, while they occupy lutetiu m sites in LuAG. There are
two inequivalent sites occupied by erbium in Er:YSO, and the YSO host has
three principal axes of polarization (39). While both orientationally resolved
and unoriented absorption spectra were obtained for Er:YSO, orientationally
resolved emission spectra were not obtained since it has been determ ined
th a t the contribution, of the two erbium centers to emission (on the tim e
scales studied in th is work) does not depend on which one is pum ped more
efficiently. (39) Moncorge et al found th a t the fluorescence bandshapes were
identical for all the pum p w avelengths studied (it is expected th a t for a
p a rticu la r w avelength one site would be excited more efficiently th a n the
other), indicating th a t energy tran sfer occurs on a sufficiently faster tim e
scale th a n the emission tran sitio n s studied, which occur on the microsecond
and millisecond tim e scales.
-
.
I
E quipm ent and Techniques
The need to conduct the various experim ents discussed earlier in this
chapter m akes a versatile spectroscopic facility necessary.
The equipm ent
“and experim ent setups used in th is project are nearly identical to those
described by Richard Martoglio (a previous m em ber of the Lee Spangler
research group at M ontana S tate.U niversity) in his thesis (84) detailing his
15
sim ilar work w ith thulium -doped m aterials. These procedures will be briefly
outlined in th is section, and all differences fro m ' the equipm ent setups
described by M artoglio will be discussed in detail.
A general equipm ent
layout is shown in Figure 2.
Absorption
Absorption experim ents were carried out using the B ruker IF S 66
Fourier Transform Spectrom eter.
The in stru m e n t,.which was described in
detail by M artoglio (84), utilizes a M ichelson interferom eter as its basic
design.
The source w as a broadband tu n g sten lam p included in the
spectrom eter.
Two different beam splitters were used depending on the
spectral range desired for the specific experim ent: a calcium fluoride
beam splitter was generally used for lower frequency m easurem ents and a
quartz beam splitter w as utilized for higher frequency spectra.
A liquid
nitrogen cooled,indium antim onide detector was utilized in the infrared,
while visible detection w as accomplished using a silicon diode. Figure 3
shows
response
curves
for
all
four
possible
detector/beam splitter
combinations. The detector/beam splitter com binations for all absorption a n d
em ission experim ents were chosen to get the best signal response for the
desired spectral region.
Coherent
cw Dye
Laser
I Coherent Innova 20 ANon Laser]
j
Coherent Sabre Ar-ion Laser
Lambda Physik
Scan Mate OPO
Sample
Off-axis
parabolic
Figure 2 . E xperim ental Set-up for Absorption and Emission.
Bruker IFS66
Step Scan FT
Spectrometer
17
Reference HeNe laser line
Si Diode/Quartz
25000
20000
InSbZCaF2
InSbZQuartz
..I— 15000
10000
Wavenumber (cm1)
Figure 3: Response Curves for D etector/B eam splitter Combinations.
Emission
An argon-ion laser provided a convenient excitation source for emission
experim ents since all the sam ples studied absorb the 488 nm line.
Two
different Ar-ion lasers were used as sources during the project. In addition to
functioning directly as an excitation source, one of the Ar-ion lasers was also
used to pump a C oherent continuous dye laser, allowing access to other
erbium absorption bands.
18
The general, emission setup is illu stra te d in Figure 2. A fter excitation
by the Ar-ion laser, sam ple emission was collected by an off-axis parabolic
m irror. The parabolic m irror was obtained from B ruker and w as originally
designed for light collection before entering the IF S 66 spectrom eter, thereby
ensuring f-m atching (f/4.5) to the IF S 66.
appropriate
distance
outside
the
The sam ple w as placed at the
interferom eter to
allow
f-matching.
Collecting the em ission in the off-axis parabolic m irror improved emission
signal-to-noise by a factor of approxim ately 40 over sim ply placing the
sam ple in front of the IF S 66 emission port, due to collection of a larger solid
angle of the em itted radiation.
Additionally, an aperture was sometimes
placed a t the IF S 66 emission port to elim inate unw anted scattered laser
light. The sam e beam splitters and detectors used for absorption were used for
the emission m easurem ents, and optical filters were som etim es placed in the
IF S 66 absorption sam ple cham ber to prevent unw anted emission signal
and/or scattered source light from reaching the detector.
Tem porally Resolved Em ission
In th is section, the equipm ent and setup for the tem porally resolved
em ission experim ents are discussed. The tem porally resolved spectroscopic
m ethod is discussed in more detail in the next section.
Since source m odulation was necessary to conduct the tim e-resolved
experim ents, the Ar-ion source (or Ar-ion pum ped continuous dye laser) was
19
m odulated using either a chopper (for m illisecond pulses) or an acousto-optic
m odulator (for microsecond pulses) for the earlier experim ents during the
in itial stages of the project.
A shorter excitation pulse w as necessary to study the relaxation
processes in these m aterials, m any of which occur on the microsecond tim e
scale or faster, th u s nanosecond pulses were used for m ost of the tem porally
resolved work. The 1064 nm output from a Coherent Infinity Nd:YAG laser
was tripled to 355 nm and used to pum p a Lam bda-Physik ScanM ate OPO.
The OPO could provide tunable output from 400 nm to nearly 2.5 microns,
and output powers used were typically 2-5 m J/pulse. W ith system repetition
rates adjustable from 0.1 to 100 Hz, the Infinity/ScanM ate combination
provided the necessary flexibility to excite each different sam ple to a num ber
of different electronic states. Additionally, the repetition rate could be set to
allow for complete relaxation of the m aterial before the next pulse was
delivered.
The general experim ent layout for the tem porally resolved emission
work w as identical to the continuous em ission setup (see Figure 2). W hen
the Infinity/OPO excitation source w as used, the beam w as focused from
approxim ately one cm to 2 mni in diam eter using a f/12 lens (based on lens
diam eter) placed betw een the OPO outlet and the sample. The sam ple was
placed at the focal point to maximize pum p power density. Additionally, the
sam ple placem ent w as such th a t the excitation beam struck the sam ple near
20
the edge to m inim ize self absorption.
The moving m irror in the B ruker
interferom eter was operated in step scan mode ra th e r th a n rapid scan mode
(this will be discussed fu rth er in the next section).
Since the m irror w as
designed to be a t a complete stop while gathering data, the system w as
susceptible to m echanical vibration. There are two m ajor differences' in the
equipm ent setup used for gathering m ost of the tim e resolved d ata in th is
study w ith the setup employed by M artoglio. (84)
First, the Infinity/O PO
source w as m ounted on a larger optical table along w ith the interferom eter.
The optical table in use w hen M artoglio gathered his d ata was too sm all to
accommodate the entire system.
Tying the entire system together on one
table greatly dim inished the effect, of building vibrations on the system,
enhancing the signal-to-noise ratio. Additionally; the setup w as moved to the
Spangler group’s tem porary laboratory facility at Scientific M aterials
Corporation, which was located‘on the ground floor as opposed to the second
floor laboratory in Gaines H all a t MSU.' The overall building vibrations were
less on. the ground floor location, again providing improved m echanical
stability and signal-to-noise ratio.
Tem porally Resolved FT Spectrosconv
In th is section, the tem porally resolved Fourier Transform technique is
briefly discussed. The technique is discussed in detail by M artoglio. (84)
21
In a sta n d ard Fourier Transform experim ent using a Michelson
interferom eter, the moving m irror is scanned continuously on the order of 4 0
kHz, known as “rapid scan” mode.
Tem poral resolution at a given
w avelength cannot be obtained in rapid scan mode because the m irror
position changes during the course of the tem poral event.
However,
operation of the IF S 66 in step scan mode allows the user to obtain tem poral
inform ation.
In step scan mode, the moving m irror is stopped at each
location and d ata is obtained. The m ethod of obtaining data is illu strated on
Figure 4 . At each m irror position, the desired num ber of d a ta points are
obtained a t a tim e interv al corresponding to the desired tem poral resolution.
This process, can be completed as m any tim es as desired for signal averaging.
D ata corresponding to a p articu lar m irror position is shown as a horizontal
column in the first p a rt of the figure. An interferogram is constructed using
the d ata from each of the different m irror positions at one m om ent in time,
known as a “tim e slice” (illustrated in the second p a rt of Figure 4). This
process is repeated for each of the tim e slices, resulting in a separate
interferogram for each m om ent in tim e. The d ata needed for each of these
interferogram s is a vertical column in the first p a rt of Figure 4 .
Each
interferogram is th e n Fourier transform ed to yield a spectrum for each tim e
slice (last p a rt of Figure 4 ) , resulting in a tem porally resolved
spectrum .
e m is s io n
m irror p osition
tim e
mirror
p osition
1
1
2
3
4
6
6
7
8
9
10
I
2
'------X--------------------------------------------------------
3
4
5
6
7
8
9
10
X
2
3
X
X
X
4
X
6
X
X
X
*
X
* ---------------------------- X
X
X
•
•
X
tim e
W avelength
Figure 4. Form ation of the Interferogram in Step-Scan Mode.
*this m od el assu m es one freq u en cy only
23
In ten sity fluctuations in the Infinity/OPO source coupled w ith the tim e
resolved FT m ethod led to broadband baseline noise- which decayed
proportionately w ith the signal over tim e (see Figure 5). The way in which
these fluctuations m anifested them selves as broadband noise can most easily
be seen by looking at the example of a single emission frequency (see Figure
.6). If the emission intensity is the sam e for each m irror position, the
interferogram will be a sine wave, which is Fourier transform ed into a single
frequency. Fluctuations in emission in ten sity lead to an interferogram th a t
is not quite sinusoidal (see Figure 6). This nonsinusoidal interferogram is
Fourier transform ed into m ultiple frequencies, corresponding to the actual
em ission frequency and noise. This effect is of course compounded in emission
from the erbium doped laser crystals, which have m ultiple emission bands as
opposed to a single em ission frequency. This effect can actually sometimes be
beneficial since the noise from intensity fluctuations is spread over the whole
spectral region ra th e r th a n being concentrated over the emission band.
However, it can be detrim ental in the case where an em ission spectrum
contains both strong and w eak bands, since noise from intensity fluctuations
in the strong bands can obscure the w eak bands.
Two actions were tak en to address these noise issues. Increased signal
averaging was the sim plest solution, although it m eant longer experim ent
tim es, leaving the experim ent more susceptible to long-term fluctuations.
Also, w hen it was desirable to exam ine a w eaker em ission band, stronger
Relative In ten sit
04
Figure 5. Time Resolved Spectrum Exhibiting Broadband System atic Noise.
25
For a pulsed monochomatic source, if the intensity of all pulses is equal, the interferogram will be a sine wave consisting of discrete points.
-X
aaaaAaaaa L
n o ise
W hen the signal is an entire emission band or bands, this effect is compounded
greatly, leading to broadband noise.
Figure 6 . Illu stratio n of system atic noise introduced by source
pulse instability
26
bands which contributed noise to the w eaker band were filtered out optically,
eith er by the use of filters or strategic selecting of detector/beam splitter
combination.
In extrem e cases including very low signal, large source intensity
fluctuations, and/or significant scattering of laser light into the detector, a
m irror position other th a n the position corresponding to the zero p a th
difference has h a d the highest intensity in the interferogram . The position
corresponding to zero p a th difference should be the m ost intense in the
interferogram
since it is the
only position "where th ere
is complete
constructive interference of all w avelengths and is the 'centerburst in a
typical interferogram . (85)
It was necessary to fix the proper zero p a th
difference position, as opposed to the m ost intense, as the centerburst, or
Fourier transform errors resulted.
^27
CHAPTER 3
EXPERIMENTAL RESULTS
This chapter contains results from absorption and emission spectra
obtained from the Er:YAG, Er:LuAG, a n d 'E r:YSO sam ples, and a brief,
in terp retatio n of the results. A more thorough discussion and exam ination of
the d ata will follow, in C hapter 4.
Absorption
Absorption spectra of Er:YAG (4,6,26,34,35,37,38,46), Er=LuAG (9),
and Er=YSO (34,36,39,40) are well docum ented in the literatu re.
These
include spectra collected at b o th room tem perature (-300 K) and at low
(liquid helium or nitrogen) tem peratures. ■While low tem perature spectra are
necessary to m ake individual S tark level assignm ents, all absorption spectra
in th is study were obtained at room tem p eratu re (-300 K),' which was
sufficient to identify pum p frequencies for emission study and to check for
any im purities or anom alies in the specific sam ples.
The
B ruker
m easurem ents.
IF S 66
Spectrom eter
was
used
for
all
absorption
M easurem ents were ta k e n over two spectral regions:
28
“visible” (9000-25000 cm-1), which utilized a q uartz beam splitter and a Si
diode detector, and IR (2000-15780 c m 1), which used a calcium fluoride
beam splitter and indium antim onide detector (see Figure 3).
Some IR
absorption spectra were also tak en over the 2000-10500 c m 1 range. A
tu n g sten lam p was used as the light source for all m easurem ents.
The
num ber of scans for signal averaging ranged from 32 to 2000, and spectral
resolution w as typically 2 c m 1.
Figure
7
contains
EriYAG
absorption
spectra.
Although
m easurem ents were tak en on several concentrations of erbium in YAG only
the 2.1% concentration d ata is shown, since varying the concentration in a
given crystal host w as found by experim ent to not affect the absorption
bandshapes. Er:LuAG absorption spectra are shown in Figure 8. Absorption
spectra for ErrYSO are shown in Figures 9-11. There are two inequivalent
sites occupied by erbium in ErrYSO, and the YSO host has three principal
axes of polarization (39). U noriented absorption spectra are shown in Figure
9. Figures 10 and 11 are infrared and visible absorption spectra for light
polarized p arallel to the three polarization axes. All the transitions seen in
Figures 7 through 11 have the ground sta te 4Ii5/2 m anifold as th eir in itial
level and term inate in the levels as indicated on the spectra. Figure 12
shows the energy level diagram containing the m anifolds involved in the
absorption bands in Figures 7-11. The sam e bands are presen t in Figures 10
Absorbanci
29
12000
10000
Wavenumber (cm 1)
Absorbance
4F 7/2
20000
15000
Wavenumber (cm1)
Figure 7. IR (top) and Visible Absorption Spectra of Er:YAG.
10000
Absorbanci
30
12000
10000
Wavenumber (cm-1)
20000
15000
Wavenumber (cm1)
Figure 8. IR (top) and Visible Absorption Spectra of Er:LuAG.
10000
31
12000
10000
Wavenumber (cm-1)
Absorbanci
2.5 •
20000
15000
10000
Wavenumber (cm1)
Figure 9. Unoriented IR (top) and Visible Absorption Spectra of Er:YS0.
32
Wavenumber (cm-1)
12000
Wavenumber (cm 1)
Figure 10. IR Absorption Spectra of Er:YSO for Light Polarized P arallel to
the <010>, D i1 and
Directions.
33
Wavenumber (cm1)
Wavenumber (cm1)
Wavenumber (cm-1)
Figure 11. Visible Absorption Spectra of Er:YSO for Light Polarized P arallel
to the <010>, Di, and D2 Directions.
34
and 11 as in Figure 9. The band ju st under 4000 c m 1 in the IR spectra is a
w ater absorption band resulting from im perfect background correction. The
sharp spikes occurring at regular w avenum ber intervals (especially noticable
in Figure 10) are noise caused by interference from AC power supplies in the
laboratory.
11/2
Figure 12. Absorption T ransitions Observed in Figures 7-11.
Comparison of Figures 7, 8 and 9 (IR spectra of Er:YAG, Er:LuAG, and
Er:YSO respectively) shows the effect of the host crystal field on the erbium
35
energy level structure. E xam ination of the spectra .shows th at, in. general,
the Er:YSO bands are about 100 c n r1 lower in energy th a n the corresponding
bands in the Er=YAG and ErrLuAG spectra.
More notably, for any of the
tran sitio n s labeled I through 8 on Figure 12, the absorption bandshapes for
Er=YSO are quite different th a n those of the erbium doped garnets. These
differences are due to the crystal fields of the hosts splitting the erbium free
ion levels differently, yielding a different S ta rk level stru ctu re in the three
host m aterials.
Since the crystal fields of the two garnet hosts are m uch
more sim ilar to each other th a n they are to YSO, the differences in the
absorption spectra of Er=YAG and Er=LuAG are m uch more subtle.
Absorption
spectra
(4,6,26,34,35,37,38,46)
and
studies
in which
individual S ta rk levels are identified (26,37) have been perform ed previously
in Er=YAG.
Good overall agreem ent w as seen betw een the m easurem ents
tak en in th is study and room tem perature m easurem ents in the literatu re
(4,6,35,38). Both bandshapes and absorption band frequencies were in good
general agreem ent. Sim ilarly, good overall agreem ent w as noted between the
Er=LuAG absorption spectra fro m ' th is study and lite ra tu re data (9).
Comparison of the Er=YAG and Er=LuAG spectra w ith d a ta obtained in the
individual energy level cold tem perature studies (4,9,26,37) shows some
deviations which are expected betw een spectra tak en a t room tem perature
and liquid helium or nitrogen tem perature. First, the room tem perature
spectra were more congested w ith features. Secondly, each m anifold,in my
36
room tsm p o rstn ro d.3t3 is bro3d.6H6d 60 - 100 cm-^ ovsr th.6 corrsspond.iiig’
band in the 4K data.
In each case, t h e ' broadening w as tow ard the low
frequency side of the bands. This was expected since population of the upper
S tark levels of the ground state 4115/2 m anifold leads to more absorption
pathw ays to each upper state manifold, broadening the absorption bands.
The resu lting new absorption pathw ays are lower in frequency, since they are
in itiated from upper S ta rk levels of the ground state manifold.
Analogously, good agreem ent w as seen the orientationally resolved
absorption spectra of ErfYSO (Figures 10 and 11) in th is study and the room
tem p eratu re d ata in the literatu re (34,36,39,40). Sim ilar broadening of the
room tem p eratu re ErfYSO spectra relative to cold tem perature spectra in the
lite ra tu re (39) w as observed.
Table 3 a t the end of th is chapter lists
individual absorption peak frequencies for the three m aterials studied. The
peak frequencies correspond to absorptions betw een individual S tark levels
and show reasonable agreem ent w ith those encountered in the literature.
(1,4,9,26,37,39,46).
Continuous Em ission
Fluorescence spectra have been obtained for ErfYSO (39), ErfYAG
(4,16,37,38,46), and Er=LuAG (9) a t a variety of erbium concentrations. The
spectra
obtained
in
th is
study
were
consistent w ith
the
literatu re
37
m easurem ents. The ErrYSO emission spectra obtained in th is study were not
o n en tatio n ally resolved along.the three polarization directions. As would be
expected, comparison of the polarized em ission spectra of ErrYSO in the
lite ra tu re (39) w ith d ata obtained in th is study shows features resulting from
all three polarizations are present in my data.
In th is study, the 488 nm line from one of two Ar-ion lasers (see
C hapter 2) provided continuous excitation of erbium in both YAG and YSO to
the 4F y/2 manifold. Additionally, the Ar-ion was used to pum p a cw dye laser,
which, w hen using the laser dye DOM, provided cw excitation to the 4Fgz2
manifold. Figure 13 contains energy level diagram s illu stratin g the emission
tran sitio n s
observed
after
4Fgz2
and
4F yzz excitation,
respectively.
N onradiative relaxations are not indicated in the energy level diagram s.
Continuous em ission d a ta was not obtained for Er:LuAG, though num erous
tem porally resolved em ission spectra were obtained for th is m aterial
(discussed in the next section).
Visible and IR em ission spectra were
obtained a t the sam e spectral ranges and using the sam e beam splitters and
detectors as the absorption spectra. The num ber of scans used for signal
averaging ranged from 64 to 2000, and resolutions of either 8 or 16 cm-1 were
obtained at room tem perature. In order to compare the stren g th of different
emission transitions in the same spectrum , it was necessary to correct for
in stru m e n t response. This was accomplished by recording a response curve
38
using a tu ngsten lamp, then elim inating the lamp contribution by dividing a
calculated intensity versus frequency curve for a blackbody at 3500 K.
Figure 13. Radiative Em ission T ransitions Observed for 4F<)/2 and 4F t/2
Excitation.
IR Emission
Figure 14 is a fluorescence spectrum of 4% Er:YAG, pum ped to the
4Fg/2 manifold by 15446 c n r1 dye laser output. Assignm ents of the emission
bands are shown on the spectra. In the bottom view, the y-axis (intensity) is
scaled to enable a closer look at the two w eaker transitions. The same
tran sitio n s were observed in the em ission spectrum of 2.1% Er:YAG. Figure
15 contains a sim ilar spectrum of 2% Er:YSO, which was excited at a slightly
lower frequency (15376 cm"1) due to the different S tark level splitting of the
YSO host. The tran sitio n s initiated from the 4111/2 manifold are not seen due
39
to the lower signal to noise ratio in th is experim ent.
41 11/2
10000
41 11/2
41 15/2
The different host
41 13/2
6000
Wavenumber (cm1)
41 13/2
41 15/2
10000
Wavenumber (cm 1)
Figure 14. IR Em ission Spectrum of Er:YAG Pum ped to the 1F;)/* Manifold.
40
... V
w v IMAi A^vzWaM
^
vk^
10000
Wavenumber (cm-1)
Figure 15. IR Em ission Spectrum of Er:YSO pum ped to the 4Fg/2 Manifold.
effects also exhibit them selves in the em ission spectra. The Er:YSO 4113/2
4115/2 em ission band does not exhibit the sharp lines seen in the Er:YAG
sam ples. This trend generally holds for the other emission bands throughout
the infrared and visible regions exam ined in this study. The apparent sharp
rise in intensity below about 2800 cm 4 is not real emission, bu t results from
imperfect correction for the in stru m en t response curve, thereby representing
the lower useful frequency lim it for the em ission spectra.
Figure 16 contains an emission spectrum of 4% ErrYAG excited to the
4F 7/2 m anifold using the 488 nm Ar-ion line. The top and bottom p arts of the
figure are simply the sam e spectrum , w ith the y-axis scaled differently to
41 ■
allow exam ination of all the emission bands.
Figure 17 features a sim ilar
tre a tm e n t of 2% Er:YSO. E xam ination of Figures 14 and 16 reveals th a t 4Fyza
excitation results in an emission tran sitio n th a t is not there under 4Fg/2
excitation.
This transition, from the 4Ss/2 to the 4111/2 manifold, occurs at
around 8000 cm"1. The analogous em ission transitions take place in Er:YSO,
and are seen in Figure 17.
Visible Em ission
Figure 18 shows visible em ission spectra of 4% Er=YAG and 2%
Er:YSO, respectively. The tran sitio n around 10000 c n r1 is the sam e 4I nz2 ->
4115/2 tran sitio n seen in the IR em ission spectra. The strong green emission at
approxim ately 18000 cm"1 is the 4S3/2
band a t 15000 c n r1 is the 4Fg/2
4Iis/2 transition, and the very weak
4115/2, emission. The large spike seen ju st
above 20000 c n r1 in both spectra is scattered 488 Ar-ion laser light. The
lower signal-to-noise ratio above 22000 c n r1 is due to decreased instrum ent
response (see Figure 2).
Tem porally Resolved Em ission
The tem porally resolved em ission m easurem ents presented in this
section were obtained using the experim ental setup described in C hapter 2.
The Infinity/OPO system provided nanosecond excitation pulses for all the
Relative Int
42
4111/2
10000
Wavenumber (cm 1)
Relative Intensity
4111/2
10000
Wavenumber (cm1)
Figure 16. IR Emission Spectrum of Er:YAG Pumped to the 4F7 /2 Manifold.
43
2500
S
1500
_2 1000
10000
6000
Relative Int
Wavenumber (cm-1)
4000
Wavenumber (cm1)
Figure 17. IR Emission Spectrum of Er:YSO Pumped to the 4F 7 /2 Manifold.
44
488 nm laser line
I
4Saza -> 4I 15/2
Relative Inti
<D
41 11/2 " 4 41 15/2
.4-1
.2
4Ssza "4 4I is/2
4F dzs -> 4Iisza
:
O-'
25000
20000
Wavenumber (cm-1)
488 nm laser line
0|L
.
25000
20000
15000
10000
Wavenumber (cm1)
Figure 18. Visible Emission Spectra of Er:YAG (top) and Er:YS0 Pumped to
the 4Ft/2 Manifold.
45
m easurem ents presented in this section. The spectral resolution was 16 cm-1
for m ost experim ents. Unless otherwise noted, the excitation repetition rate
w as set a t 15 Hz, which provided an interv al of 66.67 ms between pulses.
This repetition rate w as used to allow nearly complete relaxation and
corresponds to at least five exponential lifetim es for the slowest emission
(4111/2 -> 4115/2) decay.
Infrared tim e-resolved em ission was collected after
excitation to each of the first eight excited sta te m anifolds of erbium (H13/2
through 4Fszc). These excitation m anifolds can be seen in Figure 12.
In the following paragraphs, resu lts which are common to all the
erbium m aterials studied will be presented. W here applicable, exam ples to
illu stra te these resu lts will be provided using Er:YSO em ission spectra since
the tim e resolved spectra collected for th is m aterial showed the best overall
signal-to-noise. The spectra shown in th is section have not been corrected for
in stru m en t response.
Selective excitation at different pum p w avelengths allowed observation
of the “tu rn in g on and off’ of emission bands, helping to confirm the
originating m anifold of the emission transitions. W hen pum ped to the 4113/2
manifold, the resu ltin g emission spectrum (Figure 19) contains only one
tran sitio n , from the 4113/2 directly back to the ground sta te manifold. The
em ission occurs on the millisecond tim e scale a t around 6500 c m 1. Each
spectral “tim e slice” shown in Figure 19 is at a 250 microsecond interval.
46
Wavenumber (cm1*
Figure 19. Tem porally Resolved Em ission of Er:YSO Following Excitation to
the 4113/2 Manifold.
Excitation to any of the next three higher manifolds (4111/2,
4I<)/2,
or 4Fgzo)
yields two additional emission transitions, originating from the 4111/2
manifold and term in atin g in the
41 13/2
and 4115/2 manifolds, resulting in an
emission spectrum as seen in Figure 20. These two new emissions decay on a
considerably faster tim e scale th an the 4113/2
previous spectrum . The 4113/2
4 1 15/2
4115/2 tran sitio n shown in the
tran sitio n is still present at 6 5 0 0 c m 1,
but is still rising in intensity on the faster tim e scale of Figure 20, where
each tim e slice is five microseconds. The fact th a t the two new emissions
decay on the sam e tim e scale indicates th a t th eir em itting levels are
depopulated on the sam e time scale, so they alm ost certainly originate from
the sam e manifold. This illustrates one benefit of the tem porally resolved
method, th a t it helps in assignm ent of the em ission transitions.
47
Figure 20. Tem porally Resolved Em ission of Er:YSO Following Excitation to
the 4111/2 Manifold.
Pum ping at or above the 4S3/2 manifold yields three additional
emission bands, as can be seen in Figure 21. These three bands also occur on
the microsecond tim e scale, but decay even faster th an the emissions from
the 4111/2 manifold. These three bands, occurring at roughly 5800, 8000, and
11500 c m 1, originate from the 4S3/2 m anifold and term inate in the 4l 9/2, 4In/2,
and 4113/2 manifolds,
respectively. Additionally,
approxim ately 18000 cm 1 (4S3/2
spectrum .
an
emission band
at
4Iisza, Figure 13) can be seen in the visible
Figure 21 illu strates another benefit of the tem porally resolved
emission method. In a tim e resolved experim ent w ith tem poral resolution of
5 microseconds or higher, the 4S ^
4I<)/2 tran sitio n will be detected at 5800
48
c m 1. However, in continuous spectra such as Figure 17, this emission band
is hidden by the strong, long-lived 4I j3Z2 ^ 4Iisz2 band.
The tim e resolved
technique enables deconvolution of em ission bands th a t overlap spectrally
but have different tem poral behavior.
4S 3 /2 - > 41 13/2
12000
41 13/2 - > 41 15/2
10000
6000
Wavenumber (cnr1)
Figure 21. Tem porally Resolved Em ission of Er:YSO Following Excitation to
the 4F tz2 Manifold. The collected d ata set includes tim e from 0-130 psec, but
the d ata is truncated for visual clarity.
The weak emission from the 4Fyz2 manifold to the ground state was not
observed in the tim e resolved spectra due to its short lifetime and low signalto-noise ratio.
In the following sections, specific resu lts for the three
m aterials studied are presented.
49
Er:YSO
The 2% Er:YSO sam ple was excited a t the various frequencies as listed
in Table I.
At each excitation frequency, experim ents were run a t the
necessary tem poral resolutions and sam pling tim es to capture the tem poral
behavior of all the emission bands th a t occurred for a p a rticu la r excitation.
Table I. Excitation Frequencies for 2% Er:YSO.
Pum p Manifold
E xcitation Frequency (cm-1)
41 13/2
6508
10195
12606
15376
18409
19223
20494
22095
4Il 1/2
419/2
4Fg/2
. - 4S s/2
2H h z 2
4F 7 /2 .
4F s/2
A lthough there are two different E r3+ sites in Er:YSO, Li, Wyon, and
Moncorge (39) have shown th a t they can be considered together in the
analysis of fluorescence dynamics, since excitation of the two different sites
to different degrees resulted in the sam e emission lineshapes (see pg. 14).
Therefore, polarized tem porally resolved emission spectra were not obtained
in th is study. In the following paragraphs, results for each of the emission
tran sitio n s w hen pum ped at the different levels will be presented.
The
4Ii3/2->4Ii5/2
emission
w as
detected
at
all
pum p
levels.
E xperim ents ru n to characterize th is tran sitio n had tem poral resolution of
250 psec, and d ata w as obtained for 66.5 msec, nearly the entire tim e
50
betw een laser pulses given the 15 Hz repetition rate. Signal averaging was
five or ten coadditions, m eaning five or ten decay events were averaged at
each m irrbr position. Figure 22 contains two plots: a “decay trace”, in which
the in teg rated area under the tran sitio n (corrected for baseline noise) for a
tim e slice is plotted versus time, and a “log plot”, in which the logarithm of
the in teg rated tran sitio n area i s ' plotted instead. The integration and
baseline correction m ethods are discussed in detail in C hapter 4. In the plots
shown, m ost of the decay traces have been norm alized to a m axim um
in ten sity value of one to facilitate comparison betw een different spectra. The
d a ta shown in Figure 22 were obtained by after excitation directly into the
4113/2 manifold. Though valuable inform ation concerning relaxation processes
can be obtained by pum ping above the upper em itting level, it is best to
excite the em itting level directly w hen determ ining lifetim es to p re v e n t'
em ission level feeding processes from influencing lifetime m easurem ent. The
stra ig h t line of the log plot shows th a t the decay is exponential. An
exponential fit of the decay d ata resu lts in an experim entally determ ined
lifetim e of 8.6 msec (experim ental lifetim es for all the m aterials studied are
sum m arized in Table 4 a t the end of th is chapter). The d a ta in this plot are
tru n cated after the signal has gone below the noise level.
Two emissions originating from the 4111/2 level appear in spectra where
the 4111/2 m anifold or higher is pumped, term in atin g at 4113/2 (emission band
a t approxim ately 3500 cm-1) and 4115/2 (approxim ately 10000 c m 1). Emission
51
Normalized Int
5c
ill.
5000
log (Normalized Intensity)
0
:\
-0.5
I .i
.5 10
2 10
2.5 10
Time (microseconds)
11
X
-
TTT -rr
I l l l
X
-I
I l l l
X
-
I l l l
3.5 10
I l l l
I
l
l
l
-
X
-1.5
-2
3 10
X
-
x Xv
■ . , , ,
0
I I
5000
I IO4
I I
I
I
I
,
I
l
i
l
t
I
I
I
I
1.5 IO4 2 IO4 2.5 IO4 3 IO4 3.5 IO4 4 IO4
Time (microseconds)
Figure 22. Decay (top) and Log of Intensity for the 4Iisza -> 4Iisza T ransition in
2% Er:YSO.
52 *
from the 4Inz2 manifold has a significantly shorter lifetime th a n th a t
originating from 4113/2, and the experim ents designed to analyze these
em ission bands were tem porally resolved to 500 nanoseconds and had to tal
sam pling tim es of 120-150 microseconds.
Since th is higher tem poral
resolution required the use of a faster, 8-bit tra n sie n t recorder ra th e r th a n
the 16-bit recorder used for the slower experim ents, lower signal-to-noise
resulted, necessitating more coadditions (20 to 80 for the various pum p
levels) for signal averaging.
Figure 23 illu strates decay traces for the two
em ission bands. The m axim um intensity for both bands is norm alized to one
to show the sim ilar behavior of the two bands, which indicates th a t they
originate from the sam e manifold. E xperim entally determ ined lifetim es for
the 4Inz2 level of 2% ErrYSO ranged from 10.3 to 12.7 microseconds for all the
pum p levels analyzed, .with values of 10.3 to 10.7 (is for experim ents ru n w ith
direct 4Inzz excitation.
The intensity of the tran sitio n term inating in the
ground state is significantly higher th a t term in atin g in the 4Iiaz2 level. The
radiative branching ratio was determ ined by taking a single tim e slice from
the experim ent (an acceptable assum ption since the em ission bandshapes did
not change over time), correcting for in stru m en t response, and integrating
the area of each em ission band. The resulting branching ratio was found to
be 0.91 for the 4Inz2 -> 4Iisz2 tran sitio n and 0.09 for 4Inz2 -> 4Iiaz2. B ranching
ratio calculations for all the m aterials studied are sum m arized in Table 5 a t
the end of this chapter.
53
1
0.8
1
2c
Ti
N
15
4111/2->4115/2
4111/2->4113/2
X
"X
0.6
0.4
E
X
X
z
0.2
....
0
0
10
20
30
40
Time (microseconds)
50
60
■1*111*«*!IAiifcAA
70
Figure 23.
Decay of Emission T ransitions O riginating from the 4Ii 1/2
Manifold in 2% Er:YSO.
The log plots of the above emission transitions (Figure 24) illustrate
th a t the decays are prim arily exponential.
The mild degree of curvature
observed in Figure 24 occurred frequently in this study and was found to be
error introduced by baseline correction. The reasoning for this is discussed in
the next chapter.
D ata points are truncated after the signal reaches the
noise level.
Em ission tran sitio n s originating from the 4S;$/2 m anifold occur when
the ErrYSO w as subjected to excitation to th a t level or higher.
Spectra
obtained for ErrYSO included the four emission bands originating from the
54
'53
c
"5
CD
S
aC
Q
S
O
S
b/D
JD 4
Time (microseconds)
Figure 24. Log of Intensity of Emission T ransitions O riginating from the 4I n /2
Manifold in 2% Er.YSO.
4S3/2 manifold th a t were discussed earlier in this section. Emission spectra
obtained to study these transitions typically required tem poral resolution of
50 nanoseconds w ith 400 time slices (for a total sam pling duration of 20
microseconds) and 20 - 80 coadditions for signal averaging.
The IFS 66 spectrom eter cannot perform time resolved experim ents on
both sides of the HeNe reference frequency (15798 c m 1) sim ultaneously. The
HeNe
provides
the
reference
points
for
where
the
sam pling
(data
acquisition) is performed. Signal at higher frequencies th a n the HeNe are
undersam pled
in
step
scan
and
would
alias
if
light
frequencies on both sides were detected. In continuous scan, a phase
55
locked loop can interpolate the HeNe curve to allow more frequent sam pling
which elim inates th is problem. In step scan, two laser detectors are used:
one to lock on a HeNe zero crossing; the other (-90° out of phase) is
used to determ ine the slope to tell which h a lf w avelength is being locked
on. Therefore, the strategy used in continuous scan cannot be used in step
scan, and the four emission tran sitio n s originating from 4Ss/2 could not be
evaluated in the sam e experim ent.
The three lower frequency transitions
were observed in spectra obtained over the range of 2000 - 15700 cm-1, and
the visible tran sitio n (around 18000 c m 1) was obtained in a separate
experim ent. IR spectra were obtained after excitation to the 4Ss/2, 4F ^ , and
4F 5/2 levels. Since the silicon diode detector w as easily sa tu ra te d it was very
difficult to detect emission transitions in the spectral region close to the
excitation source, therefore, we were not able to obtain a tim e resolved
em ission spectrum containing the 4Ss/2
4115/2 tran sitio n after pum ping the
4S3/2 manifold. However, this tran sitio n w as observed in tem porally resolved
spectra after excitation to the 4F7/2, and 4F 5/2 manifolds.
Figure 25 shows
decay traces of the three IR emission bands following 4S3/2 excitation. The
4S3/2 -> 4Ii3/2 tran sitio n has the best signal-to-noise ratio, and an exponential
fit based on th is tran sitio n yields a lifetim e of 3.5 microseconds. The 4S3/2 -> ■
4111/2 tran sitio n is the w eakest and the figure illu strates the effect of the lower
signal-to-noise on the decay trace. Figure 25 also contains a log plot of the
decays.
Again, the slight curvature in the log traces is attributable to
56
' baseline correction.
Branching ratios were calculated for these radiative
tran sitio n s based on spectra obtained after 4F 7/2 excitation, since a visible
spectrum was obtained a t th a t excitation level.
ErrYAG
The 4% ErrYAG sam ple w as excited a t the frequencies as listed in
Table 2. Additionally, emission spectra were obtained from 2.1% ErrYAG
following excitation to the 4Fg/2 manifold. The same emission transitions were
observed for the sam e pum p m anifolds as in ErrYSO.
R esults for the .
fluorescence decays for the 4113/2, 4111/2, and 4Ss/2 levels are discussed in the
following paragraphs.
The 4113/2
4Ii5/2 emission occurred on a sim ilar tim e scale in ErrYSO
and ErrYAG, and experim ents ru n to evaluate th is tran sitio n were set up
w ith sim ilar tem poral resolution, sam pling duration, and num ber of
coadditions. Figure 26. shows the integrated, results of the transition after
Table 2. Excitation Frequencies for 4% ErrYAG.
Pum p Manifold
E xcitation Frequency (cm"1)
4113/2
4111/2
4Fg/2
4S 3/2
4F 7/2
6797
10347
15445
18797
20494
57
S'
'5e
js
-tiGJ
yCS
4S3/2 -> 419/2
4S3/2 -> 4111/2
4S3/2 ~> 4113/2
S
S
2;
Time (microseconds)
4 S3/2 > 419/2
4S3/2 -> 4111/2
4S3/2 ~> 4113/2
Time (microseconds)
Figure 25. Decay (top) and Log of Intensity of Emission Transitions
Originating from the 4S:3/2 Manifold in 2% Er:YSO.
58
4F9/2 excitation for the two erbium concentrations studied. The decay profile
for this tran sitio n in EriYAG is sim ilar regardless of excitation manifold
since the emission rise time is 2-3 orders of m agnitude shorter th an the decay
tim e for all the excitation levels studied.
2% EivYAG
4% Er:YAG
0
1 104
2 104
3 104
4 104
5 104
Time (microseconds)
6 104
7 IO4
Figure 26. Decay of the 4113/2 -> 4Ii5/2 T ransition for 2.1% and 4% Er:YAG.
The experim entally determ ined lifetim es for this tran sitio n is on the
order of 6.5 - 7 milliseconds for the two sam ples. A difference in lifetime for
the two concentrations could not be confirmed w ithin experim ental error.
While the traces shown in Figure 26 would seem to indicate a shorter lifetime
in the 4% sam ple, in another sim ilar set of experim ents the decay rate was
slightly faster in the 2.1% sam ple.
Figure 27 illu strates log plots for the
59
decay traces shown in Figure 26. The slight curvature seen in the log plots
again is attrib u tab le to baseline correction.
2.1% Er:YAG
4% ErYAG
,
.
I
,
3 10
4 10
Time (microseconds)
Figure 2 7 . Log of Intensity of the 4113/2
and 4% Er:YAG.
->
4115/2 Emission T ransition in 2 . 1 %
The two emission transitions originating from the 4111/2 manifold decay more
slowly in Er:YAG th a n in Er:YSO.
E xperim ents ru n to evaluate these
tran sitio n s were typically had I msec tem poral resolution and 500 msec total
sam pling tim e. All other experim ental p aram eters were sim ilar to those for
the Er:YSO experim ents. Figure 28 shows the integrated tran sitio n data and
log plot of the fluorescence after 4111/2 excitation. The calculated lifetime is
67.3 microseconds based on the higher intensity 4111/2
the w eaker 4111/2
->
->
4115/2 emission, w ith
411,3/2 emission showing good agreem ent. The emission
60
lifetime is several tim es longer th a n th a t of the sam e tran sitio n in Er:YSO.
In both m aterials, m ultiphonon relaxations dom inate relaxation from the
4111/2 level, and the m echanism appears to be more efficient in Er:YSO. Decay
resu lts for 2.1% ErrYAG were not obtained for excitation to th is manifold, bu t
experim ents ru n on both concentrations w ith 4Fgys excitation do not show a
strong concentration dependence in the 2-4% concentration range.
.
Figure 29 illu strates in tegrated tran sitio n d ata and a log trace of the
visible 4Ssza
4Iisza emission. Decay from th is level is also exponential, as
shown by the log trace. The IR tran sitio n s from the 4Ssza level show sim ilar
tem poral behavior. An exponential fit of th is decay process for 4% ErrYAG
yields a calculated lifetime of 10.7 microseconds, which is longer th a n the
analogous process in ErrYSO (approxim ately 3.5 microseconds). The tem poral
resolution for the experim ents ru n to analyze these tran sitio n s was 250 nsec,
w ith a to tal sam pling tim e of 150 microseconds.
O ther experim ental
p aram eters were the sam e as those for the ErrYSO experim ents.
ErrLuAG
Time resolved em ission spectra were obtained for 0.5% ErrLuAG after
excitation to the 4Iisza, 4Inza, and 4Ssza m anifolds, using pum p energies of
6525, 10385, and 18448 cm-1, respectively. As w ith the other samples, the
pum p m anifolds were selected since they are the em itting levels. The sam e
61
'
I '
1
£
4111/2 -> 4115/2
4111/2->4113/2
C
c
-3
QJ
N
H
S
•-
Z
Time (microseconds)
'
'
I '
4111/2->4115/2
4111/2 ~> 4113/2
Time (microseconds)
Figure 28. Decay (top) and Log of Intensity of Emission Transitions
Originating from the 411 1 /2 Manifold in 4% ErrYAG.
62
£
'3
c
CU
C
-d
N
15
3
O
Time (microseconds)
'fafl
Se
-a
I
cs
E
Z
Time (milliseconds)
Figure 29. Decay (top) and Log of Intensity of Emission Transitions
Originating from the 4S 3 /2 Manifold in 4% Er:YAG.
63
tran sitio n s were observed in 0.5% E r:LuAG as in the other m aterials studied.
Since the tem poral behavior of the em ission transitions in E r:LuAG was close
to th a t in Er:YAG, the same experim ental param eters were used in the
E rL uA G experim ents as in the ErY A G experim ents. D etails of the resu lts
for each of the emission transitions follow.
The integrated 4113/2
4115/2 tran sitio n shown in Figure 30 shows th a t
this tran sition also occurs on the millisecond time scale.
The signal was
w eaker for th is m aterial th an in the other m aterials studied due to the lower
(0.5%) erbium concentration, resulting in decreased signal-to-noise.
decay w as also found to be exponential w ith a lifetime of 5.7 milliseconds.
5000
I IO4 1.5 IO4 2 IO4 2.5 IO4 3 IO4 3.5 IO4 4 IO4
Time (microseconds)
Figure 30. Decay of the 4113/2
4115/2 Transition for 0.5% ErLuAG.
This
64
Figure 31 shows integrated decay and log plots for the 4Inz2 -> 4I i5Z2
transition, following excitation to the 4Inz2 manifold. The tem poral behavior
of the 4IIiz2
4I i3/2 tran sitio n is sim ilar, bu t the signal-to-noise ratio is
slightly lower due to the lower signal intensity of the transition.
The
experim ental lifetime is 67.7 microseconds, and the log plot i s . again
indicative of exponential behavior.
The three infrared emission tran sitio n s in itiated from the 4Ssz2 level
were also observed in the Er:LuAG sam ple.
The in teg rated trace of the
tran sitio n term in atin g at the 4Iisz2 m anifold is shown in Figure 32. This
em ission occurred after direct excitation to the 4Ssz2 manifold. The log plot
showed the .emission decay to be exponential, and the exponential fit on the
decay d a ta yielded a calculated lifetim e of 12.3 microseconds: A ttem pts to
obtain the visible em ission spectrum after excitation to the 4Fyz2 m anifold
failed due to the relatively low signal stren g th of the low concentration
sam ple and satu ratio n of the silicon diode detector by the excitation pulse.
However, th is visible tran sitio n could be seen by the eye.
The rise tim e of the 4Iiszs -> 4Iisz2 tran sitio n has a exponential rise tim e
sim ilar to th a t of the 4Inz2
4Iiszs decay lifetime, indicating th a t the m ain
channel of population of the 4Iiszs level is through the 4Inz2 level. The rise for
all th ree m aterials studied is shown in Figure 33.
65
5
c
'35
QJ
15
S
, ■I
Time (microseconds)
QJ
C
XJ
N
OS
S
biD
't , ,
■ ■ I .
iO
150
, 2i
Time (microseconds)
Figure 31. Decay (top) and Log of Intensity of the 4111/2
0.5% Er:LuAG.
->
4115/2 Transition in
66
' I 1 1
Normalized Inti
•"SS
Time (microseconds)
log (Normalized Intensity)
1 ' I r
, I . .
Time (microseconds)
Figure 32. Decay (top) and Log of Intensity of the 4S 3 /2
0.5% Er:LuAG.
411 3 /2 Transition in
67
4 %
E r Y A G
0 .5 %
2 %
7
E r L u A G
E r Y S O
/Z
.
I
I
,
0
150
2
Tim e (microseconds)
Figure 33.
Hosts.
Rise of the 4113/2
4115/2 T ransition for E r3+ in Three Crystal
A ttem pts were made to study the population of the 4I n /2 and 4S3/2
levels by exam ination of the rise of em ission bands initiated from these
levels.
However, the rise tim es of tran sitio n s initiated from both levels is
faster th a n the 180 nanosecond response tim e of the infrared detector.
Additionally, the rise tim e of the 4S&2 level w as found to be less th a n 5 ns, the
response time of the visible detector. The detector response tim es currently
lim it our ability to study these rapid rise tim es.
68
Table 3. A bsorption T ransition Frequencies in E r3+ Doped M aterials (cm-1).
m anifold
Er=YSO
<010>
6428
6452
6470
6508
6538
6592
6699
6727.
6752
6768
6784
Er=YSO
Dl
6429
6470
6498
6509
6538
6549
6565
6583
6595
6623
6700
6725
6767
Er=YSO
D2
6470
6508
6538
6547
6558
6566
6596
6621
6696
6727
6767
6798
6849
4Il 1/2
10157
10183
10193 '
10209
10235
10252
10270
10283
10296
10315
10343
10382
10100
10115
10156
10174
10195
10209
10227
10233
10251
10268
10293
10315
10333
10344
10114
10136
10158
10183
10196
10209
10227
10233
10253
10261
10286
10294
10304
10316
10332
10343
10384
10175
10191
10206
10223
10231
10250
10262
10279
10296
10307
10331 .
10336
10347
10353
10365
10388
10393
10407 ■
10411
10180
10208
10230
10251
10262
10315
.1 0 3 2 6 .
10337
10348
10360
10365
10380
10385
10403
10417
4I d/2
12197
12218
12322
12361
12257
12279
12225
12240
12220
'12245
41 13/2
Er=YAG
-
6522
6525
6535
6542
6601
6703
6721
6741
6760
6778
6797
6819
6859
6878
Er=LuAG
6493
6519
6525
6541
6549
.6559
6602
6721
6741
6764
6783
6814
6830
6852
6884
69
Table. 3. Continued
)
manifold
4Igzs
4Fg/2
Er: YSO
<010>
12253
12277
.12320
12352
12378
12421
12432
12448
12464
12493
12534
12560
12577
. 12606 .
Er: YS O
Dl
1.2421
12445
12463
12527
12543
12568
12607
12645
ErrYSO
.. D2
12322
12349
12358
12375
12423
12444
12464 ■
12491
12508
12527 .
12563
12572
12606
12644
15072
15095
15133
15158
15168
15192
• 15215
15274
15296
15332
15356
15376
15454
15491
15033
15070
15087
15119
15131
15167
15179
15191
15215
15274
15298
15316
15356
15372
15395
15490
15000
15051
15118
15133
15156
15168
15179
15193
15215
15274
15301
15317
15333
15355
15373
15491
18182
18228
18243
18267
18283
18245
18278
18304.
18329
18351
•
4Sszs
18225
15246
18266
18284
18327
Er: YAG
ErrLuAG
12281
12299
12446
12462
12503
12510
12551
12679
12691
12735
12753
12267
12291
12456
12476
12497
12523
12543
12693
12714
12767'
15207
15220 •
15230
15246
15262
•15279
15286
. 15292
15306
'
15331
15351
15387 •
15435
15444
15464
15490
■
15509
18311
18385
18391
18432
18450
15218
15240
15263
15284
15296
15318
15351
15399
15443
15463
15476
15484
15516
18319
. 18395
18416
18448
70
Table 3. Continued
manifold
ErrYSO
<010>
18366
18382
18410
ErrYSO
Dl
18327
18366
ErrYSO
D2
18363
18382
18408
ErrYAG
■-
2Hl 1/2
19040
19044 .
19070
19077
19127
19134
19139
19158
19197
19225
19235
19260
19307
18942
18968
19002
.19040
19048 ■
19065
19083
19088
19095
19121
19139
19161
19172
19196
19219
19231
19252
19267
19308
18918
18942
18987
19004
19046
19066
19076
19084 .
19094
19124
19138
19160
19178
19196
19224
19260
19268
19306
19013
19032
19050
19069
19090
19110
19126
19146
19262
19279
19304
19319
19338
19357
19015
19036
19059
19068
19076
19090
19112
19272
19284
. 19317
19339
19348
19365
%
20262
20299
20394
20423
20495
20521
20569
20631
20260
20300
20342
20387 .
20396
20425
20468
20497
20519
20567
20609
20259
20343
20369
20387
20395
20425
20469
20496
20520
20610
20638
20653
20433
20447 .
20486
20543
20564
20580
20618
20630
20640
20671
20442
20645
20486
20535
. 20595
20615
20639
20659
4S 3/2
.
ErrLuAG
71
Table 4. E xperim ental Lifetim es of T ransitions in E rbium Doped M aterials
Level
M aterial
41 13/2
2% Er=YSO
2.1% Er=YAG
4% Er=YAG
0.5% Er=LuAG
2% Er=YSO
4% Er=YAG
0.5% Er=LuAG
2% Er=YSO
4% Er=YAG
0.5% Er=LuAG
41 11/2
4S 3/2
E xperim ental Fit
Lifetime fusee)
8640
6600
6500
5700
10.5
67.3
67.7
3.5
10.7
12.3
1/e Lifetime
fpsec)
8500
6800
6500
5800
12.0
73.5
70.0
3.7
11.3
11.8
Table 5. R adiative B ranching Ratios Calculated from E xperim ental D ata.
M aterial
2% Er:YSO
Level
4S 3/2
4% Er:YAG
0.5% Er:LuAG
41 11/2
T ransition
4111/2 *4115/2
4111/2 -3» 4113/2
4S 3 / 2 4115/2
4S 3 / 2 4113/2
4S 3/2 ^ 4111/2
4S 3 / 2 419/2
4S 3/2 ^ 4F<)/2
4S 3/2 ^ 4115/2
4S 3 / 2 ^ 4113/2
4S 3/2™^ 4111/2
4S 3 / 2 419/2
4 S3/2™^ 4Fg/2
4111/2
4I l5 /2
41 11/2
41 13/2
Branching Ratio
0.91
0.09
0.62
0.36
0.01
0.01
0- (not detected)
. 0.72
0.26
0.01
0.01
0 (not detected)
0.95
0.05
72
CHAPTER 4
DISCUSSION OF RESULTS
In th is chapter, the m ethod of generating th e experim ental decay
curves shown in the previous chapter is described. Additionally, kinetic
m odeling of the d ata is presented and rate constants are determ ined. A new
m ethod of determ ining non-radiative rate s and branching ratios utilizing the
in ten sity "volume" under three-dim ensional tim e and frequency resolved
em ission bands is developed.
Values obtained using th is m ethod are
compared to non-radiative rate s and branching ratios determ ined from
comparison of experim ental emission lifetim es w ith calculated radiative
lifetim es using M orrison's code.
G eneration of E xperim ental Decay Curves '
The experim ental decay curves w ere generated from the tem porally
resolved spectra by integration using the software supplied w ith the B ruker
spectrophotom eter. The software gives the option of num erous integration
techniques; of these, the m ethod outlined here gave the best results for the
d a ta in th is study.
73
Figure 34 is a single tim e slice from a tem porally resolved' emission
spectrum . The area betw een lines A and B is obtained by integration and
contains both signal and baseline noise. The baseline noise contribution is
estim ated by defining a spectral region on each side of the emission band
(regions C to D and E to F) th a t are each equal in w idth to the emission band,
and averaging the two in tegrated areas.
This baseline noise is th en
subtracted from the in itial integration to give the in tegrated signal.
The
average of two baseline regions was used to improve the results for spectra
th a t had a “slan t” in baseline intensity, since the goal w as to subtract the
baseline noise contributing to the area u nder the em ission band. While th is
m ethod gave the best results of the m ethods tried, it did not give ideal resu lts
for all spectra and sm all baseline correction errors still occurred.
These
baseline correction errors were found to lead to the slight curvature noted in
some of the log plots of the in tegrated decay traces, as discussed in the next
section.
Discussion of Long Term C urvature in Log Plots
A slight degree of curvature w as noted in m any of the plots when the
logarithm of the in tegrated tran sitio n intensity w as plotted versus tim e.
Obviously, for purely exponential decay, the log plot would yield a straig h t
line. An example of th is curvature is shown in Figure 35.
Both convex and
74
concave curvatures were observed. Figure 35 represents a typical degree of
curvature; in a few instances it was considerably sharper while in other
instances no curvature was noted. This curvature is attrib u tab le to baseline
error introduced by the system atic baseline noise caused by source pulse-topulse fluctuations th a t was discussed in C hapter 2. Since the pulse-to-pulse
in ten sity fluctuations of emission transitions are Fourier transform ed across
the entire spectral dim ension as noise, the presence of one band adds noise to
VVa ~ vvw«~aAi~wvn
Wavenumber (cm 1)
Figure 34. Illustration of Integration M ethod Used to G enerate Decay
Traces. The baseline correction is obtained by averaging the areas between
lines C and D and lines E and F under the spectrum . This baseline
correction value is then subtracted from the integrated area under the
spectrum between lines A and B to obtain the corrected integral value.
75
'S
C
■
2
ti
aca
5
o
%
be
JD
Time (microseconds)
Figure 35. C urvature Observed in Logarithm ic Plots of Em ission Decays.
the baseline of other emission bands in the spectrum , causing error in decay
traces of bands (especially those w ith different tem poral behavior) th a t are
m anifested as curvature in log plots of decay profiles. While the fact th a t the
noise is spread out over the entire spectrum with the Fourier transform
m ethod can be an advantage over dispersive methods, it is a disadvantage in
this instance since it h u rts the quality of the tem poral data.
ru n when the 4113/2
- > 41 15/2
Experim ents
transition is filtered out significantly altered the
curvature seen in log plots of the emission transitions from the 4111/2 level.
Two approaches to baseline correction dem onstrate the connection
between baseline error and the observed curvature.
First, the software
accompanying the B ruker spectrom eter allows several different m ethods of
76
integration and baseline correction of spectra. In teg ratin g the sam e spectra
using a variety of m ethods was found to introduce and/or change the
curvature in the log plots, prim arily due to differences in baseline correction.
The integration m ethod described earlier in th is chapter w as chosen since, of
the m ethods tried, it least frequently resulted in nonzero baselines in the
in teg rated decay traces and led to the best agreem ent in tem poral behavior
betw een tran sitio n s w ith the sam e em itting level.
Second, addition or
subtraction of a sm all constant value from the decay trace after integration
w as successful in reducing and/or elim inating the curvature.
Figure 36
shows the d ata from Figure 35 after th is correction was im plem ented. As can
be seen by comparison of Figures 35 and 36, the degree of curvature has been
reduced substantially. This correction typically only caused a sm all variation
in the exponential lifetime, up to a m axim um of 4%. It is also possible th a t
nonlinear detector response could contribute to the observed curvature,
particularly in the earlier portions of the decay event w hen emission signal
was stronger.
Since the depopulation of the excited states studied w as found to be
dom inated by exponential processes (with the observed curvature explained
by in stru m en tal or signal processing errors ra th e r th a n physical processes),
the modeling described in the next section assum es exponential decay for all
the relaxation processes.
,
77
a,
Q
)
S
N
CC
E
bO
JD
i
Time
Figure 36. D ata from Figure 35 After Addition of a C onstant Value to the
Decay Trace D ata for Baseline Correction. Note the reduction in curvature
compared to Figure 35.
Kinetic Modeling of E xperim ental D ata
In th is section, modeling perform ed to explain the decay processes
responsible for the experim ental results is discussed. Sets of coupled
differential equations describing the population and depopulation of the
energy
levels
studied
were
w ritten
and
num erically
solved.
Both
spontaneous radiative and nonradiative (multiphonon) relaxation processes
are considered in the model.
Sim ilar approaches to modeling for Er:YAG
(4,28,29,43,52), other erbium -doped garnets (4), and EriYAlOs (20,21) can be
78
found in the literatu re.
In this study, two separate cases were modeled:
decay following 4111/2 and 4Sazg excitation. These two pum p manifolds were
considered since they are initial levels for radiative processes.
4Ii 1/2 excitation
The decay transitions considered for the case of 4111/2 excitation are
shown in Figure 37. The radiative processes (which are the sam e ones th a t
were observed experim entally) are indicated by solid lines, while the wavy
lines indicate m ultiphonon relaxation. It is assum ed th a t the dom inant
m ultiphonon transitions are those th a t resu lt in decay from one manifold to
the one directly below it, therefore m ultiphonon processes such as the one
4f „ „
4Iot
Figure 37. Decay Processes Following 4Inzg Excitation.
79
from the 4Inza to the 4Iiszs manifold are not included. For brevity, the 4Inza,
4Iiszs, and 4Iiszs m anifolds will be referred to as levels 2,1, and 0 respectively,
as indicated on Figure 37. Rate equations for the three manifold populations
can be expressed as follows:
= ~ (4>0 +
^21
+
^21 ) ^ 2
^-^- = (A2\ +W2[)N2 - ( A w +Ww)N\
(I)
= (Ao + ^lO )-^l + A o^2
M = population of level i
Aij. = rate constant for spontaneous radiative emission from m anifold i
toy
Wi; - m ultiphonon emission from m anifold i to j
The excitation pulse is short com pared to the lifetim es of most of the
processes, so behavior directly after excitation will be modeled.
In order to evaluate the above set of equations, values are needed for
the rate constants Aij and Wij. The experim entally determ ined lifetime n is
related to the rate constants as follows:
'- i
>0
>0
■
w here ki is the overall rate constant for decay from manifold i.
80
Therefore, for manifold 2 (neglecting Wso):
— = &2 =A 20 + A21 + W21
X
(3)
2
E xam ination of the three dim ensional em ission spectra for the -4Inza pum p
manifold shows th a t m ost of the decay from levels 2 to I is nonradiative.
This can be seen by com paring the to tal volume under the emission
tran sitio n s (see Figure 20) for the radiative em issions for level 2 to I (3500
cm-1) and I to 0 (6500 cm-1).
The volume under the l-> 0 tran sitio n is
significantly larger th a n the 2-> I transition, and the integration over the
to tal volume of a band gives inform ation about radiative depopulation. Since
the to tal population entering a level by both radiative and nonradiative
m eans should equal the to tal population leaving a level, the dom inant
process feeding level I m ust be nonradiative. This “to tal volume balance”
around level I m ay be stated m athem atically as follows:
Total volume of feeding processes - T otal volume of depopulating processes .
A 21 + W 21 —Aio + Wio
(4)
where the bold typeface indicates a to tal volume integral. The Ay and Wij
symbols correspond to the sam e tran sitio n s as the analogous Ay and Wy
symbols in the rate equations.
Strictly speaking, W 21 and Wio are not
volumes since they do n o t , correspond to radiative em ission processes.
Instead, each W value is num erically equal to the total, volume of a
hypothetical radiative em ission process betw een the indicated in itial and
81
term in al levels w ith the sam e probability as the nonradiative rate being
evaluated. Total volume integral values for the radiative processes (Aij) were
determ ined by first in tegrating under the tran sitio n in the spectral
dimension, after the spectrum was corrected for in stru m en t response. This
in teg ratio n w as done for each tim e slice in a tem porally resolved spectrum . A
plot of these integ rated areas versus tim e gives a decay trace like th a t shown •
in Figure 22. Next, the decay trace w as fit to an exponential decay equation
which w as th en in tegrated over tim e to yield the to ta l volume integral.
In teg ratio n of baseline noise over the long exponential “ta il” was found to
introduce significant error, so the exponential fit equation was integrated
ra th e r th a n the experim ental decay trace. E quation 4 can be rearranged to
solve for Wzi:
W n — Wio + Aio - A 21
(5)
R esults specific to the three m aterials modeled (4% Er:YAG, 2% Er:YSO, and
0.5% Er:LuAG) are presented in the following paragraphs.
For the 4% ErY A G sample, the experim entally determ ined lifetime n
w as found to be 6.5 ms.
The theoretical radiative lifetim e n rad, obtained
from a com putational m ethod derived by M orrison et al. (37,78-80) is 13 ms.
U sing E quation 2 and the fact th a t
Ti rad
= I / A 10 , it can be found th a t Aio -
Wio for th is set of experim ental and theoretical radiative lifetimes., The to tal
volume in tegrals of the radiative processes (Aij) can be determ ined using the
procedure described in the previous paragraph. E quation 5 can th en be used
82
to solve for W2 1 . The branching ratio for a radiative or nonradiative process
is defined as the probability th a t depopulation of a level will occur by th a t
process. For example, the branching ratio for the radiative
e m is s io n
of level
2 to level I can be determ ined as follows: .
B a = ------- ^
^
-------A21+A20+W21 A 22+ A 2g+W2j
'
76)
Note the branching ratios can be calculated using either rate constants or
to tal volume integrals. The branching ratios for depopulating level 2 using
th is to tal volume integ ral m ethod are listed in Table 6.
To check the effectiveness of the to tal volume integral calculations, the
branching ratios for level 2 were also calculated using the experim ental and
theoretical radiative (again obtained using the M orrison com putational
method) lifetimes. The relative branching ratio of A 20 and A22 was calculated
by using the experim ental radiative branching ratio presented in Table 5.
The resu lts of these branching ratio calculations are also listed in Table 6
(under “.Using Theoretical Radiative Lifetim e”) and show good agreem ent
w ith the to tal volume integral calculations.
The total volume integral approach was also used to determ ine
branching ratios for level 2 for the 2% ErfYSiO sam ple and the 0.5% ErrLuAG
sam ple.
For Er:YSO, Wio was determ ined using theoretical radiative
calculations perform ed by Moncorge et al (39). For Er:LuAG, since the LuAG
83
Table 6. B ranching Ratios for Decay from 4Inz2 Manifold in 4% Er:YAG
T ransition
U sing Total Volume In teg ratio n
0.992
Using Theoretical RediAtivA
Lifetime
0.993
Asi
7.69X 10-4
7.04 X 10-4
Aso
7.08X10-3
6.50X10-3
crystal host is sim ilar to YAG, it w as assum ed th at, as was the case w ith
Er:YAG, Ww w as approxim ately equal to Aio. Once Wio w as determ ined for
each m aterial, the branching ratios were determ ined using the same to tal
volume procedure used for ErY A G and are presented in Table 7.
Once the branching ratios were obtained, the rate constant A ij (or Wij)
can easily be obtained using the equations:
AJ = b Aij
■ ^iJ = b Wuk,.
where ki
(7)
is the overall experim ental rate constant.
Table 7. B ranching Ratios for 4Inz2 M anifold in 2% E rY SO or 0.5% Er:LuAG
T ransition
2% E rY SO
0.5% Er:LuAG
Wiu
0.997
0.984
Asi
1.85 X !O'4
1.37 X 10-3
A so
2.42 X 10-s
1.51 X 10-2
84
Using this equation, Ay and Wij were calculated for all three materials
modeled. Using these values and Mathematica, the system of simultaneous
differential equations was solved. Tahle 8 contains the fate constants used in
the differential equations.
Table 8. Rate Constants for Relaxation following 4111/2 Pumping (psec1)
Transition
4% ErtYAG
2% ErfYSO
0.5% Er:LuAG
A io
7.69 X10-s
1.16 XlO-4
8.77 X 10-s
Wio
7.69 XlO-s
A 20
1.06 XlO-4
A 21
W 21
8.77 X10-s
-
1.15 XlO-s
2.25 XlO-4
1.73 XlO-s
2.04XlO-s
1.48 XlO-2
9.30 XlO-2
1.47XlO-2
2.26 XlO-4
The lack of entry for Wio for 2% Er:YSO results from the theoretical
radiative lifetime value of 5.7 ms for
Ti,
which is shorter than the
experimentally observed lifetime of 8.6 ms.
Moncorge et a l (39) have
attributed this situation to lifetime lengthening due to radiation trapping.
The allocation of probability for relaxation via radiative emission or
multiphonon processes does not affect the ' solution of the differential
equations, so the relative contribution of radiative and nonradiative decay
from level one has not been explored, further in this study. Decay curves for
both manifolds in all three
materials
are
shown compared . to the
85
experim ental d ata in Figures 38 through 41. The decay curves are shown as
solid lines, while the actual experim ental d ata points are shown. Both linear
and logarithm ic plots are shown for 4% Er:YAG, while only logarithm ic plots
are shown for the Er:LuAG and Er:YSO samples.
. The decay curves
generated by the model show good overall agreem ent w ith the experim ental
data.
Sum m ary of Total Volume In tegral M odeling M ethod '
. The steps involved in the kinetic modeling calculations described in
the last section are sum m arized as follows:
1. R ate equations for the population of each manifold included in the
model are w ritten and include rate constants for spontaneous
radiative emission
(A y )
and m ultiphonon relaxation (Wij).
2. Total volume balances are w ritten around energy levels th a t are
being both populated and depopulated in the model. For the case of
4111/2 excitation, the only level m eeting this description is level I
(411 3 /2 ).' The balance is w ritten using the principle th a t the sum of
the total volumes of the processes populating a level is equal to the
sum of the processes depopulating the level. Total volume integrals
are signified using bold typeface symbols, i.e. Aio.'
3. The total volume integrals Ay are determ ined from experim ental
data.
86
4. The “to tal volume integral” Wjo is estim ated by comparing the ■
experim ental lifetim e and theoretical radiative lifetim e of level one
and the relationships:
I
^ rad
4o
' I .
= A10+ JVm
^ Iexp
W10 _ w m
A 10
4o
5. The total volume integral balance from Step 2 is solved for upper
level nonradiative processes, i.e. W 2 1 .
6. B ranching ratios
are
determ ined ■using E quation
6 (sim ilar
equations can be w ritten for Wg).
5
_
^21
A21+A20+JV21
_
'^ 2 1
____
A 2J+ A 20+JV2j
( 6)
7. Rate constants Ay (or Wij) are th e n calculated using Equation 7:
4/ = V
(7)
8. The rate constants determ ined in Step 7 are p u t into the rate
equations from Step I and solved using M athem atica.
Normalized Intensity
87
10000
20000
30000
40000
30000
40000
log (Normalized Intensity)
Time (microseconds)
20000
-6
-7
Time (microseconds)
Figure 38. Linear (top) and Log Plots of Decay from the 4Iiazz Manifold (Level
I) of 4 % Er:YAG. The solid lines are modeled decay, and the points are
experimental data.
Normal i zed Intensity
88
0.8
J __
Time (microseconds)
>>
C/l
C
U
- 0 .5
C
1O
(L)
_N
C8
S
Ui
O
%
- 1 .5
Oti
O
- 2 .5
Time (microseconds)
Figure 39. Linear (top) and Log Plots of Decay from the 411 1 /2 Manifold (Level
2) of 4% Er:YAG. The solid lines are modeled decay, and the points are
experimental data.
log (Normalized Intensity)
89
-
0.2
-0 .4
-
0.6
-
0.8
-
1.2
log (Normalized Intensity)
Time (microseconds)
JT)000
15000
20000
25000
30000
35000
-6
-7
Time (microseconds)
Figure 40. Logarithmic Plots of Decay from the 4L 1/2 Manifold (top) and 411,3/2
Manifold of 0.5% Er:LuAG. The solid lines are modeled decay, and the points
are experimental data.
90
Time (microseconds)
Figure 41. Logarithmic Plots of Decay from the 411 1 /2 Manifold (top) and 4 11 3 /2
Manifold of 2% Er:YSO. The solid lines are modeled decay, and the points
are experimental data.
91
4S 3/2 excitation
The decay tran sitio n s considered for the case of 4Ss/2 excitation are
shown in Figure 42.
As was the case w ith the 4Inz2 model, the radiative
processes are indicated by solid lines, the wavy lines indicate m ultiphonon
relaxation, and the manifolds .will be referred to as levels 0 through 5 in
order of increasing energy.
Rate equations for the populations of the six
m anifolds are expressed as follows:
dt
( 8)
- -(W sa + 4 b + ^ 52 +4si + A5a) N 5
— I- = Rf,,AT,
at
^
at
= Aa N ^ W a N i -W n N,
~~T~ = 4-52^5 + W iN 3 ~ (A20 + W21 +A2])N 2
at
I
- W = A sqN 5 + (A10 +Ww)N^ +A20N 2
at
Ni - population of level i
Aij - rate constant for spontaneous radiative em ission from manifold i
to ;
Wij - m ultiphonon emission from m anifold i to j
Total volume balances around levels 1,2,3,and '4 (sim ilar to E quation 4) can
be stated as follows:
92
A si
+ W 2 1 + A 2 1 = A io + Wio
A s2
+ W32 = A21 + W21 + A»0
(9)
As,3 + Wio = Woo
Wsi = Wio
The Wio values for each m aterial were the sam e ones used in the calculations
for 4111/2 excitation in the previous section.
Once Wzo (and therefore Wzo)
were determ ined, .the values of" the other Wy could be calculated by
rearran g in g the E quations (9) and solving them sequentially as follows:
Wsi = Wio = Wzo + Aio + Aoo - A si -Aso - Aso
Woo —Aso + Wzo
Woi = Aio + Wzo - A si - Aoi
The Aij are calculated from the experim ental d ata using the same m ethod
utilized for 4111/2 excitation (described in the previous section).
R esults
specific to the three sam ples modeled are presented in the following
p aragraphs.
For the 4% Er:YAG sample, branching ratios for the 4Iim and 4Ss/2
m anifolds are calculated using equations analogous to E quation 6 and are
listed in Table 9.
The branching ratios were also calculated using the
experim ental and theoretical radiative lifetim es for comparison w ith the to ta l
volume in tegral calculations. The relative branching ratio of Aso, A s i, A 5 2 ,
and A s s w as obtained from Table 5 while the relative branching ratio for A 2 0
93
X
Figure 42. Decay Processes Following 4S3/2 Excitation.
and A 21 was obtained from emission d ata following 4S3/2 pumping, to
elim inate potential effects from population in other levels th a t m ight cause
deviation from the 4111/2 manifold branching ratio after direct I n /2 excitation.
These branching ratios are also listed in Table 9 under “using theoretical
radiative lifetim e”. For both levels, the total volume integral m ethod predicts
less nonradiative depopulation th an would be expected from comparision of
the experim ental and theoretical lifetimes.
The total volume integral approach was also used to determ ine
branching ratios for the 2% Er:YSO and the 0.5% Er:LuAG sam ples.
The
Er:YSO results are presented in Table 10, and the Er:LuAG results are in
94
Table 9. B ranching Ratios for Decay from the 4S3/2 and 4Inz2 Manifolds in 4%
ErrYAG following 4S3/2 Excitation.
T ransition
Using Total Volume
Integration
0.834
U sing Theoretical Radiative
Lifetime
0.993
A 21
4.22X10-3
3.32 X IO-4
A 20
' 0.162
6.87X10-3
W 54
0.750
0.985.
Wi%
A
53
1.57X10-3
1.65 X IO-4
A
52
3.27X10-3
1.97 XlO-4
A si ■
0.0656
4.05 X 10-3
0.180
0.0111
A.5 0
Table 11.
integrals
The ErrYSO branching ratios calculated from total volume
are
compared
w ith
those
calculated
by
comparison
of
experim entally determ ined lifetim es w ith theoretical radiative lifetim es
calculated by Moncorge et al (39).
The branching ratios from the two
m ethods showed improved agreem ent over the ErrYAG resu lts (see Table 10).
The ra te constants Ay and Wy were calculated for all three m aterials
modeled using E quation 7 and the overall experim ental ra te constants h. and
are listed in Table 12. These values were used in the set of sim ultaneous
differential equations which were th en num erically solved using M athm atica.
95
Table 10. B ranching Ratios for Decay from the 4Ss/2 and 4Inz2 M anifolds in
2% Er:YSO following 4S3/2 Excitation.
T ransition
W21
Using- Total Volume
Integration
0.997
A 21
1.93 X IO-4
1.68X10-4 .
A 20
2.89X 10-4
1.71 X 10-3
W54
0.992
0.994
A53
7.04 X 10-5
8.06 X10-G
As2
1.23 X IO-4
9.20 X 10-5
A 51
2.86X10-3
2.21X10-3
Aso
4.97X10-3
3.83 X 10-3
U sing Theoretical Radiative
Lifetime
0.998
The Aio and Wio values for all three m aterials are the sam e as those used in
the 4Inz2 model (see Table 8). The W32 and W43 values of 3.5 (.isec1 for 2%
ErrYSO were obtained from experim ental lifetime d ata for 1% ErrYSO tak en
after direct excitation into the respective levels (39). Since sim ilar d ata for
W32 was not available for ErrYAG or ErrLuAG, the sam e value was used for
these m aterials. As long as W32 has a significantly higher value th an IZt2
(the sum of A 20, A 21, and W21), which corresponds to the significantly shorter
lifetime of the 4Igz2 level compared to the 4Inz2 level, error in the estim ated
value of W32 will have little effect on the resu lts presented by the model. The
96
value for W43 used for Er:YAG and EriLuAG of 0.833 jLisec1 was obtained from
experim ental d ata tak en using a 3% ErrYAG sam ple (52).
Table 11. B ranching Ratios for Decay from the 4S3/2 and 4L m Manifolds in
0.5% ErrLuAG Following 4S3/2 Excitation.
T ransition
U sing Total Volume Integration
Win
0.993
A 21
5.13 X lO -4
A 20 .
6.58X10-3
W54
0.996
A 53
4.17 X 10-5
A 52
5.06X10-5
A si
1.02 X 10-3
A so
2.79 X IQ-3
The m odel-generated logarithm ic decay curves for all three em itting
levels of each m aterial are compared w ith experim ental d a ta in Figures 43
through 47. Both lin ear and logarithm ic plots are shown for the 2% Err^SO
sample, while only logarithm ic plots are shown for the ErrYAG and ErrLuAG
m aterials.
The decay curves generated by the model show good overall
agreem ent w ith the experim ental d ata w ith a few notable exceptions. Since
97
Table 12. Rate C onstants for R elaxation following 4Ssza Pum ping (psec1).
T ransition
4% Er:YAG
2% Er:YSO
0.5% Er:LuAG
Aio
7.69X10-5
1.16 X 10-4
8.77 X 10-5
m
7.69X10-5
o
8.77 X 10-5
-
A bo
2.15 X 10-3
2.70 X IO-4
9.71 X 10-5
A bi
6.30X10-6
1.80 X 10-5
7 .5 8 XlO-G "
1.25 X 10-2
9.30 X 10-2
1.47 X 10-2
3.50
3.50
3.50
0.833
3.50
0.833
Aso
1.63 X 10-2
1.38X10-3
2.27 X IO-4
Asi
5.96X10-3
7.95 X 10-4
8.28 X 10-5
2.97X10-4
3.42 X 10-5
1.43 X ,10-4
1.96 X 10-5
3.39 XlO-G
6.82X10-2
0.276
0.810
W sb
m
o
A sb
Ass
•
.
4.12 X 10-6
only first order term s are listed in the ra te equations, the logarithm ic decay
plots calculated using the model are stra ig h t lines.
C urvature in the
experim ental d ata (attributed prim arily to baseline correction error earlier in
th is chapter) is responsible for long term deviation from predicted behavior in
several of the log plots. Also, the calculated decay curve for the 4Iiaza level in
4% Er:YAG (see top of Figure 46) differs significantly from the experim ental
98
data. The Aio and Wio values used in the model were the sam e as those used
in the 4Iim model, and correspond to a lifetim e of 6.5 msec for the 4Iisz2 level,
while the lifetim e determ ined using the experim ental d ata from the top of
Figure 46 is 10.6 msec.
This longer experim ental lifetim e is closer in
agreem ent w ith the 9 msec lifetime level obtained on a 3% Er:YAG sam ple by
Zhekov et al (52) following excitation to the 4Sgzs level.
It is possible th a t
cross relaxation processes occurring at the higher excitation level (and not at
the lower one) are partially responsible for the longer experim ental lifetime,
though if th is were so these effects would be expected to result in more
curvature in the logarithm ic plot th a n is seen. It is also possible th a t the
wide range of experim entally determ ined lifetim es resu lts from radiation
trapping, where trap p in g of the em itted photons lengthens the apparent
emission lifetime. D eviations of th is order, expected to have been caused by
radiation trapping, were seen by Moncorge et al in low concentration Er:YSO
sam ples (39). More work will be needed to determ ine the cause or causes of
th is deviation.
Finally, the predicted and experim ental logarithm ic decay
curve from the 4Inz2 level in 0.5% Er:LuAG show poor long term agreem ent.
E xam ination of a lin ear plot of the d a ta shows this deviation is attributable
to baseline correction error.
Normalized Intensit
99
10000
20000
30000
40000
50000
60000
50000
60000
Time (microseconds)
log (Normalized Intensity)
I
0000
20000
30000
40000
Time (microseconds)
Figure 43. Linear (top) and Log Plots of Decay from the 4 11 3 /2 Manifold (Level
I) of 2% Er:YSO. The solid lines are modeled decay, and the points are
experimental data.
100
Normalized Intensity
I
0.8
Time (microseconds)
CO
C
OJ
C
-a
u
N
S
Uc
O
Z
OO
O
Time (microseconds)
Figure 44. Linear (top) and Log Plots of Decay from the 411 1 /2 Manifold (Level
2) of 2% Er:YSO. The solid lines are modeled decay, and the points are
experimental data.
Normalized Inten
101
Time (microseconds)
Figure 45. Linear (top) and Log Plots of Decay from the 4 S 3 /2 Manifold (Level
5) of 2% Er:YSO. The solid lines are modeled decay, and the points are
experimental data.
102
log ( N o r m a l i z e d I n t e n s i t
I
|00
20000
30000
40000
50000
60000
log ( N o r m a l i z e d I nt e n s i t y )
Time (microseconds)
-4
-6
-8
( N o r m a li z e d Intensity)
Time (microseconds)
Time (microseconds)
Figure 46. Log Plots of Decay from the 4Ii3/2 (top), 4I n /2 (center), and 4Sgyg
Manifold of 4% Er:YAG. The solid gray lines are modeled decay, and the
black points are experim ental data.
Iog ( N o r m a l i z e d I n t e n s i f
103
1
10000
20000
30000
40000
50000
60000
log ( N o r m a l i z e d I n t e n s i t y )
Tim e (microseconds)
log ( N o r m a l i z e d I n t e n s i t y )
Time (microseconds)
Time (microseconds)
Figure 47. Log Plots of Decay from the 4113/2 (top), 4111/2 (center), and 4S 3/2
Manifold of 0.5% E r:LuAG. The solid gray lines are modeled decay, and the
black points are experim ental data.
104
CHAPTERS
SUMMARY
Spectroscopic analysis and kinetic m odeling of relaxation processes
have been completed for 2.1% and 4% Er:YAG, 2% Er:YSO, and 0.5%
Er=LnAC at room tem perature.
IR and visible absorption, continuous
emission, and tem porally resolved emission spectra were obtained for all the
m aterials studied using Fourier transform spectroscopy.
Experim ental
emission lifetim es determ ined from the tem porally resolved spectra were
compared w ith theoretical radiative lifetim es obtained from quan tu m m echanical calculations using M orrison’s code to generate rate constants for
the radiative and nonradiative processes involved in relax atio n .' The rate
constants were also determ ined via a newly developed m ethod (using “total
volume in tegrals”).
The rate constants were plugged into sets of coupled
differential equations and num erically solved using the fifth order RungeK u tta m ethod in M athem atica. Each of these components of the study are
discussed briefly in the following paragraphs.
The experimental setup was for the most part effective in obtaining
the necessary data for the investigation.
The. multiplex advantage of the
Fourier transform method was particularly of value. In addition to providing
105
the usual benefit of reducing experim ental time, it allowed the m onitoring of
several em ission bands at once in the tem porally resolved experim ents. This
capability greatly facilitated determ ination of branching ratios and-enabled
calculation and balance of the total volume integrals used in the modeling
calculations.
Additionally, it aided in assignm ent of emission transitions
since tran sitio n s originating from the sam e m anifold have a sim ilar decay
profile. The tunable source also aided in assignm ent and kinetic modeling by
allowing the “tu rn in g on and off’ of upper level emission bands by adjusting
excitation w avelength: Furtherm ore, the system enabled compilation of room
tem p eratu re absorption and emission spectra, including tem porally resolved
emission, following excitation into eight different excited states of erbium in
the crystal hosts. These capabilities enabled compilation of the data needed
to characterize the erbium doped m aterials studied.
Em ission tran sitio n s were observed from three different manifolds:
4113/2, 4111/2, and 4S3/2.
Of these, the la tte r two had m ultiple .term inal
manifolds, resulting in.tw o emission bands from 4Iim- and four bands'from
the 4S3/2 level. E xperim ental emission lifetim es for all three levels were
determ ined directly from the tem porally resolved spectra. The 4113/2 lifetim es
were found to be 6.6, 6.5, 8,6, and 5.7 msec for 2.1% and 4% Er:YAG, .2%
Er:YSO, and 0.5% Er.LuAG, respectively. Lifetim es of .67.3, 10.5, and 67.7
psec were observed for 4% Er:YAG, 2% Er:YSO, and 0.5% Er:LuAG,
respectively for the 4Iim state. E xperim ental lifetim es of 10.7, 3.5, and 12.3
106
l-isec were observed for the 4Ssza state in 4% Er:YAG, 2% Er:YSO, and 0.5%
EriLuAG. These experim ental lifetim es provided overall rate constants used
in the kinetic modeling.
Once excitation had tak en place, relaxation w as dom inated by
m ultiphonon emission and radiative emission, both exponential processes.
Therefore only exponential relaxation processes were included in the kinetic
model. The lack of nonexponential relaxation processes w as probably largely
due to the relatively low erbium concentration of the m aterials studied. It is
well established th a t cross relaxation and. upconversion occur in erbium
doped into a variety of crystal hosts; however, these effects are more
pronounced..at .higher concentrations (often 10% or more).
S om e'gradual,
long-term curvature attrib u tab le to baseline correction error was seen in the
logarithm ic decay plots.
One of the disadvantages of the Fourier transform m ethod is the
baseline noise introduced by the in ten sity fluctuations (this can also be of
benefit as it removes noise from the signal and spreads it all- over the
spectrum ) which can m ake baseline correction difficult. One possible way to
address th is difficulty is to use filters to reduce the baseline noise. However,
th is use of filters comes a t the expense of the advantages provided by the
ability to observe m ultiple emission bands sim ultaneously.
Q uantum -m echanical calculations (using M orrison’s code) allowed
calculation of theoretical radiative lifetim es. Comparison of the experim ental
9
107
lifetim es (due to both radiative and nonradiative relaxation processes) w ith
the theoretical lifetim es for each level yielded the relative contribution of
radiative and nonradiative processes to relaxation. This, combined w ith the
radiative branching ratios obtained from the spectra, enabled determ ination
of radiative and nonradiative rate constants th a t could be utilized for kinetic
modeling.
A lternatively, the. radiative and nonradiative rate constants were
determ ined using the “to tal volume integ ral” method.
The sim ultaneously
tim e and frequency resolved spectra were integrated to provide intensity
“volumes” under the three dim ensional em ission bands. “B alances” w ritten
around each energy level allowed for determ ination of branching ratios and
rate constants. The advantage of th is m ethod is th a t a theoretical radiative
lifetime is only needed for the first excited state (to provide the nonradiative
contribution to relaxation from the level) ra th e r th a n every energy level
involved in the model.
A lternatively, the nonradiative contribution to
relaxation from the first excited state could be provided by comparison of cold
(liquid helium ) and room tem perature experim ental lifetim es. Comparison of
the to tal volume integral m ethod w ith the m ethod utilizing theoretical
lifetim es for all levels showed good agreem ent betw een the two methods.
Finally, the rate constants obtained by these m ethods were plugged
into a set of coupled differential equations and num erically solved using the
fifth order R unge-K utta m ethod in M athem atica. M odeling w as completed
108
for all th ree host m aterials for the cases of excitation to the 4Inz2 and 4Ssz2
levels. These levels were chosen since they were upper state levels th a t were
in itial levels for em ission transitions. Good agreem ent w ith the experim ental
tem poral behavior was obtained (including emission rises as well as decays in
the case of the 4Sszs model) indicating the validity of the results.
109
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115
APPENDIX A
PROCEDURES USED IN QUANTUM-MECHANICAL MODELING OF
LASER MATERIALS
116
Introduction
This appendix describes the procedures used in the operation of
com puter program s developed by M orrison et al to predict energy levels and
radiative tran sitio n probabilities, branching ratios, and lifetimes.
The
m ethod has the capability to predict S tark levels for a rare e arth ion in a
crystalline host and calculate theoretical radiative tran sitio n intensities
(either line-to-line or manifold-to-manifold).
The tran sitio n intensity
calculations enable determ ination of theoretical branching ratios and
radiative lifetim es. The theoretical background of the m ethod is discussed
briefly in C hapter I and in more detail in References 37, 39, and 78-83. The
calculations become less valid as rare e a rth concentration in the crystalline
host is increased, since they do not consider the effects of the rare e arth ion
on the host. The following sections outline the functions and relationship of
the com puter program s th a t do the calculations and give directions for
ru n n in g each program .
Program File O rganization
The calculations can be accomplished w ith the use of the three
FORTRAN program s LATSUM, ROTATE, and NPHASE.
LATSUM uses
host x-ray d ata to calculate values for the crystal field-components A nm a t the
117
rare e a rth site in the host crystal lattice". It works by using point charges and
completing a sum m ation over the crystal lattice. ROTATE takes the
Anm
output from LATSUM and rotates it to th e coordinate system necessary for
use in the program NPHASE, providing a new, equivalent set of suitable
Anm .
The program NPHASE can be ru n in different modes of operation depending
on w hat inp u t d ata is available and w hat output is desired from the.
calculations.
NPHASE can use known experim entally determ ined energy
levels and free ion wavefunctions (already contained in the code) to fit a set of
crystal field p aram eters
Bnm
to account for host effects on the rare e a rth ion.
A lternatively, it can be used to use a set of B n m and a free ion wavefunction to
calculate energy levels, radiative lifetimes, tran sitio n intensities,
and
branching ratios for a laser m aterial. These different modes lend them selves
to a num ber of com putational strategies;, flowsheets for two commonly used
strategies are shown in Figures A l and A2. The strategy shown in Figure A2
is useful if calculations are desired for a m aterial for which experim ental
d a ta is not readily available (Er:GGG, for example), b u t reliable d ata is
available for the sam e active ion in a sim ilar host (Er:YAG, for example)
118
O peration of the Program s
LATSUM
The x-ray d ata necessary to ru n LATSUM are tab u lated for m any
commonly used crystalline hosts in Reference 79.
Unless otherwise noted,
the input d ata necessary to ru n LATSUM are contained in this reference.
x-ray
d ata for
YAG
E xperim ental energy
levels for Er:YAG
NPHASE
LATSUM and
ROTATE
NPHASE
Calculated energy levels, transition
energies, etc. for Er:YAG
Figure A l. Flowsheet for Calculations w hen E xperim ental Energy Level
D ata are Available.
An example using LATSUM to determ ine
Anm
for YAG follows. Once
the program is running, it prom pts the user for a title, simply for
identification purposes; therefore, any input is suitable. Next, the program
prom pts for input of the space group num ber corresponding to the host space
119
E xperim ental levels
for Er:YAG
NPHASE
x-ray data
for YAG
LATS UM,
ROTATE
ErY A G
Input
centroids of
ErYAG
manifolds
x-ray
d ata for
GGG
LATSUM,
ROTATE
NPHASE
GGG
Calculated
energy levels,
tran sition
intensities,
etc. for
E rG G G
Figure A2. Flowsheet for Calculations when E xperim ental Energy Level
D ata for the Rare E arth Ion in a Sim ilar Host are Available.
120
group in the In tern atio n al Tables (from #1 to #230, corresponding to the
possible num ber of space groups).
For YAG, this num ber is 230.
The
program th en asks for the x-ray d ata necessary for the crystal (a,b,c,a,(3,y in
th is YAG example). The user should th en type in the
p aram eters
in
the
sam e
order
in
w hich
t h e y . were
values of the
listed,
i.e.
12,12,12,90,90,90. Next, the user is prom pted for the num ber of sites. Since
YAG has one yttriu m site, two alum inum sites, and one oxygen site, the
num ber 4 should be entered. The program th en requests the label for site I.
The user can enter Y for yttrium : (The order is arbitrary: A ll, A12, or O would
also be suitable entries.) In response, the program asks for x,y,z,q,ct for site
I. Reference 79 supplies the x,y, and z param eters; the ionic charge should
be entered for q, and a denotes the polarizability for the ion.
The
corresponding inp u t is 0,0.25,0.125,3,0. The polarizability inp u t is som ewhat
a rb itrary (80). After receiving th a t input, the program asks for the label for
site 2, an appropriate response would be A ll.
NPHASE will th en again
request values for x,y,z,q,a; the appropriate response is 0,0,0,3,0. Similarly,
the program will th en ask for labels and x,y,z,q, and a for sites 3 and 4,
respectively. Once d ata for the crystal sites is entered, NPHASE gives the
u ser the option of prin tin g n earest neighbor distances.
The program then
asks for the centering point in the un it cell. A point m ust be chosen such th a t
the to tal dipole m om ent of the u n it cell is zero. The choice will be sym m etry
dependent; for YAG 0,0,0 is suitable input.
For more detail, see Reference
121
80. Finally, the program asks for the origin site, which is the site at which
you wish to evaluate the
A nm .
This will be the rare e a rth site, which in the
case of the YAG exam ple will be the yttriu m site, arb itrarily chosen as site I.
Therefore, the inp u t to th is query is I: At th is point, LATSUM will run, and
will autom atically create the output file FOR009.DAT; it is recom mended to
renam e the output file after LATSUM is run.
ROTATE
ROTATE tak es the Amn output from LATSUM and ro tates it to the
coordinate system necessary for use in the program NPHASE, which requires
th a t the crystallographic c axis is p arallel to the sym m etry axis of the rare
e a rth site. As a result, th e program provides a new, equivalent set of A n m for
the “correct” coordinate system .
W hen ROTATE is started, it first asks for the nam e desired for the
output file.
Any nam e is suitable here; I norm ally nam ed these files the
crystal form ula followed by the extension ROT , as in YAG.ROT. Secondly,
the program will prom pt for the
Anm
components (which unfortunately m ust
be entered m anually a t th is tim e) by asking for the tensor ra n k k, where k=n
in
Anm .
A im .
A response of “1” will cause the program to ask , one by one, for all
After all
A im
are entered, the program will again prom pt for the tensor
rank, at which tim e 2 can be entered, causing the com puter to ask for the
A sm .
W hen all
Anm
have been entered in th is fashion, the user can enter “0”
122
in response to the tensor ran k prom pt, which signals to the program th a t all
the
Anm
are entered. After completion of th is task, the program requests the
input of three rotation angles—a,|3, and y. The program will th en prin t out a
set of ro tated
Anm ,
and ask w hether the user w ants to try another set of
rotation angles. Before actually trying rotation angles, it is helpful to enter
0,0,0 as the first set of “rotation angles”, since th is resu lts in a prin to u t of the
un ro tated
Anm
Anm ,
giving the user the opportunity to check the m anually entered
for accuracy. Different sets of rotation angles should be tried until the
properly ro tated
Anm
are obtained (criteria for properly rotated
A nm
are
described in the next paragraph); after which time, “no” can be typed when
the program asks about trying another set of angles. The output will th en be
saved to the filenam e suggested by the user.
The properly rotated
A nm
can be obtained by using the following
procedure:
1. Choose the correct
A nm
component to evaluate, for each set of
rotation angles: the
Anm
to evaluate shouldihave even, nonzero n
and m, and it should not be forced to have a value of zero by
' sym m etry. Of the components th a t m eet these two criteria, the one
w ith m inim um n and m should be chosen.
2. The correct s e t'o f rotation angles-w ill rotate the coordinate axes
such th a t p articu lar
Anm
chosen (i.e.
A
22 )
from step one become real
and positive. Different sets of rotation angles should be tried u ntil
123
th is criteria is met.
R otation angles to try are given by the
following equation:
angle = ± —arctan lmA
n
RealA nm J
where n is a positive integer.
3. The angles given by the equation in Step 2 can be tried in any
com bination for rotation angles a ,(3, and/or y.
NPHASE
NPHASE is the program th a t does the bulk of the calculations and can
be ru n in different modes of operation, depending on w hat d a ta is available
and where the user is in the com putational analysis (see Figures A l and A2).
W hen the program is started, it first prom pts for a title. The choice of a title
is a rb itrary and has no bearing on the operation of the program . Once the
title is entered, the user is directed to enter the sym m etry group label at the
rare e a rth site of the crystal host. For g arnet hosts, the proper label is D2.
Sym m etry group labels are available in Reference 79.
The program then'
asks w hether the user w ants to fit experim ental energy levels. An answ er of
“yes” will direct NPHASE to accept inp u t of S tark level energies in cm-1,
which it will use to fit a set of crystal field p aram eters B nm to account for the
host crystal field splitting of the free ions energy levels into manifolds. The
program allows the user to specify the num ber of iterations to be used in the
124
fit and to choose w hether the fit should be done using all the S tark levels or
ju st the centroids of the manifolds.
NPHASE requires
etc) and odd n
Anm
the identity of the rare e a rth ion (Er, Nd, Tm,
Bum ,
to complete calculations of theoretical S tark levels,
radiative lifetim es, and line-to-line tran sitio n probabilities.
come eith er from a previous ru n of NPHASE th a t fits
energy levels or from rotated
Anm
and the relationship
Bnm
The
B nm
can
to experim ental
B nm = PnAnm .
The
Bnm
can eith er be entered m anually or by selecting a file already containing the
p aram eter values. If the
Bnm
are entered m anually, NPHASE offers the
option of saving them to a file of nam e “filenam e.BNM.PARM”. NPHASE
will th en prom pt the user for the identity of the rare e a rth ion. After the ion
is selected, NPHASE will suggest a set of “default sta te s” which are the
m anifolds (listed by term symbol) to be included in the calculations. The user
is given the option of adding or deleting m anifolds from consideration in the
calculations.
Once NPHASE has these input p aram eters, it will do the calculations
and save the output as file REHOME.DAT.
S tandard output includes
energies of the centroids of the m anifolds included in the calculations.
Optional o utput includes radiative lifetim es for each manifold, branching
ratios for tran sitio n s betw een manifolds, Stark-level-to-Stark-level transition
probabilities (both electric, dipole and m agnetic dipole), and. the Judd-O felt
125
p aram eters f2k. The optional output is selected by answ ering questions posed
by the program .
MONTANA STATE UNIVERSITY - BOZEMAN