Two-center holographic
3.4 Theory
3.4.2 Parameters of the model
read-out beam, and neff is the effective index of refraction for the read-out beam that depends on the polarization of that beam. Finally, we can calculate the diffraction efficiency from !::::.n by using Kogelnik's formula [52].
Bleaching
As mentioned before, the bleaching experiment is performed to find the dynamics of electron transfer from Fe to Mn traps by a homogeneous red beam. When a sensitized crystal is illuminated with a strong red light, electrons are excited only from Fe traps to the conduction band. Some of these electrons are trapped by Fe centers, others are trapped by Mn centers. Those electrons that are trapped in Mn centers can not be re-excited to the conduction band. Therefore, the electron concentration in Fe traps decreases with time. The decrease in electron concentration in the Fe traps results in a decrease in the absorption of red light. The governing equations for the analysis of bleaching dynamics are
aNMn at aNie
at aNie aNMn
7it+-at
_~ eaj ax = 0 '(3.30) (3.31) (3.32)
where we assume adiabatic approximation
(Wl- =
0) in Equation (3.32). The initial conditions for Nie and NMn depend on the sensitization process, but the condition Nie+
NMn=
NA must be always satisfied. We can find n as a function of NMn and Nie by putting aNMn/at and aNie/at from Equations (3.30) and (3.31), respectively, into Equation (3.32). Putting NMn=
NA - Nie and the formula found for n in Equation (3.31), we can obtain a differential equation for Nie asqFe'RSFe'RI~I'lvln(Nlvln - NMn) _ Nie c:::: _ aNie .
I'lvln(Nlvln - N lvln )
+
I'Fe(NFe - NFe ) Tb (3.33) To simplify the calculations, we assume that the absorption of the red light in the crystal is not large. Therefore, we can consider the red light intensity (h) m Equation (3.33) as a constant. We further assume that Nie«:
NFe and NMn NA . This is a good approximation when we neglect the beginning of the bleaching curve [Figure 3.3 (b)], and consider the dynamics of the latter part of the curve.11 12
I I
Figure 3.7: Transmission of an optical beam through a very thin portion of the crystal with thickness 6.z. h and 12 are incident and transmitted light intensities.
With all these approximations, the bleaching time constant Tb in Equation (3.33) becomes a constant independent of the position inside the crystal. The bleaching speed normalized to bleaching light intensity (IR) can be represented as
(
N
)
1 IFe Fe -1
-1-
=
(qFe,RSFe,R) 1+
(N N ) .Tb R IMn Mn - A
(3.34)
The right-hand side of Equation (3.34) can be found experimentally by fitting the latter part of the bleaching curves at different intensities with monoexponential formulas. The important fact is that in the bleaching experiment, we measure the transmitted light intensity. Therefore, we need to find the relation between light transmission and electron concentration in Fe traps (Nie). For this purpose, consider a very thin portion (6.z) of the crystal as shown in Figure 3.7. The transmitted intensity from this portion,
h ,
is related to the incident intensity, h , as(3.35) or
dIll
(3.36)Integrating Equation (3.36) from z
=
0 to z=
L, with L being the thickness of the crystal, results in(3.37)
where Ii and It are incident [1(z = 0)] and transmitted [1(z = L)] intensities, re- spectively. The experimental bleaching curve (as shown in Figure 3.3 (b)) can be approximately represented by
I t
=
J. -I /;:,.1 ~ - J. -I /;:,,10 exp(--) t ,Tb2
(3.38)
where /;:,.1 is the absorption change due to electron transfer from Mn to Fe centers, and can be represented from Equation (3.37) by
/;:,.1
=
Ii [1 - exp ( -SFe,Rhl/1L
Nie(Z)dZ) ] (3.39)The time constant Tb2 for the variation of /;:,.1 can be calculated by taking the derivative of both sides of Equation (3.39)
d/;:,.1
dt 1is Fe,Rhl/ exp
(
-SFe,Rhl/ 01 L ) 1L
Nie(z)dz 0 dN-~~ (z) dz h1
L dNie(z) d-SFeR 1/ d z
, 0 t /;:,.1
1 - exp (SFe,R1W
1L
Nie(Z)dZ)/;:,.1
(3.40)
Assuming the optical density of the crystal for red light to be small (sFe,Rlw JoL Nie (z) dz
«
1), we can simplify the denominator of the right-hand side of Equation (3.40). Fur- thermore, we can use dNie/ dt c::: - Nie/Tb from Equation (3.33) to rewrite Equa-
tion (3.40) as
dD.I dt
r
L dNi.(z) dzJo
dt D.I1L
Ni.(z)dz1
o L -Ni.(z)dz -1 Tb D.I1L
Ni.(z)dz)-D.I
(3.41 )
Comparing Equations (3.40) and (3.41) results in Tb
=
Tb2. Therefore, we can calcu- late T!2 /h
from bleaching experiments and put it in Equation (3.34) to obtain one equation for the unknown parameters. Figure 3.8 shows the variation of the bleach- ing speed (1/Tb2) with red light intensity (h). The solid line in Figure 3.8 shows the linear fit to the experimental data. Using the slope of this line, we can obtain one equation for the unknown parameters as3.44. (3.42)
Sensitization
As mentioned before, the sensitization experiment is performed to find the dynamics of electron transfer from Mn to Fe traps by a homogeneous UV beam. The increase in electron concentration in the Fe traps results in an increase in the absorption of red light. The governing equations for the analysis of sensitization dynamics are
aN
Mn
at
aNi.
at
aNi. aN
Mn
----at + ----at
- -1 aj e -ax -
-0,
(3.43) (3.44) (3.45)
....
' N ..,
P
4
3
1
o o
•
1000 2000 3000 4000
Bleaching Intensity (W/m2)
Figure 3.8: Variation of the bleaching speed (1/Tb2) with bleaching intensity (h). The solid line shows a linear fit to the experimental data.
where we assume adiabatic approximation
(fJJt
= 0) in Equation (3.45). The initial conditions are Ni.(t=
0)=
0 and NMn(t=
0)=
N A. Using N Mn=
NA - Ni. from Equation (3.45), we can find the differential equation for the electron concentration in the Fe traps asaNi. -qFe,UVSFe,UVl'Mn(NI-'ln - NMn)Ni. + I'FeqMn,UVSMn,Uv(NFe - Ni.)NMn at I'Fe(NFe - N Fe ) + I'Mn(NMn - N Mn )
x1uv. (3.46)
The major complication in finding an analytic solution for Equation (3.46) is the high absorption of the UV light. The measured absorption coefficient of the crystal at 365 nm is a = 9 mm-I. Equation (3.46) is a point form formula, i.e., it is valid at any point inside the crystal. To use the experimental results of the sensitization experiment for the calculation of the material parameters, we need to find from Equation (3.46) the total transmission of a weak red beam. For this purpose, consider the very thin portion (6.z) of the crystal as shown in Figure 3.7. Using
Equation (3.39), we can calculate the initial slope of the transmitted intensity ratio vs. time as
d(Id dt Ii) 1 t=o -- _ SFe,R . h V
1L
0dN}~(z)
dt 1 t=O z. d (3.47)Substituting dNff't(z) It=o from Equation (3.46) into Equation (3.47), and replacing luy(z) by luyo exp( -aUYz), we obtain
d(Id Ii) It=o
=
(SFe,Rhv)QMn,UySMn,UyNA Iuyo dt 1 + I'Mn(JVMn - NA ) aUYI'FeNFe
(3.48)
The left-hand side of Equation (3.48) can be calculated from the experimental re- sults. Figure 3.9 (a) shows the variation of the initial sensitization slope (d(Id{ Ii) It=o) with Iuyo. Figure 3.9 (a) shows that initial sensitization slope varies linearly with the UV intensity Iuyo as Equation (3.48) suggests. Replacing the slope of the line fitted to the experimental data ([d(Id{ Ii) It=ol! luyo) in Equation (3.48) results in one equation for the unknowns as
(3.49)
Another equation can be obtained using the saturation value of the transmitted red light in the sensitization experiment. Figure 3.9 (b) shows the variation of the ratio of the transmitted to incident red intensity after 3 hours of sensitization vs. UV intensity Iuyo. This value is related to the electron concentration in Fe traps after 3 hours of sensitization. The complication in the theoretical calculation is due to the variation of Nie within the thickness of the crystal (as a result of the large UV absorption). We first replace N Mn by NA - Nie in Equation (3.46), and rewrite the
(a)
(b)
3 r , ---~---~---
~
'"
u..,
CI) III
o ...
:;2
CI) CI) c.I/)
en c
;;1 'N 'iii
c CI) I/)
c;; ;;:
- 0 c
•
•
o 50 100 150 200
Sensitizing Intensity (W/m2)
1.1 r--- ---
.c
...
en~ 0.9
en
en 'N c E ~ 0.7CI) I/)
I
0.5 [
o
• • •
•
50 100 150
~J
200Sensitizing Intensity (W/m2)
Figure 3.9: Variation of (a) initial sensitization speed
([d(Id~
Ij) 11=0]1 Iuvo) , and (b)sensitization strength (ratio of the final transmitted red power to the initial trans- mitted power) with sensitizing intensity (Iuvo). Solid line shows a linear fit to the experimental data.
equation as
aNie _ Nie
+
'YFeqMn,uvs~'ln,uvNFeNAIuvo exp( -cxuv z)+
O([Nie]2) at T(Z) 'YFeNFe+
'YMn(NMn - NA)+
("(Mn - 'YFe)NFeNie,final - Nie T(Z)
where T(Z) is the space-dependent sensitization time constant defined by
1
T(Z) [
- qFe,UVSFe,UV'YMn(NMn - NA )
+
'YFeqMn,UVSMn,Uv(NFe+
NA )]'YFeNFe
+
'YMn(NMn - NA )+
("(Mil - 'YFe)NFe(3.50)
xIuvoexp(-cxuvz). (3.51)
Therefore, we can approximately express Nie(z) as
Nie(z, t)
=
Niefinal , [1 - exp (-Atexp(-cxuvz))] , (3.52)where
(3.53)
Using Equation (3.42) and assuming NA '::: 0.9NMn (we check this assumption later), we can calculate A ~ O.Ols-l . To calculate the total transmitted red light through the crystal, we again divide the crystal into very thin portions as we did before (Figure 3.7). Considering
It
=
Ii exp(l
L -cx(z, t)dZ) , (3.54) withcx(Z, t)
=
SFe,RhvNie(z, t) , (3.55)and using Equation (3.52) yield
1
1£
- In(It/Ii)
=
Niefinal [1-exp(-Atexp(-Quyz))]dz.SFe,Rhv ' 0
(3.56)
The calculation of the integral in Equation (3.56) can be simplified by defining a new variable u
=
At exp( -QUyz). Applying this change of variable results in1
o ·£ [1-exp(-Atexp(-Quyz))]dz =L ( l - - -QUY 1 L .j.
AtAt
exp( -auv £) exp( U u) du ) (3.57)For sensitization of about 3 hours, we can calculate At C:::' 108. For our L = 0.85 mm thick crystal with QUY
=
9 mm-1 at 365 nm, we have Atexp( -QuyL) C:::' 0.05.The integral in Equation (3.57) can be calculated using the tabulated Exponential Integral Function. The upper bound of this integral can be replaced with 00. The interesting property of the integral is that it is not a sharp function of the lower bound in the range of values that are relevant to our experiment. For example, for lower bounds (At exp( -QuyL)) of 0.04, 0.05, and 0.07, the integral is equal to 0.65L, 0.68L, and 0.72L, respectively. This justifies most of the approximations we made in this calculation. Using 0.7 L as the value of the integral in Equation (3.56), and using It/Ii
=
0.9 from Figure 3.9 (b), we obtain Nie,final C:::' 1.16 X 1024 m-a To obtain the third equation for calculating the model parameters, we apply the saturation (or steady-state) condition to Equation (3.46). At steady-state, the time derivatives are replaced with zero, and all variables are replaced with their final values. Therefore, we put 8Nie/8t = 0, Nie = Nie , final C:::' 1.16 X 1024 m-3, and NMn = NA - Nie , final into Equation (3.46), and rearrange different terms to obtain(3.58) We now need to solve a system of three equations [Equation (3.42), Equation (3.49), and Equation (3.58)] for three unknowns (qMn,UySMn,UY, rMn, and NA)' To do this, we use Equations (3.42) and (3.49) to replace the first two variables in terms of N A
in Equation (3.58). This results in a second-order equation for iVA as
(3.59)
Equation (3.59) results in two solutions for iVA and, therefore, two sets of solutions for our three unknowns are obtained. It turns out that only one of these sets results in acceptable recording and read-out response as evidenced by experimental results.
The values for the three unknowns found by solving Equations (3.42), (3.49), and (3.58) are
I'Mn
3.1 x 1024m-3 3.55 x 10-5m2/ J
8.51'Fe = 1.32 x 1O-13m-3s- 1 .
(3.60) (3.61) (3.62)
The value of iVA agrees with our assumption (iVA ':::: 0.9iVMn ). We use
iV. ... ,
qMn,UVsMn,UV, and I'Mn from Equations (3.60)-(3.62) as the initial values in the simu- lation of sensitization, bleaching, and holographic recording and read-out curves, and then fine tune these values by trying to get the best fits to the experimental results.
The final values of iVA, qMn,UVSMn,UY, and I'Mn are shown in Table 3.1.