Effect of non-linear absorption on characteristics of laser-induced luminescence
This paper deals with the methodology of laser-induced luminescence at high excitation levels. For a case when a photodetector collects integral (over the volume) luminescence power, a method is proposed for processing the experimentally-measured luminescence power and optical transmittance as some...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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| Цитувати: | Effect of non-linear absorption on characteristics of laser-induced luminescence / S.E. Zelensky, B.A. Okhrimenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 2. — С. 51-57. — Бібліогр.: 15 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1206552025-02-23T19:58:02Z Effect of non-linear absorption on characteristics of laser-induced luminescence Zelensky, S.E. Okhrimenko, B.A. This paper deals with the methodology of laser-induced luminescence at high excitation levels. For a case when a photodetector collects integral (over the volume) luminescence power, a method is proposed for processing the experimentally-measured luminescence power and optical transmittance as some functions of the incident laser power. The method is based on the analysis of relative increments of luminescence and transmitted laser power. Considered are two examples of mechanisms of non-linear laser-induced luminescence, namely: saturated molecular luminescence and luminescence excited via two-photon absorption. 2005 Article Effect of non-linear absorption on characteristics of laser-induced luminescence / S.E. Zelensky, B.A. Okhrimenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 2. — С. 51-57. — Бібліогр.: 15 назв. — англ. 1560-8034 PACS 78.55.-m https://nasplib.isofts.kiev.ua/handle/123456789/120655 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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This paper deals with the methodology of laser-induced luminescence at high excitation levels. For a case when a photodetector collects integral (over the volume) luminescence power, a method is proposed for processing the experimentally-measured luminescence power and optical transmittance as some functions of the incident laser power. The method is based on the analysis of relative increments of luminescence and transmitted laser power. Considered are two examples of mechanisms of non-linear laser-induced luminescence, namely: saturated molecular luminescence and luminescence excited via two-photon absorption. |
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Zelensky, S.E. Okhrimenko, B.A. |
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Zelensky, S.E. Okhrimenko, B.A. Effect of non-linear absorption on characteristics of laser-induced luminescence Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Zelensky, S.E. Okhrimenko, B.A. |
| author_sort |
Zelensky, S.E. |
| title |
Effect of non-linear absorption on characteristics of laser-induced luminescence |
| title_short |
Effect of non-linear absorption on characteristics of laser-induced luminescence |
| title_full |
Effect of non-linear absorption on characteristics of laser-induced luminescence |
| title_fullStr |
Effect of non-linear absorption on characteristics of laser-induced luminescence |
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Effect of non-linear absorption on characteristics of laser-induced luminescence |
| title_sort |
effect of non-linear absorption on characteristics of laser-induced luminescence |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2005 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120655 |
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Effect of non-linear absorption on characteristics of laser-induced luminescence / S.E. Zelensky, B.A. Okhrimenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 2. — С. 51-57. — Бібліогр.: 15 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT zelenskyse effectofnonlinearabsorptiononcharacteristicsoflaserinducedluminescence AT okhrimenkoba effectofnonlinearabsorptiononcharacteristicsoflaserinducedluminescence |
| first_indexed |
2025-11-24T20:20:33Z |
| last_indexed |
2025-11-24T20:20:33Z |
| _version_ |
1849704456974761984 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
51
PACS 78.55.-m
Effect of non-linear absorption on characteristics
of laser-induced luminescence
S.E. Zelensky, B.A. Okhrimenko
Taras Shevchenko Kyiv National University, Physics Department,
6, prospect Academician Glushkov, 03680 Kyiv, Ukraine;
E-mail: zele@univ.kiev.ua
Abstract. This paper deals with the methodology of laser-induced luminescence at high
excitation levels. For a case when a photodetector collects integral (over the volume)
luminescence power, a method is proposed for processing the experimentally-measured
luminescence power and optical transmittance as some functions of the incident laser
power. The method is based on the analysis of relative increments of luminescence and
transmitted laser power. Considered are two examples of mechanisms of non-linear laser-
induced luminescence, namely: saturated molecular luminescence and luminescence
excited via two-photon absorption.
Keywords: luminescence, absorption, non-linearity, laser induction.
Manuscript received 14.02.05; accepted for publication 18.05.05.
1. Introduction
Laser-induced luminescence is a powerful tool for
investigations of physical properties of molecules and
impurity centers in various matrixes. The possibilities of
this method are enhanced significantly, if the laser
radiation power is sufficiently high to initialize various
non-linear processes in the studied object. For example,
such processes may be a saturated absorption, spectral
hole burning, multiphoton absorption, excited-state
absorption, etc.
When investigating the laser-induced non-linear
processes, the choice of a suitable parameter for
characterization of the investigated process is a critical
factor. In the traditional non-linear optics, the formalism
of non-linear susceptibility tensors is widely used to
characterize coherent phenomena (generation of optical
harmonics, four-wave mixing, etc.). However, for a
number of non-coherent phenomena (saturation, non-
linear absorption, etc.), the non-linear susceptibility
approach seems to be inconvenient as it results in
complicated equations with the loss of clearness.
Choosing a suitable parameter of non-linearity is also
impeded by the fact that the experimentalists usually
deal with the integral characteristics of luminescence
and absorption, with an essentially non-uniform spatial
distribution of the laser power.
Consider a simple example. As is theoretically well
substantiated for a two-photon mechanism of excitation
of luminescence, it is expedient to approximate the
dependence of the luminescence intensity on the
excitation one using the following power function
I = const Fγ (1)
where γ = 2 for this instance. Whereas the γ-parameter is
a constant, and two-photon absorption is usually a low
probable process, the integral (over the volume)
luminescence power can be also approximated using the
function (1). The introduced parameter γ is a convenient
characteristic of the investigated process; it can be easily
determined from the experimental data (as a slope of the
curve I(F) in a log-log scale) and provides the
information on the number of photons engaged in the
multiphoton absorption transition. It should be also
noted that γ is a dimensionless parameter which can be
calculated using the experimental data (I and F)
measured in suitable arbitrary units.
It is not a rare case in laser spectroscopy when the
laser-induced emission is characterized with a non-linear
dependence of its power on the excitation intensity. In
many these cases, the attempt to approximate the
experimental data the function (1) leads to the
conclusion that γ is not a constant but depends on the
excitation laser power γ = γ(F). For example, such
behavior is observed in the following cases: the saturated
luminescence, the laser-induced incandescence [1, 2],
the luminescence excited by sequential absorption in
YAG:Nd3+ [3], etc. In the mentioned examples, γ-para-
meters possess non-integer values and provide the useful
information on the investigated laser-induced non-linear
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
52
processes. It should be also noted that the dependence
γ(F) makes difficulties for theoretical calculations and
results in significant errors caused by the non-uniformity
of laser power distribution within the sample volume [4].
In this paper, we analyze the effect of non-linear
absorption of the excitation laser radiation on the
experimentally measured luminescent characteristics.
Using the above-mentioned γ-parameters both for the
luminescence intensity and for the transmitted laser
power, we propose a method for processing the
experimental data that provides a possibility to
distinguish the absorption and luminescence non-
linearities.
2. General considerations
Consider the interaction of a laser beam with light-
absorbing centers or molecules in a transparent matrix.
Assume the surface power density of laser radiation is
uniformly distributed within the cross-section area of the
laser beam. Denote the surface density of the laser
power (expressed in photons ⁄cm2s) for incident and
transmitted laser beams by F0 and Fd, respectively.
Denote the optical transmittance of a layer with the
thickness d as T = F0/Fd. Let the molecules absorb laser
photons and emit the luminescence ones with a quantum
efficiency η and the average photon energy ћω. dF
signifies the difference of the surface density of the laser
power passing through the layer of the thickness dz. PL
identifies the integral power of luminescence emitted in
all directions from the total volume V = Sd, where S is
the cross-section area of the laser beam. Knowing the
number of laser photons absorbed in the elementary
volume Sdz is –SdF, we obtain the following expression
for PL as a function of F0
( ) ( )
( )0
0
L 0 d
dF F
F
P F S F Fω η= − ∫h . (2)
If the quantum efficiency of luminescence is
independent of the excitation power density, then (2) can
be reduced to the following form
PL(F0) = ћωηS [F0 − Fd(F0)]. (3)
For the integral luminescence, the following parameter
of non-linearity can be defined as a ratio of relative
increments
L L
L
0 0
d
d
P P
F F
γ = . (4)
By a little algebra, the expression (3) yields
( )
L
1 1
1
T T
T
γ
γ
− +
=
−
, (5)
where the non-linearity parameters for the optical
transmittance γT and transmitted laser power density γF
are defined similarly to (4)
0 0 0 0
dd 1 1
d d
d d
T F
F FT T
F F F F
γ γ= = − = − . (6)
It follows from the expression (5) that γ-parameters
of integral luminescence (4) and transmittance (6) are
interrelated. As γT and T are functions of F0, then γL is
also a function of F0 according to (5). For example, in a
simple case of power-independent absorption (T = const,
γT = 0, γF = 1), we obtain γL = 1, that means an ordinary
linear response of luminescence intensity to the
variations of the excitation power.
It should be emphasized that the expression (5) is
derived without any assumption concerning the
mechanism of absorption of laser radiation by the
luminescence centers. So, it could be both linear and
non-linear absorption, for example, single-photon
absorption according to Bouguer's law, multiphoton
absorption, saturated absorption, etc. In this sense, the
expression (5) seems to be a universal formula.
As seen from the expression (2), the non-linear
properties of integral luminescence are determined by
several factors. (i) The non-linearity of absorption
effects on the luminescence. According to (2), the power
of luminescence depends on the upper limit of
integration Fd that, in its turn, depends on F0. That is
why the value γL and its dependence on F0 are deter-
mined by the values T and γT, and by their behavior with
changing F0. (ii) The quantum efficiency of lumines-
cence can be a function of the excitation laser power
η = η(F), which also influences on the properties of
integral luminescence according to (2).
As far as the expression (5) is derived from the
assumption η = const, it accounts for the non-linearity of
integral luminescence caused by the non-linearity of
absorption. This fact provides a principal possibility to
separate the effect of non-linear absorption from that of
a non-constant quantum efficiency. If an experiment
reveals the violation of expression (5), it can be
considered as an indication of laser-induced changes in
the luminescence quantum efficiency. Below we
consider a couple of examples illustrating the use of the
expression (5).
2.1. Saturated absorption
Consider a simple two-level model of molecular
luminescence by means of the following balance
equation
2 2
1
d
d
n nFn
t
σ
τ
= − , (7)
where σ is the absorption cross-section, τ is the
luminescence lifetime, n1 and n2 are the occupancies of
the ground and excited states, respectively, n1 + n2 = N is
the numerical density of molecules. Using a stationary
approximation, the luminescence power emitted from an
elementary volume dV can be written as
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
53
2
L
sat
d d dn N FP V S z
F F
ω ω
τ τ
= = ⋅
+
h h , (8)
where Fsat = (στ)−1. Besides, the difference of the laser
power surface density dF passing through the layer of
the thickness dz can be written [5] as a non-linear photon
transport equation
sat
1
sat
d d dFF Fn z FN z
F F
σ σ= − = −
+
. (9)
By substituting (9) into (8) and integrating the
luminescence power over the whole volume, V = Sd, we
obtain the integral power of luminescence as follows
( )
0
0
L
sat sat
d 1
dF
F
SFFP S T
F F
ωω
στ στ
= = −∫
h
h . (10)
By differentiating the expression (10), we can make
sure of that the expression (5) holds true.
The solution of the transport equation (9) can be
given in the following implicit form
0
0
0 sat sat
exp expd dF F FT
F F F
⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
, (11)
where T0 is the low-signal transmittance, i.e., the
transmittance without saturation. The parameter of the
transmission non-linearity γT can be easily derived from
(11) as follows
0 sat
sat
1 1
1T
d
F F
F F
γ +
= −
+
. (12)
The expressions (12) and (5) yield the following
expression for γ-parameter of integral luminescence
L
sat
1
1 dF F
γ =
+
. (13)
As seen from (13), γL depends on F0 due to the
dependence Fd(F0) that can be calculated by numerical
solution of the equation (11). The results of calculations
performed for the expressions (11)–(13) are given in
Fig. 1a, curves 1-3.
It is worth noting that the above expressions are valid
under the condition of uniform distribution of the laser
power density across the beam and in the stationary
approximation. As mentioned in [4], if the investigated
process is characterized with a power-dependent
parameter of non-linearity, significant errors arise due to
the non-uniformity of spatial distribution of the laser
power. Thereinafter, by means of computer simulation,
we consider the effect of the mentioned error-causing
factors on the validity of the expression (5).
For modelling the non-uniform distribution of the
power density across the laser beam, we use the
following Gauss function
( ) ( )2 2
0 expF r F r −= − Δ ,
where γ is the distance from the beam axis, Δ is the beam
radius. Then the expression (3) is transformed into the
following
( ) ( )2 2
L 0
0
exp 2 ddP F r F r r rωη π
∞
−⎡ ⎤= − Δ −⎣ ⎦∫h . (14)
The distribution of transmitted laser power Fd(γ) is
an unknown function to be calculated.
The parameter γL of integral luminescence was
calculated by numerical integration of the expression
(14). While integrating, the unknown function Fd (γ) was
determined by numerical solution of the equation (11) by
substitution of F0exp(–r2Δ–2) for F0. The calculated
dependence γL(F) is shown in Fig. 1a, curve 6. Fig. 1a
also presents the calculated curves γF = 1 + γT and T
(curves 5 and 4, respectively). While calculating γT and
T, the transmitted laser power Fd (γ) was integrated over
the beam cross-section using (11).
By substitution of the calculated γT and T into (5),
we obtain the dependence γL(F0) that can be compared
with that calculated using (14). The results of
calculations show that both mentioned curves γL(F0)
coincide within the accuracy of approximation
ΔγL ≤ 0.0001. The coinciding curves γL(F0) are plotted in
Fig. 1a as the single curve 6. Thus, the calculations
confirm the validity of the expression (5) in the case of
non-uniform distribution of the laser power density
across the beam.
Now we consider the expression (5) in the case of
non-stationary excitation of molecular luminescence.
Taking into account that photodetectors often operate in
the integrating mode, i.e., measuring an integral of
optical pulse over time, we calculate the energy of
luminescence and laser pulse as follows
2d dL
V
t V nωε
τ
∞
−∞
= ∫ ∫
h ,
(15)
d dF
S
t S Fε
∞
−∞
= ∫ ∫ .
While integrating, the integrands 2n and F were
calculated numerically from the equations (7) and (11).
The temporal shape of the laser pulse was given by the
following function
( ) ( ) ( )2 2
i, exp 4ln 2F r t F r t τ −= − , (16)
where τi is the laser pulse duration. Calculations were
performed for two examples of luminescence lifetime,
τ = 6 ns and 30 ns, and for the laser pulse duration
τi = 10 ns. The results of calculation are shown in
Fig. 1b. Again we calculated γL(F0) by two ways from
the expressions (5) and (15) and both the curves prove to
coincide. In Fig. 1b, these coinciding curves are plotted
as a single one (curve 9 for τ = 6 ns and curve 12 for
τ = 30 ns). Thus, we demonstrate the validity of the
expression (5) in the non-stationary case.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
54
As seen from Fig. 1, the non-uniformity of laser
power distribution and the non-stationary behavior of
occupancies cause significant changes in the curves
γL(F0), γF(F0), and T(F0) (for example, the γL-values
calculated for different models are several times
different for the same F0). Nevertheless, the expression
(5) remains valid.
For experimental verification of the expression (5),
we measured the laser-induced luminescence and
transmittance of an aqueous solution of Rhodamine 6G.
As an excitation source, the second harmonic of a Q-
switched YAG:Nd3+ laser was used (wavelength
λ = 532 nm, τi = 10 ns). The luminescence was detected
at the fixed wavelength 585 nm (through a single grating
monochromator). The low-signal transmittance was
T0 = 0.1. The luminescent track was completely located
within the operating field of the photodetector.
Measurements were performed at room temperature. The
results of measurements are plotted in Fig. 2 where open
circles represent γL calculated in accord to the definition
(4) by using luminescence power experimental data, and
filled circles are result of calculations according to the
expression (5) with the experimental data taken for T
and γL. As seen from the figure, the curves 3 and 4 are in
good agreement, which substantiates the validity of the
expression (5).
2.2. Two-photon absorption
Now we consider the mechanism of two-photon
absorption of laser radiation by molecules or
luminescence centers. This mechanism implies
transitions through the intermediate virtual states of
molecules with simultaneous absorption of a couple of
photons. When a powerful laser radiation interacts with
molecules that absorb laser light by the two-photon
mechanism the decrease of the propagating laser beam
power can be described as follows [5]
2d dF F zβ= −
with the well-known solution
( ) ( ) 1
0 01F z F F zβ −= + , (17)
where β defines the probability of two-photon transitions
in the given centers/molecules. From the expression
(17), the parameter of non-linearity can be easily derived
( ) 1
01F F dγ β −= + . (18)
Suppose the two-photon absorption excites
luminescence of molecules. For an elementary volume
dV, the power of luminescence emitted in all directions
is proportional to the square of laser power density,
dPL = const F2dV. Then the integral power of
luminescence is
SddFFPL
1
0
2
0 )1(const −+= β .
Fig. 1. The results of calculation of optical transmittance T
(1, 4, 7, 10), parameters of non-linearity γF = γT + 1 (2, 5, 8,
11) and γL (3, 6, 9, 12) as functions of F0 for uniform (1–3)
and Gauss (4–6) distribution of the laser power density across
the beam, for the stationary (1–6) and non-stationary (7–12)
approximations with τ = 6 ns (7-9) and 30 (10–12).
Fig. 2. Optical transmittance T (1) and parameters of non-
linearity γF (2) and γL (3, 4) of aqueous solution of
Rhodamine 6G as a function of the excitation laser power.
Filled circles are the values of γL calculated with (5) using
the experimental data T(F0) and γT (F0).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
55
Comparing this expression with (17), we see that
dL FFP 0~ . This fact yields the following simple relation
for the parameters of non-linearity
γL = 1 + γF = 2 + γT. (19)
The expression (19) seems to be unusual as it
predicts that γ-parameter of integral luminescence
excited via two-photon absorption can be different from
its well-known value of 2. This difference can take
place when γF ≠ 1 or γT ≠ 0, i.e., when the two-photon
absorption essentially changes the distribution of the
laser power within the luminescent volume. This is an
example of the effect of non-linear two-photon
absorption on the characteristics of luminescence.
However, this case is difficult to observe experimentally,
as the probability of two-photon absorption is usually
low (in most of molecules and impurity centers).
Now we consider the situation when a luminescence
center absorbs two laser photons in sequence (step by
step) through an intermediate stationary state. Such a
sequential absorption is usually much more probable
than two-photon absorption through a virtual state.
Sequential absorption can result in the excitation of
luminescence together with the significant non-linear
depletion of the laser beam.
Consider the following model of sequential
absorption in an impurity center (Fig. 3) where three
energy levels (1, 2, and 3) are involved. In Fig. 3, σ12 and
σ23 are the absorption cross-sections, τ2 and τ3 are the
luminescence lifetimes, Q represents the non-radiative
relaxation. Suppose the probability of relaxation exceeds
the rate of laser excitation, FFQ 2312 ,~ σσ . Denote the
occupancies of levels 1, 2, and 3 as ,,, 321 nnn
respectively. Suppose the laser-induced decrease of
occupancy of ground state is negligible, i.e.,
Nnnn ≈132 ~, . Besides, suppose the occupancy of the
first excited state is proportional to the laser power
density, n2 ~ F. (Though this supposition seems to be
intuitively obvious, it requires some argumentation,
which will be given thereinafter.)
According to the above model, the decrease of power
of the propagating laser beam can be described
( )2d dF F F zα β= − + , (20)
where α = σ12N is the absorption coefficient, β is a
constant proportional to the cross-section σ23. By
integrating (20) we obtain
( ) ( )( ) ( )
1
0 01 1 exp expF z F F z zβ α α
α
−
⎡ ⎤= + − − −⎢ ⎥⎣ ⎦
, (21)
( )
1
01 1 expF F dβγ α
α
−
⎡ ⎤= + − −⎡ ⎤⎣ ⎦⎢ ⎥⎣ ⎦
. (22)
From (21) when z d= and (22), it follows
1
0 FT T γ− = . (23)
According to the model (Fig. 3), the power of
luminescence from the level 3 emitted from a unit
volume is proportional to the square of the excitation
laser power. Then for the integral power of
luminescence from the level 3, taking into account the
expression (21), the integration over the volume leads to
the following relation dL FFP 0~ , hence it follows that
the parameters of non-linearity obey the expression (19),
similarly to the case of two-photon excitation.
Now we consider some reasoning in respect to the
relation n2 ~ F. First, in the case of stationary
approximation ( iττ ~2 ), the following balance equation
2 2
12 23 2
2
d 0
d
n nFN Fn
t
σ σ
τ
= = − −
shows that the relation n2 ~ F requires proportionality
between n2 and F under the condition that the third term
in the right-hand side of the equation is negligible as
compared with the second term. This implies
F < (σ23τ2)−1. This condition can be easily fulfilled
experimentally by limiting the laser power; however, it
will limit the range of observable changes of optical
transmission at a level of several percents. With such
limitations, the calculations of γ-parameters using the
experimental data become difficult because of the
fluctuations of the laser power. Second, if the stationary
approximation break down (τ2 ≥ τi), then the above-
mentioned condition of limitation of the laser power is
written as follows F < (σ23τi)−1 with the same
consequences. Moreover, without the stationary
approximation, calculations of optical signals require
integration over time, which makes it practically
impossible to obtain a simple analytical relation for the
parameters of non-linearity. Thus, we note that the above
expressions for the parameters γT and γL, obtained from
Fig. 3. Optical transitions in a luminescence center.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
56
the model of sequential absorption of two laser photons
should be considered as a first approximation.
For experimental verification of the main relations
used in this section, the luminescence of YAG:Nd3+
crystals was investigated. As known, the absorption and
luminescence spectra of YAG:Nd3+ crystals contain a
large number of narrow lines in the wide spectral range
[6]. Most of these lines are assigned to the transitions
between the energy levels (split in the crystal field) of
impurity ions Nd3+ with the electron configuration 4f 3.
In Fig. 3, the lower level 1 represents 4I9/2 level of Nd3+
ion. In this work, the third harmonic of YAG:Nd3+ laser
(wavelength λ = 355 nm) was used to excite
luminescence of YAG:Nd3+ crystal. For this excitation
wavelength, the level 2 in Fig. 3 corresponds to the level
2P3/2, and the level 3 − to the level 2F25/2 of Nd3+ ion [7].
High excited states, from which the non-radiative
relaxation occupies the level 2F25/2 , are assigned to
4f 25d configuration of Nd3+ ion [8]. f → fd transitions in
Nd3+ ions are relatively strong, hence the appropriate
cross-section σ23 is enough high to make a chance for
easy experimental observation of significant non-linear
absorption. The rate of non-radiative relaxation (Q in
Fig. 3) can be approximately estimated as 109 s−1 [9].
The luminescence from 2F25/2 level includes a number of
lines in the visible spectral range, all of them are
characterized with the lifetime τ3 ≈ 3 μs. In this paper,
the luminescence from 2F25/2 level was detected at the
wavelength close to 401 nm that corresponds to 2F25/2 →
→ 2H9/2 transition in Nd3+ ion.
The results of experiments are shown in Figs 4 and 5.
As seen from the figures, the increase of laser power
causes significant decrease of the parameters γL at
λ = 401 nm (curve 1 in Fig. 4) and γF (curve 1 in Fig. 5).
Such behavior is in agreement with the above-
considered model. Besides, the agreement between the
parameter γF and the normalized transmittance T/T0
(Fig. 5, curves 1 and 2, respectively) supports a validity
of the expression (23).
To verify the expression (5), we used Fig. 5
presenting the plots of appropriate combinations of
experimental data (1 − TγF) (curve 3) and (1 − T)γL at
λ = 401 nm (curve 4). The agreement of curves 3 and 4
confirms the validity of the expression (5).
As seen from Figs 4 and 5, the observed decrease of
γL at λ = 401 nm with F0 is larger than the appropriate
decrease of γF. This fact contradicts to the expression
(19). According to (19), it is expected that the curve
γF(F0) should coincide with the curve γL(F0) being
shifted along the ordinate by a unity. The observed
disagreement between the theory and experiment can be
explained if we suppose that the conditions of
experiments do not fulfil the above-mentioned relation
F < (σ23τi)−1. The results of the following experiment
confirm the validity of this supposition. We investigated
luminescence from the level 2 (Fig. 3) that corresponds
to 2P3/2 level of Nd3+ ion. This luminescence was
detected at the wavelength 740 nm that corresponds to
2P3/2 → 4F5/2 transitions. As seen from Fig. 4 (curve 2),
γL at λ = 740 nm decreases with F0, which indicates
violation of the relation n2 ~ F at high levels of laser
excitation. Thus, we conclude that the observed decrease
of γL at λ = 401 nm with F0 (Fig. 4, curve 1) is caused
Fig. 4. Parameters γL of integral luminescence of
YAG:Nd3+ crystal, measured at the wavelengths 401 nm (1)
and 740 (2), as functions of the excitation laser power.
Fig. 5. Parameter γF (1, filled circles), normalized
transmittance T/T0 (2, triangles), (1 − T γF) (3, filled circles),
and (1 − T )γL at λ = 401 nm ( 4, triangles) of YAG:Nd3+
crystal as functions of the excitation laser power.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 51-57.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
57
by at least two reasons: (i) sequential absorption without
the saturation of the transition 1→ 2 (according to (19)),
and (ii) saturation of the transition 1→ 2 (violation of
the relation n2 ~ F).
Thus, the results presented in Figs 4 and 5 confirm
the validity of theoretical relations between the
parameters of non-linearity, at least to a first
approximation.
3. Concluding remarks
This paper presents the analysis of the effect of non-
linear absorption on the characteristics of laser-induced
luminescence. For characterization of degree of non-
linearity, both for luminescence and transmittance, we
propose to use similar dimensionless parameters, γL (4)
and γT (6) or γF, which can be easily calculated from the
experimental data. For the case of the constant quantum
efficiency of luminescence, the universal expression (5)
is derived, which gives a useful relation for the
mentioned parameters of non-linearity of absorption and
luminescence. The validity of the expression (5) is
verified for two examples of non-linear absorption and
luminescence: saturated molecular luminescence and
luminescence of impurity centers excited via sequential
absorption of two laser photons.
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