Characteristics of optical limiting in media with nonlinear absorption and scattering
A comparative study is performed for characteristics of optical limiting in the media with nonlinear absorption and scattering with the use of nanosecond-scale laser pulses. Two methods are proposed to analyze of experimental nonlinear transmittance curves. The experiments revealed differences in th...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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| Цитувати: | Characteristics of optical limiting in media with nonlinear absorption and scattering / S.E. Zelensky, O.S. Kolesnik, O.V. Kopyshinsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 74-79. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860258160676175872 |
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| author | Zelensky, S.E. Kolesnik, O.S. Kopyshinsky, O.V. |
| author_facet | Zelensky, S.E. Kolesnik, O.S. Kopyshinsky, O.V. |
| citation_txt | Characteristics of optical limiting in media with nonlinear absorption and scattering / S.E. Zelensky, O.S. Kolesnik, O.V. Kopyshinsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 74-79. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | A comparative study is performed for characteristics of optical limiting in the media with nonlinear absorption and scattering with the use of nanosecond-scale laser pulses. Two methods are proposed to analyze of experimental nonlinear transmittance curves. The experiments revealed differences in the shape of nonlinear transmittance curves in the media with different physical mechanisms of optical limiting. For optical limiting in suspensions of light-absorbing particles, it is concluded that the fifth-order nonlinear susceptibilities should be taken into account.
|
| first_indexed | 2025-12-07T18:52:18Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 74-79.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
74
PACS: 42.62.-b, 42.65.-k
Characteristics of optical limiting in media with nonlinear
absorption and scattering
S.E. Zelensky, O.S. Kolesnik, O.V. Kopyshinsky
Taras Shevchenko Kyiv National University, Physics Department
6, prospect Glushkova, 03680 Kyiv, Ukraine E-mail: zele@univ.kiev.ua
Abstract. A comparative study is performed for characteristics of optical limiting in the
media with nonlinear absorption and scattering with the use of nanosecond-scale laser
pulses. Two methods are proposed to analyze of experimental nonlinear transmittance
curves. The experiments revealed differences in the shape of nonlinear transmittance
curves in the media with different physical mechanisms of optical limiting. For optical
limiting in suspensions of light-absorbing particles, it is concluded that the fifth-order
nonlinear susceptibilities should be taken into account.
Keywords: optical limiting, laser-induced absorption, nonlinear scattering, nonlinear
transmittance curves, carbon black suspensions.
Manuscript received 24.05.05; accepted for publication 25.10.05.
1. Introduction
One of the effects of self-action of laser beams
propagating through nonlinear media is the dependence
of optical transmittance (or the extinction coefficient) on
the laser intensity. Several physical mechanisms are
known to be responsible for laser-induced changes of
optical transmittance at high levels of the laser power.
For example, saturated absorption in atoms and
molecules due to the decrease of occupance of lower
energy levels is the well-known phenomenon that is
widely used in quantum electronics and laser
spectroscopy [1, 2]. Similar behavior of transmittance
curves is observed in the case of non-stationary self-
induced transparency [1, 2], however, its non-stationary
physical mechanism differs from the mechanism of
simple saturated absorption. Besides, the enhancement
of optical transmittance is observed for powerful laser
beams interacting with aerosols [3].
It is not a rare occasion when a laser pulse of a
moderate power density demonstrates power-dependent
self-induced attenuation during propagation through
various media. In these cases, the laser-induced decrease
of optical transmittance is observed, i.e. the nonlinear
medium tends to limit the transmitted laser power. Such
effects are called optical limiting that has promising appli-
cations in the field of laser safety of photodetectors, etc.
One of the mechanisms of optical limiting is the
laser-induced absorption (excited-state absorption of
atoms and molecules, absorption by photochemical
products, absorption by laser-ionized atoms, molecules,
impurity centers, etc.) [4-10].
Interaction of powerful laser pulses with suspensions
of light-absorbing submicron and nano-size particles
(carbon black suspensions, carbon nanotube suspen-
sions, etc.) demonstrates effective broadband optical
limiting [11-15]. Unlike the nonlinear absorption, the
primary mechanism of optical limiting in such colloidal
media is the laser-induced nonlinear light scattering, i.e.,
the increase of scattering cross-section with the increase
of the laser power.
It should be also mentioned that the absence of laser-
induced changes of optical transmittance does not grant
the absence of non-linearity in the processes being
investigated. For example, when the cross-sections of
linear and non-linear absorption coincide, the
experiments show the laser power-independent optical
transmittance accompanied by essentially nonlinear
luminescence (see, for example, [16]).
Self-induced changes of optical transmittance of
laser beams can be treated as non-coherent nonlinear
optical phenomena by using nonlinear susceptibilities of
odd orders, ( )12 −nχ [2]. For example, to a first
approximation, in a great number of instances, laser-
induced absorption can be interpreted by the presence of
the non-zero imaginary part of the third-order suscep-
tibility tensor, ( ) ( ) 0Im 3 ≠+−= ωωωωχ . It should be
noted that the phenomena of nonlinear absorption can be
treated using the microscopic models accounting for
optical transitions in atoms and molecules, without the
use of formalism of optical susceptibilities.
In the present paper, we perform a comparative study
of peculiarities of optical limiting of nanosecond-scale
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 74-79.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
75
laser pulses in different media with excited state
absorption and with nonlinear scattering. It is shown that
the analysis of the shape of nonlinear transmittance
curves provides useful information about the physical
mechanisms of powerful laser radiation interaction with
matter. We propose two methods to analyze the shape of
nonlinear transmittance curves. These methods enabled
us to reveal the nonlinearities of high orders (fifth and
higher) in the experimental transmittance data.
2. Experimental details
The experiments were performed using the harmonics of
Q-switched YAG:Nd3+-lasers (1064 nm, 355 nm,
266 nm) with pulse duration from 10 to 20 ns in
different experiments, with the pulse repetition rate of
(0.5…2) s−1. Each laser pulse was processed separately;
the data averaging was implemented into the software.
The laser beams had smooth bell-shaped cross-beam
distributions of the power density. Optical transmittance
was measured as a ratio of energies of incident and
transmitted laser pulses. All measurements were carried
out at the room temperature.
The investigations were performed using the
following samples. (i) Y3Al5O12 garnet crystals doped
with Nd3+ ions. A segment of standard laser rod with a
thickness of d =5 mm was used. (ii) Borate glasses of the
following formula K2O · 6B2O3 – 0.1% wt. SnO. The
glasses were melted in platinum crucibles in air at the
temperature of ~1300 K . (iii) Aqueous suspensions of
submicron light-absorbing particles. Diluted and filtered
gouache paint was used. A nephelometric estimation of
the mean particle size is 0.1 μm. To eliminate errors
caused by laser-induced fading, the investigated
suspensions were pumped through the optical cell in
such a manner that each laser pulse interacted with a
fresh portion of suspension.
3. Results and discussion
Suppose the intensity (the surface power density, F) of
laser radiation propagating through a nonlinear medium
satisfy the following equation
( ) dzFNFdF α−= (1)
where α is the extinction cross section which depends on
the laser intensity, N is the concentration of absorption
(scattering) centers, z is the coordinate along the laser
beam. To a first approximation, consider the interaction
of laser radiation with a medium that can be
characterized by the third-order optical nonlinearity (i.e.
the laser-induced dipole momentum can be expressed as
( ) ( ) EEEEP
rrrrr
:31 χχ +⋅= ). As is known for a third-order
nonlinear media, the extinction cross section can be
presented as a linear function of laser intensity as
follows
( ) FF βαα += 0 (2)
where 0α is the linear cross section, β is the coefficient
proportional to the imaginary part of the third-order
optical susceptibility, ( )3Imχ .
Denote the surface density of the incident (z = 0) and
transmitted (z = d) laser powers as 0F , dF . For a sample
thickness d, Eqs (1) and (2) yield the following
expression for the laser power dependency of the optical
transmittance
( ) ( ) ( )
1
00
0
0
0
0
0 11
−
⎥
⎦
⎤
⎢
⎣
⎡
−+== FTT
F
FFFT d
α
β (3)
where the low-signal transmittance, 0T , satisfies
Bouguer’s law, ( )NdT 00 exp α−= .
Expression (3) shows that the transmitted laser
power, dF , demonstrates nonlinear response on
variations of the incident laser power, 0F . For further
analysis, we introduce the following parameter of
nonlinearity
00 FdF
FdF dd
F =γ . (4)
The dimensionless parameter Fγ can be easily
calculated from the experimental data as a local slope of
( )0FFd curve plotted in a log-log scale. In the case of
linear absorption, the transmittance is a constant, and the
transmitted power, dF , is proportional to the incident
power, 0F , hence the parameter of nonlinearity equals to
unity, Fγ =1. As far as 0TFFd = , obviously, the value
of Fγ can be determined via numerical differentiation
of the experimental curves ( )0FFd or ( )0FT as follows
11
00
+=+= TF FdF
TdT γγ (5)
According to the definition (4), the expression (3)
yields
( ) ( )
0
0
0 T
FTFF =γ . (6)
Thus, for media with third-order nonlinearities we
conclude that the parameter of nonlinearity, Fγ , and the
normalized transmittance, 0TT , behave similarly with
the increase of the incident laser power, 0F . The
calculated curves ( )0FFγ and ( ) 1
00
−TFT coincide with
the curve 1 plotted in Fig. 1. The validity of the
expression (6) can be easily checked up experimentally.
However, before doing so, consider possible causes
violating the expression (6).
The expression (6) is derived with the assumption of
uniform distribution of the laser intensity across the
beam. However, as is known, the neglect of non-
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 74-79.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
76
uniformity of laser power cross-beam distribution causes
significant errors in nonlinear laser spectroscopy [17,
18]. To check the validity of the expression (6) with
non-uniform laser beams, a computer simulation seems
to be suitable. Numerical calculations were performed
for the ( )0FFγ and ( ) 1
00
−TFT dependencies with the
Gauss cross-beam distribution of the laser power. The
results of calculations are given in Fig. 1, curves 2 and 3.
As is seen from the figure, in the region of small changes
of the transmittance ( 0TT > 0.7), the difference
between the curves ( )0FFγ and ( ) 1
00
−TFT does not
exceed uncertainties of typical pulsed laser experiments,
whereas at high levels of the laser power the mentioned
curves differ significantly. Therefore, the expression (6)
can be applied for analysis at low levels of optical
limiting (with keeping the approximate condition
0TT > 0.7).
Consider several examples of media with optical
limiting at moderate levels of the pulsed laser power
(below 100 MW·cm-2). The first example – YAG:Nd3+
crystals. Spectral characteristics of Y3Al5O12:Nd3+
crystals in the visible and near UV regions are
determined by optical transitions within 4f 3
configuration of Nd3+ ions [19, 20]. At the wavelength
355 nm, the absorption of laser radiation provides an
occupance of the metastable 2P3/2 level of Nd3+ ion via
4I9/2→2P3/2 transitions. With the increase of the laser
intensity, the occupance of 2P3/2 level increases
significantly, hence the transitions from 2P3/2 to high
levels of 4f 25d configuration become actual [21].
Optical transitions within f-f configuration are forbidden,
whereas f-d transitions are allowed and have large
oscillator strengths. That is why, the excited state
absorption from the 2P3/2 level to the levels of the 4f 25d
configuration is easily observed experimentally as a
nonlinear transmittance. The results of experiments are
given in Fig. 2a. As is seen from the figure, the
normalized transmittance and the parameter of
nonlinearity practically coincide; this fact agrees with
the expression (6). Thus, the excited state absorption in
Y3Al5O12:Nd3+ crystals can be approximated to the
model ( ) const3 =χ (the expressions (1) and (2)).
As a second example of the media with optical
limiting, consider alkali-borate glasses doped with the
mercury-like ions (Tl+, Pb2+, In+, Sn2+). Optical
characteristics of such glasses in UV and visible spectral
regions are determined by the dopants [22, 23]. The
mercury-like ions in glasses form wide absorption and
luminescence bands. At high levels of UV laser
excitation, such glasses demonstrate optical limiting via
the mechanisms of excited state absorption and
absorption by ionized centers [9, 16, 22-24]. In the
present paper, the glass composition was chosen to
minimize the laser-induced ionization as compared with
other glasses. The results of experiments are given in
Fig. 2b. As is seen from the figure, 0TT and Fγ
coincide within the accuracy of measurements, which
substantiates the validity of third-order approximation.
Fig. 1. Normalized transmittance T/T0 (solid curves) and
parameter γF (points) calculated according to equations (1), (2)
(plots 1-3) and (1), (7) (plots 4, 5) for uniform (1, 4, 5) and
Gauss (2, 3) cross-beam distribution of laser intensity.
Fig. 2. Normalized transmittance T/T0 (solid lines) and
parameter γF (circles) in YAG:Nd3+ crystals (a) and in
K2O·6B2O3-SnO glass (b) as functions of laser power density
at wavelengths 355 nm (a) and 266 nm (b).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 74-79.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
77
Besides, it seems surprising, the above-mentioned
nonlinear transmittance curves of different objects –
crystals and glasses – coincide (compare the graphs in
Fig. 2a, b).
Finally, consider the third example – aqueous
suspensions of black-body particles – where optical
limiting is caused by the laser-induced nonlinear
scattering. There are two primary mechanisms of the
nonlinear scattering in suspensions: (i) vaporization of
liquid around the laser-heated suspended particles, and
(ii) production of laser plasma in the neighborhood of
particles [11-15]. Presently, relative contributions of the
mentioned two mechanisms to optical limiting are not
completely clear. In [11, 25], a simple model is
developed for optical limiting in light-absorbing
suspensions; the model takes into account laser-induced
vaporization of surrounding fluid.
The results of optical limiting experiments with an
aqueous suspension are given in Fig. 3. As is seen, the
obtained curves T/T0 and γF are essentially different:
first, γF one rapidly decreases with 0F from 1 to ~0.8,
then it slowly decreases to ~0.75, whereas the decrease
of transmittance is smooth. Such behavior of the T/T0
and Fγ dependencies contradicts to the expression (6)
and can not be explained within the framework of the
assumption of ( ) const3 =χ .
To a second approximation, assume the investigated
medium characterized by nonlinear susceptibilities of the
third and fifth orders. Taking account of ( )5χ generates
a new summand in the right-hand portion of the equation
(1), proportional to the third power of F, due to the
following expression
( ) 2
0 FFF μβαα ++= (7)
where the coefficient μ depends on χ(5). Calculations
performed with (1) and (7) showed a significant
difference in the laser power dependency of 0TT and
Fγ . The curves 4 and 5 in Fig. 1 were calculated for the
case of β = 0, ≠μ 0. As is seen from the figure, taking
account of ( )5χ makes the dependency ( ) 1
00
−TFT more
abrupt than in the case of ( )3χ . Besides, Fγ decreases
with 0F more quickly than the transmittance does.
The mentioned peculiarities of 0TT and Fγ curves
calculated in the ( )5χ approximation agree with the
results of the experiments given in Fig. 3. In the region
of onset of optical limiting, the non-coinciding
experimental curves 0TT and Fγ , with Fγ below
0TT , can be considered as a signature of the
( )5χ nonlinearity.
Concerning the role of high-order nonlinearities in
the processes of interaction of nanosecond-scale laser
pulses with carbon black suspensions, its importance is
also confirmed by the results of experiments with
degenerate four-wave mixing [26].
It should be noted, application of the considered
procedure (the expression (6)) to the experimental data
on optical limiting published in [27, 28] gives results
similar to those described in the present paper. Namely,
optical limiting in the media with nonlinear scattering
mechanism (carbon black suspensions, carbon nanotube
suspensions) show the signs of ( )5χ nonlinearity,
whereas fullerene solutions demonstrate the ( )3χ
nonlinearity caused by the excited state absorption
mechanism.
Fig. 3. Normalized transmittance T/T0 (1) and parameter γF (2)
as functions of laser intensity F0 (1064 nm, 20 ns) in aqueous
suspension of black-body particles. Low signal transmittance
T0 ≈ 0.7.
Fig. 4. Nonlinear transmittance curves of aqueous suspension
of black-body particles for laser pulses 20 ns, 1064 nm.
Numbers near the curves represent concentrations of particles
in relative units.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 74-79.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
78
The observed change from quick to slow decrease of
Fγ with the increase of 0F (see Fig. 3) is not explained
by the proposed simple ( )5χ model. It is plausible to
suggest that the mentioned turn of ( )0FFγ curve
indicates hidden changes in the physical processes
responsible for observed optical limiting in the
investigated suspensions.
Now consider another method of analysis of
nonlinear transmittance curves. The equation (1) can be
integrated along the laser track as follows
( )∫−=
d
dzNzT
0
ln α . (8)
The integral in (8) can be presented in the following
form ( )∫
Nd
Nzd
0
α . Then, a derivative of (8) with respect
to the parameter N yields the following expression
( ) ( )
N
T
d
F
Δ
Δ
⋅−=
ln1α (9)
where the value of extinction cross section, α ,
corresponds to the laser power density dFF = . The
expression (9) gives the intensity dependence of the
extinction cross section, which can be obtained from the
experimental data (nonlinear transmittance curves)
measured using suspensions (or solutions) of different
concentrations.
As an example of applying the expression (9),
consider optical limiting in carbon black suspensions. A
series of nonlinear transmittance curves measured with
suspensions of different particle concentrations is given
in Fig. 4. These data were treated according to (9), and
the nonlinear part of the extinction cross section, 0αα − ,
is plotted against 0F in Fig. 5 in a log-log scale. The
data depicted in Fig. 5 correspond to the region of the
incident laser intensity, 1.25 < 0lg F < 1.9, where the
Fγ curve is located below the transmittance curve
(within the margins marked by dashed lines in Fig. 3). A
straight solid line in Fig. 5 is drawn with a slope of 2. As
is seen, the slope of the plot in Fig. 5 is close to 2 (linear
fit gives the value 1.7). This circumstance indicates that
optical limiting in carbon suspensions is primarily
determined by the fifth-order optical nonlinearities.
Therefore, both of the considered methods of analysis of
nonlinear transmittance curves lead to the similar
conclusions.
4. Concluding remarks
In this paper, we performed a comparative study of
optical limiting caused by excited state absorption and
nonlinear scattering of (10…20) ns laser pulses in
crystals, glasses, and suspensions. The results obtained
give us grounds to conclude that differences in the
mechanisms of laser radiation interaction with matter
manifest themselves in the differences of shape of
nonlinear transmittance curves. To reveal the mentioned
differences, we propose two simple methods based on
differentiation of nonlinear transmittance curves. Using
the formalism of nonlinear optical susceptibilities, we
conclude that the proposed methods of analysis provide
to reveal the fifth-order optical nonlinearities in the
investigated processes.
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| id | nasplib_isofts_kiev_ua-123456789-120972 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T18:52:18Z |
| publishDate | 2005 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Zelensky, S.E. Kolesnik, O.S. Kopyshinsky, O.V. 2017-06-13T11:34:26Z 2017-06-13T11:34:26Z 2005 Characteristics of optical limiting in media with nonlinear absorption and scattering / S.E. Zelensky, O.S. Kolesnik, O.V. Kopyshinsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 74-79. — Бібліогр.: 28 назв. — англ. 1560-8034 PACS: 42.62.-b, 42.65.-k https://nasplib.isofts.kiev.ua/handle/123456789/120972 A comparative study is performed for characteristics of optical limiting in the media with nonlinear absorption and scattering with the use of nanosecond-scale laser pulses. Two methods are proposed to analyze of experimental nonlinear transmittance curves. The experiments revealed differences in the shape of nonlinear transmittance curves in the media with different physical mechanisms of optical limiting. For optical limiting in suspensions of light-absorbing particles, it is concluded that the fifth-order nonlinear susceptibilities should be taken into account. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Characteristics of optical limiting in media with nonlinear absorption and scattering Article published earlier |
| spellingShingle | Characteristics of optical limiting in media with nonlinear absorption and scattering Zelensky, S.E. Kolesnik, O.S. Kopyshinsky, O.V. |
| title | Characteristics of optical limiting in media with nonlinear absorption and scattering |
| title_full | Characteristics of optical limiting in media with nonlinear absorption and scattering |
| title_fullStr | Characteristics of optical limiting in media with nonlinear absorption and scattering |
| title_full_unstemmed | Characteristics of optical limiting in media with nonlinear absorption and scattering |
| title_short | Characteristics of optical limiting in media with nonlinear absorption and scattering |
| title_sort | characteristics of optical limiting in media with nonlinear absorption and scattering |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120972 |
| work_keys_str_mv | AT zelenskyse characteristicsofopticallimitinginmediawithnonlinearabsorptionandscattering AT kolesnikos characteristicsofopticallimitinginmediawithnonlinearabsorptionandscattering AT kopyshinskyov characteristicsofopticallimitinginmediawithnonlinearabsorptionandscattering |