International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
Semi Classical Analysis of Thulium Doped
Optical Fiber Amplifier
A. Prof. Abdul K. Hussein Dagher
Department of Physics, College of Education, the University of Mustansiriyah, Baghdad, Iraq.
ABSTRACT
A theoretical model simplified to three levels continuous-wave laser system which is based on density matrix model,
describes the interaction process between optical fields and optical fiber materials of amplifiers. Semi-classical model
gives a simple approach expression for gain which depends on some fiber parameters such as input power, number of
doped atoms per unit volume, fiber length, and core radius.
Keywords: fiber amplifier, optical fiber, semi classical model, and gain characterization.
1. INTODUCTION
Optical fibers attenuate light during propagation like any other material. In case of silica fibers the attenuation constant is
quite small, particularly in the wavelength range 1.0-1.6 µm, where it is typically less than 1dB/km with the minimum
value of about 0.2dB/km occurring near 1.55µm. In long – haul fiber optic communication systems, where transmission
distances are about 500km and may exceed thousands of kilometers for undersea light wave system, this attenuation
cannot be ignored. In practice, loss limitations are overcome by periodic generation of the optical fiber amplifier. Kakkar
and et al in 2006 present a theoretical analysis of inherently gain flattened, fluoride based thulium doped fiber amplifier
with 20dB net gain over 32nm wide bandwidth from 1604nm to 1636 nm [1].
In 2007 Du Ge-gue and et al on- off gain was measured, the gain varying with pump power and with signal wavelength
was studied in detail [2]. Pearson and et al 2008 reported a high power widely tunable Tm-doped fiber master oscillator
power amplifier system generating over 100Wof linearly – polarized output with a > 190nm tuning range [3]. Chun Jiang
and Li Jin in 2009 present for the first time a theoretical model of Er+3, Tm+3 &Pr+3 co doped fiber pumped with both 980
nm& 800 nm lasers. The rate and power propagation equations of the model are solved numerically and the dependence
of the gains at 1310,1470,1530,1600, and 1650nm windows on fiber length is calculated [4]. Peterka and et al in 2010
investigate performance of the proposed laser at around 810 nm in three different hosts: fluoride glass (ZBLAN), standard
silica and silica modified by high alumina codoping, using a comprehensive numerical model of TDF [5].
In 2011 Kulkarni and et al, a mid – IR super continuum fiber laser based on a thulium doped fiber amplifier is
demonstrated. A continuous spectrum extending from ~1.9 to 4.5 µm is generated with ~ o.7W time average power in
wavelength beyond 3.8µm [6]. Emami and et al in 2012 study a numerical model for different transverse thulium
distribution profiles characterizing the fibers used in thulium amplifiers [7]. Peng Wan and et al in 2013 a high energy,
high power ultrafast laser system based on Tm doped fiber at low repetition rates was successfully developed. Pulse
energy of up to 15µJ and average power of up to 15.6Wwere achieved [8]. With an increase of information traffic in the
optical telecommunication systems, the exploration of S-band is becoming an important issue in the wavelength-divisionmultiplexing network system. While most of the Er+3 doped fiber amplifiers (EDFA) utilized now is composed of silica,
the glass materials for the Tm+3 doped fiber amplifiers (TDFA) should be non silica, because the non-radiative loss
becomes an issue due to its small energy gap of the initial level to the next lower level.
A fluoride-based TDFA is now in practical use, which has a gain band around 1.45–1.49 µm (S-band) by single
wavelength pumping with 800nm- laser[9]. The gain shifted TDFA by dual-wavelength pumping scheme attracts a great
interest, because they can utilize the band TDFA. The wavelength region between C-band
EDFA and the conventional development of the S-band amplifiers are important
because few other amplifiers can
operate in this unexplored gap of the low loss window. The key principle of this pumping scheme is the use of an
auxiliary pumping laser at 790nm to control the population inversion factor between the initial 3H4 and the terminal 3F4
level, in addition to the main pumping laser source at 1050nm, which is used for the excited state absorption (ESA) [10].
TDFA is basically a four-level amplifier. The first pump transition uses 790 nm and excites thulium ions from the ground
state to level c. The next step is a stimulated emission process, which ends up in level a. At this point the system would
terminate, since the lifetime of level (a) in fluoride glasses according to the measurement is about 2.7 ms, due to the low
phonon energy of these glasses. Thus a second pump transition is needed to depopulate level (a). One possibility is the
excited state absorption (ESA) at 1055 nm, thulium ions in higher level will undergo a fast non-radiative decay ending in
the upper amplifier level (c), so the energy loop is closed. In practice the large lifetime of level (a) leads to the fact, that
the second pump is much more important than the first one and the TDFA is even working when the 790 nm pump is
omitted, although with lower power conversion efficiency. In a pumped TDFA, level (a) due to its large lifetime plays the
Volume 3, Issue 1, January 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
role of the ground state and in fact, with the pump configurations discussed above, the TDFA behaves as a quasi-threelevel amplifier [11].
2. MATHEMATICAL MODEL
A density matrix model for optical fiber amplifier in which the interaction process between optical field and optical fiber
material will be presented, the system assumes that only states c, b, and a are significant by the external field. The
population will be distributed only in N = ρ aa  ρ
  where, N is the total number of ions per unit volume,
bb
cc
ρ CC is the population in c state, ρ bb is the population in b state, and ρ aa the population in a state as in figure 1.
Total energy defined as H= Ho+ H', where Ho is the internal forces of the system, and H' is a small perturbation, H'= E
(t), for Ho >>H' density matrix equation can be written [13]-[14].
1
  
(1)
 i
 (   H     H  )
t
 ( a , b ,c )
Where,  is the density matrix, H is the Hamiltonian, let I, k =a, b, and c, then time derivative of density matrix
coefficients are.
  aa
1
  aa H aa   aa H aa   ba H ab   ab H ba   ca H ac   ac H ca 
(2)
t
i
 
  ba
1
  aa H ba   ba H aa   ba H bb   bb H ba   ca H bc   bc H ca 
t
i
  ca
1

 aa H ca   ca H aa   ba H cb   cb H ba   ca H cc   cc H ca 
t
i
  bb
1
  ab H ba   ba H ab   bb H bb   bb H bb 2   cb H bc   bc H cb 
t
i
  cb
1

 ab H ca   ca H ab   bb H cb   cb H bb   cb H cc   cc H cb
t
i

  cc
1
  ac H ca   ca H ac   bc H cb   cb H bc   cc H cc   cc H cc 
t
i
Let, Hbb=Eb and Hcc=Ec, where Eb, Ec are the energy of laser levels b and c respectively, H cb
6 can be written as [14].
 cb i
 (  cc   bb )  cb E x  ( E c  E b )  cb  (  ca  ab   ab  ca ) E x 
t
 cb
i

(  cc   bb )  cb E x   cb  cb  (  ca  ab   ab  ca ) E x 
t
(3)
(4)
(5)
(6)
(7)
   cb E x ( t ) , equation
(8)
(9)
We subtract equation 5 from equation 7 to obtain equation represents the difference (cc-bb) which multiplied by the
number of atoms N, it represent the population inversion Nc-Nb.
*
d
i *
( cc  bb )  [( cb  cb )2cbEx  (ca   ca )caE x  (ba  *ba )baE x ]
dt
Volume 3, Issue 1, January 2014
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Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
*
 nm
where
is the complex conjugate of
 nm , folding equation 9
a rate of destruction of
 cb by such collisions in
terms of a time constant Tcb the dynamic equation become [13].

cb  1
i

 icb  cb  (cc  bb )cbE x  (ca ab  abca )E x 
t  Tcb

(11)
We also fold in the fact that the population differences, cc-bb are being maintained at some equilibrium value (cc-bb)o
by some pumping process, which works against a natural decay rate(1/), then.
d
(  ) (  )o i *
*
*
(ccbb) cc bb cc bb  [(cbcb)2cbEx (ca ca)caEx (baba)baEx]
dt

Let,
E x (t )  Eox cos wt 
Eox it
e  e it
2


(12)
the term Eo e-i t "rotates" in near synchronism with the natural
response. The other part of the cosine, e+it , rotates in the opposite (phasor) sense and has little impact on
a slowly variation amplitude for along with fast time scale synchronism with E (t),
ρcb =ρcb
*
,
 cb . Assume
 cb = σcb (t) e-iωt [13].We know that
hence σcb (t) = σ*cb (t). With a careful harmonic balance of equations 11 and 12,
 cb
vary as the driving
*
frequency  according to equation 11 then the product terms E ( t )(  cb
  cb ) have a slowly variation or "zero
frequency" component and additional variation at 2 that appears in the expression for cc-bb.
If one chases this harmonic reasoning around the loop a few times, then it becomes clears at odd harmonics, 3, 5 and
so on, appear in the expression for cb, and even harmonics 0, 2, 4, and so on appear in the difference cc-bb. Taking
only the first terms of the harmonics sequence for each quantity [15].
 d

 1

 cb  
 i (  cb   )   cb  e  i  t 

 T cb

 dt

E
 E


 i  cbx ox e  i  t  cbx ox e  i  t  (  cc   bb )
2
2


(13)
The last term is neglected in the rotating wave approximation, and this step enables one to cancel common factor
 i t
so e
.
d
1
 cb  (
 i ( cb   )) cb   i  32 (  cc   22 )
dt
T cb
where  
(14)
 E ox
is the Rabi frequency [13]. Keeping only zero frequency terms of equation 12 yields.
2
(    bb )  (  cc   bb ) o
d
(  cc   bb )  cc

dt

*
*
2 i  cb (  cb
  cb )  i  ca (  ca   * ca )  i  ba (  ba   ba
)
(15)
Let us pick the simplest of all cases–a steady state so that d/dt=0 thus equation 14 can be solved for cb, in terms of the
difference in populations.

cb


cb
(
cb
(  cc  
)
1
) i
T cb
*
 cb   cb
 i
2  cb
Tcb
bb
,
*
cb


cb
(  cb
(  cc   bb )
1
)i
T cb
 cc   bb

( cb
1
)  2
Tcb
2

(16)
(17)
Now we substitute equation 17 into equations 15 and solve this population differences in terms of the equilibrium values.
Volume 3, Issue 1, January 2014
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Volume 3, Issue 1, January 2014
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 cc   bb 
1
1
2
4  cb
Tcb ~
g ( w cb )
( 
cc
  bb ) o 
(18)
   aa
   aa
2  2ca  Tca ~cc
 2  2ba  Tba ~bb
g ( w ca )
g ( w ca )
g~ (  )
where
~
g ( cb ) 
is the line shape function [16]-[17].
2
1 / Tcb
2
  ) 2  1 / Tcb
( cb
(19)
Equation 18 can be written in simple form;
 cc   bb 
1
1
2
4  cb
Tcb ~
g ( cb )
( 
cc
  bb ) o  R cb 
(20)
where
R cb  2  2ba  T ba
 bb   aa
   aa
2
 2  ca
 T ca ~cc
~
g (  ba )
g (  ca )
To find an expression for 
 cb 
cb
,
(21)
the population difference in equation 20 should be used then.
 cbx E ox Tcb
( cb   ) Tcb  i
(  cc   bb ) o  R cb
2
2
1  4  2cbTcb  ( cb   ) 2 Tcb


(22)
The complex susceptibility could be defined as a real and imaginary part [13].
/ 
 // 
2N
Re(  )
 o E ox
(23)
2 N
Im(  )
 o E ox
(24)
Substituting equation 22 in equations 23 & 24 we get.
/
cb

2
2
cb
Tcb
(cb )Tcb
cbx
Tcb
(cb )Tcb
o
NRcb

N

cb
2
2 2
2
2
 o 1 4cbTcb  (cb ) 2 Tcb
o 1 4cbTcb  (cb ) Tcb
//
cb

(25)
  2cbTcb
Nocb
 2 cbTcb
NRcb

2
2 2
2
2
o

1  4cbTcb  (cb  ) 2 Tcb
1  4cbTcb  (cb  ) Tcb
o
o
o
Where, N cb  N(  cc   bb ) . The gain coefficient g0 () is related to

//
by
(26)
g 0 ( )  
k / //
 [13], thus we
n2
have finally arrived at a derivation of the saturated gain coefficient that has its roots firmly planted in quantum theory. If
2
one works with the total dipole moment (for unpolarized light) [  x ] 
g0 () 
2
2
  1 cb
Tcb
 1 cb
Tcb 
Nocb



2
c
n
3

c n 3o 1 42cb Tcb  (cb )2 Tcb
o

2
and K
3
/

n
c


NRcb


2
2 2
 1  4cb Tcb  (cb  ) Tcb 
[18] one obtain.
(27)
3. RESULTS AND DISCUSSION
Assuming that all the core of the fiber is doped uniformly with ions, the analysis of variation of amplifier gain based on
some parameters were presented. Table 1 presents the data that are used in simulation program for thulium doped fiber
amplifiers at 800 nm pumping wavelength.
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Volume 3, Issue 1, January 2014
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Table 1: typical TDFA parameters, with (transition 3 H 4  3 F4 ) Tm +3 doped fluoride glass [7], [11], [19]
Symbol
Definitions
Value
p
se
s
p
cb
a
L
PP
PS
Nt
pump absorption cross sections
Signal emission cross sections
Signal Wavelength
Pump wavelength
E3-E2 transition lifetime
Core radius of the fiber
Length of the fiber
Pump power
Signal power
Total doping
3.44*10-25 m2
3.3*10-25 m2
1470 nm
800 nm
0.35 mS
2.5 m
(1-30) m
(0-30) W
1*10-3 mW
(1-3)*1026 m-3
The gain characteristics of thulium amplifier are shown in figure2.The saturated gain exceeded (7.5 ) dB from (1450 to
1520 nm) , corresponding to a pumping power of (1.5 ) W, concentration is (4.16*1024 m-3), core radius is (2.5µm) and
fiber length is (3m). We extended the tuning range to (1440nm) and to (1530nm) outside the signal wavelength. Altho
the gain characteristics with respect to signal wavelengths were nearly identical, the most efficient wavelength was
around (1470 )nm in terms of gain deactivation. These results are in agrement with experemantal works [20], [1], and
[21].
Further investigatation of the performance at wavelengths (1400, 1470, and 1540nm) as shown in figure3.
Larger than (2.5 )dB of gain at a concentration of (3*1024 m-3), core radius of (2.5µm) and fiber length (3m). As effective
area increase pumping power shold be increased for gain coefficent stays unchanged, figure 4 demonstrat that, for doping
(3*1024 m-3), (1.5W) and (5m) length gain is larger than (12dB) but decreases as core radius increased, after (5µm) of
core radius gain values is clambed at less than 1dB.
Threshold gain value for (1µm) core radius are different as the pumping power different. While gain is increased with
fiber length, to a certain value then saturation is started as shown in figure 5.
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Volume 3, Issue 1, January 2014
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Less than (4dB) gain for (3*1024m-3) concentration and (2.5µm) core radius.Gain can be increased if pump power or/and
fiber length are increased, these results are in coincedence with results of [21], [6].Effects of doping level are
demonstrated in figure6, it is clear the increment in concentration means high gain, with (2.5µm) and (5m) core radius
and fiber length respectively gain is less than (4dB), Optical fiber amplifier as short as possibel doping level shold be
increased.
Any increase in doping level causes gain reach saturation. Figure7 illstrates that befor the doping level (3*1024 m-3) gain
is sharply increased but after this value of doping level, the increment in gain is very small therfore one can said the gain
is saturated.
For pumping power of ( 0.5W and1.5W) the doping level shold be larger than (4*1024m-3 and 1*1024m-3) respactively in
order to gain >0 as it clear in fig. 7. These results symmetrical with results of [22]-[23].
4. CONCLUSIONS
In this work a theoretical expression for gain was achieved in a fluoride-based thulium-doped fiber. Although the
pumping scheme and the set of parameters for the fiber were not optimized, the measured gain of (>9.5 dB) allowed us to
validate the numerical model developed for TDFA. Then, we can estimate the opto-geometric parameters to reach a
probable gain.
Fiber length decreases when thulium ion densities increase, according to the result, it is possible to design amplifiers with
high gain for amplifier length as short as few meters by increasing thulium ion density and vice versa.
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Volume 3, Issue 1, January 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
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